text:
High School Mathematics Extensions/Primes/Modular Arithmetic/text

Introduction
Modular arithmetic connects with primes in an interesting way.

Modular arithmetic is a system in which all numbers up to some positive integer, n say, are used. So if you were to start counting you would go 0, 1, 2, 3, ..., n - 1 but instead of counting n you would start over at 0. And what would have been n + 1 would be 1 and what would have been n + 2 would be 2. Once 2n has been reached the number is reset to 0 again, and so on. Modular arithmetic is also called clock-arithmetic because we only use 12 numbers to tell standard time. On clocks we start at 1 instead of 0, continue to 12, and then start at 1 again. Hence the name clock-arithmetic.

The sequence also continues into what would be the negative numbers. What would have been -1 is now n - 1. For example, consider modulo 7 arithmetic, it's just like ordinary arithmetic except the only numbers we use are 0, 1, 2, 3, 4, 5 and 6. If we see a number outside of this range we add 7 to (or subtract 7 from) it, until it lies within that range.

As mentioned above, modular arithmetic is not that different to ordinary arithmetic. For example in modulo 7 arithmetic, we have
 * $$3 + 2 = 5 \!$$
 * $$5 + 6 = 11 = 4\!$$
 * $$5 - 6 = -1 = 6 \!$$

and

\begin{matrix} 3 \times 5 = 15 = 1\\ 5 \times -6 = -30 = 5\\ \end{matrix} $$

We have done some calculation with negative numbers. Consider 5 × -6. Since -6 does not lie in the range 0 to 6, we need to add 7 to it until it does. And -6 + 7 = 1. So in modular 7 arithmetic, -6 = 1. In the above example we showed that 5 × -6 = -30 = 5, but 5 × 1 = 5. So we didn't do ourselves any harm by using -6 instead of 1. Why?

Note - Negatives: The preferred representation of -3 is 4, as -3 + 7 = 4, but using either -3 and 4 in a calculation will give us the same answer as long as we convert the final answer to a number between 0 and 6 (inclusive).

Exercise
Find in modulo 11

1.
 * -1 × -5

2.
 * 3 × 7

3. Compute the first 10 the powers of 2
 * 21, 22, 23, ..., 210

What do you notice? Using the powers of 2 find
 * 61, 62, 63, ..., 610

What do you notice again?

4.
 * $$\sqrt{4} $$

i.e. find, by trial and error (or otherwise), all numbers x such that x2 = 4 (mod 11). There are two solutions, find both.

5.
 * $$\sqrt{9} $$

i.e. find all numbers x such that x2 = 9 (mod 11). There are two solutions, find both.

Inverses
Consider a number n, the inverse of n is the number that when multiplied by n gives 1. For example, if we were to solve the following equation
 * $$5x = 3 \pmod{ 7} \!$$

the (mod 7) is used to make it clear that we are doing arithemetic modulo 7. We want to get rid of the 5 somehow. Multiplying it by something to turn it into a 1 would do the job. Notice that
 * $$3\times 5 = 15 = 1 \pmod{ 7} \!$$

because 3 multiplied by 5 gives 1, we say 3 is the inverse of 5 in modulo 7. Now we multiply both sides by 3 So x = 2 modulo 7 is the required solution.

Definition (Inverse)
 * The inverse of (a number) x is a number y such that xy = 1. We denote the inverse of x by x-1 or 1/x.

 Inverse is unique

From above, we know the inverse of 5 is 3, but does 5 have another inverse? The answer is no. In fact, in any reasonable number system, a number can have one and only one inverse. We can see that from the following proof

 Suppose n has two inverses b and c
 * $$b = b \times 1 = b (nc) = (bn)c = 1 \times c = c \!$$

From the above argument, all inverses of n must be equal. As a result, if the number n has an inverse, the inverse must be unique.

An interesting property of any modulo n arithmetic is that the number n - 1 has itself as an inverse. That is, (n - 1) × (n - 1) = 1 (mod n), or we can write (n - 1)2 = (-1)2 = 1 (mod n). The proof is left as an exercise at the end of the section.

Existence of inverse
Not every number has an inverse in every modulo arithmetic. For example, 3 doesn't have an inverse mod 6, i.e., we can't find a number x such that 3x = 1 mod 6 (the reader can easily check).

Consider modulo 15 arithmetic and note that 15 is composite. We know the inverse of 1 is 1 and of 14 is 14. But what about 3, 6, 9, 12, 5 and 10? None of them has an inverse! Note that each of them shares a common factor with 15!

As an example, we show that 3 does not have an inverse modulo 15. Suppose 3 has an inverse x, then we have
 * $$3x = 1 \pmod{15} \!$$

We make the jump from modular arithemetic into rational number arithmetic. If 3x = 1 in modulo 15 arithmetic, then
 * $$3x = 15k + 1 \!$$

for some integer k. Now we divide both sides by 3, we get
 * $$x = 5k + \frac{1}{3} \!$$

But this cannot be true, because we know that x is an integer, not a fraction. Therefore 3 doesn't have an inverse in mod 15 arithmetic. To show that 10 doesn't have an inverse is harder and is left as an exercise.

We will now state the theorem regarding the existence of inverses in modular arithmetic.

Theorem
 * If n is prime then every number (except 0) has an inverse in modulo n arithmetic.

Similarly
 * If n is composite then every number that doesn't share a common factor with n has an inverse.

It is interesting to note that division is closely related to the concept of inverses. Consider the following expression
 * $$6 \times 3^{-1} \pmod{7} \!$$

the conventional way to calculate the above would be to find the inverse of 3 (being 5). So
 * $$6\times 3^{-1} = 6\times 5 = 30 = 2\pmod{7} \!$$

We write the inverse of 3 as 1/3, so we think of multiplying 3-1 as dividing by 3, we get
 * $$6\times \frac{1}{3} = \frac{6}{3} = 2 \pmod{7} \!$$

Notice that we got the same answer! In fact, the division method will always work if the inverse exists.

However, the expression in a different modulo system will produce the wrong answer, for example
 * $$6 \times 3^{-1} \pmod{9} \!$$

we don't get 2, as 3-1 does not exist in modulo 9, so we can't use the division method.

Exercise
1. Does 8 have an inverse in mod 16 arithemetic? If not, why not?

2. Find x mod 7 if x exists:
 * $$x = 2^{-1} \!$$
 * $$x = 3^{-1} \!$$
 * $$x = 4^{-1} \!$$
 * $$x = 5^{-1} \!$$
 * $$x = 6^{-1} \!$$
 * $$x = 7^{-1} \!$$

3. Calculate x in two ways, finding inverse and division
 * $$x = 28\cdot 7^{-1} \ \ \mbox{(mod 29)} \!$$

4. (Trick) Find x
 * $$x = 5^{99} \times (40 + 3^{-1}) \ \pmod{11} \!$$

5. Find all inverses mod n (n ≤ 19)

Coprime and greatest common divisor
Two numbers are said to be coprimes if their greatest common divisor (gcd) is 1. E.g. 21 and 55 are both composite, but they are coprime as their greatest common divisor is 1. In other words, they do not share a common divisor other than 1.

There is a quick and elegant way to compute the gcd of two numbers, called Euclid's algorithm. Let's illustrate with a few examples:

Example 1:
 * Find the gcd of 21 and 49.

We set up a two-column table where the larger of the two numbers is on the right hand side as follows

We now compute 49 (mod 21) which is 7 and put it in the second row smaller column, and put 21 into the larger column.

Perform the same action on the second row to produce the third row.

Whenever we see the number 0 appear on the smaller column, we know the corresponding larger number is the gcd of the two numbers we started with, i.e. 7 is the gcd of 21 and 49. This algorithm is called Euclid's algorithm.

Example 2
 * Find the gcd of 31 and 101

Example 3
 * Find the gcd of 132 and 200

Important to note
 * 1) The gcd need not be a prime number.
 * 2) The gcd of two different primes is 1.  In other words, two different primes are always coprime.

Exercise
1. Determine whether the following sets of numbers are coprimes
 * 5050 5051
 * 59 78
 * 111 369
 * 2021 4032

2. Find the gcd of the numbers 15, 510 and 375

info -- Algorithm
 An algorithm is a step-by-step description of a series of actions when performed correctly can accomplish a task. There are algorithms for finding primes, deciding whether 2 numbers are coprimes, finding inverses and many other purposes. You'll learn how to implement some of the algorithms we have seen using a computer in the chapter ../../../Mathematical Programming/.

Finding Inverses
Let's look at the idea of inverse again, but from a different angle. In fact we will provide a sure-fire method to find the inverse of any number. Let's consider:
 * 5x = 1 (mod 7)

We know x is the inverse of 5 and we can work out it is 3 reasonably quickly. But x = 10 is also a solution, so is x = 17, 24, 31, ... 7n + 3. So there are infinitely many solutions; therefore we say 3 is equivalent to 10, 17, 24, 31 and so on. This is a crucial observation.

Now let's consider
 * $$216x \equiv 1 \ \ \mbox{(mod 811)}$$

A new notation is introduced here, it is the equal sign with three strokes instead of two. It is the "equivalent" sign; the above statement should read "216x is EQUIVALENT to 1" instead of "216x is EQUAL to 1". From now on, we will use the equivalent sign for modulo arithmetic and the equal sign for ordinary arithmetic.

Back to the example, we know that x exists, as gcd(811,216) = 1. The problem with the above question is that there is no quick way to decide the value of x! The best way we know is to multiply 216 by 1, 2, 3, 4... until we get the answer, there are at most 811 calculations, way too tedious for humans. But there is a better way, and we have touched on it quite a few times!

We notice that we could make the jump just like before into rational mathematics:

\begin{matrix} 216a & = & 1 + 811b\\ 0 & \equiv & 1 + 163b &\pmod{216}\\ \end{matrix} $$ We jump into rational maths again

\begin{matrix} 216c &=& 1 + 163b \\ 53c &\equiv& 1 &\pmod{163}\\ \end{matrix} $$ We jump once more

\begin{matrix} 53c &=& 1 + 163d \\ 0 &\equiv& 1 + 4d&\pmod{53}\\ \end{matrix} $$

Now the pattern is clear, we shall start from the beginning so that the process is not broken:

\begin{matrix} 216a & = & 1 + 811b\\ 216c &=& 1 + 163b \\ 53c &=& 1 + 163d \\ 53e &=& 1 + 4d \\ e &=& 1 + 4f \\ \end{matrix} $$

Now all we have to do is choose a value for f and substitute it back to find a! Remember a is the inverse of 216 mod 811. We choose f = 0, therefore e = 1, d = 13, c = 40, b = 53 and finally a = 199! If f is chosen to be 1 we will get a different value for a.

The very perceptive reader should have noticed that this is just Euclid's gcd algorithm in reverse.

Here are a few more examples of this ingenious method in action:

Example 1

Find the smallest positive value of a:

\begin{matrix} 33a & \equiv & 1 \ \ \mbox{(mod 101)}\\ 33a &=& 1 + 101b\\ 33c &=& 1 + 2b\\ c &=& 1 + 2d\\ \end{matrix} $$ Choose d = 0, therefore a = 49.

Example 2 Find the smallest positive value of a:

\begin{matrix} 27a & \equiv & 1 \ \ \mbox{(mod 821)}\\ 27a &=& 1 + 821b\\ 27c &=& 1 + 11b\\ 5c &=& 1 + 11d\\ 5e &=& 1 + d\\ \end{matrix} $$ Choose e = 0, therefore a = -152 = 669

Example 3 Find the smallest positive value of a:

\begin{matrix} 34a & \equiv & 1 \ \ \mbox{(mod 55)}\\ 34a &=& 1 + 55b\\ 34c &=& 1+ 21b\\ 13c& =& 1 + 21d\\ 13e& =& 1 + 8d\\ 5e &=& 1 + 8f\\ 5g& = &1 + 3f\\ 2g& =& 1 + 3h\\ 2i& = &1 + h\\ \end{matrix} $$ Set i = 0, then a = -21 = 34. ''Why is this so slow for two numbers that are so small? What can you say about the coefficients?''

Example 4 Find the smallest positive value of a:

\begin{matrix} 21a & \equiv & 1 \ \ \mbox{(mod 102)}\\ 21a &=& 1 + 102b\\ 21c &=& 1 + 18b\\ 3c &=& 1 + 18d\\ 3d &=& 1\\ \end{matrix} $$ Now d is not an integer, therefore 21 does not have an inverse mod 102.

What we have discussed so far is the method of finding integer solutions to equations of the form:
 * ax + by = 1

where x and y are the unknowns and a and b are two given constants, these equations are called linear Diophantine equations. It is interesting to note that sometimes there is no solution, but if a solution exists, it implies that infinitely many solutions exist.

Diophantine equation
In the Modular Arithmetic section, we stated a theorem that says if gcd(a,m) = 1 then a-1 (the inverse of a) exists in mod m. It is not difficult to see that if p is prime then gcd(b,p) = 1 for all b less than p, therefore we can say that in mod p, every number except 0 has an inverse.

We also showed a way to find the inverse of any element mod p. In fact, finding the inverse of a number in modular arithmetic amounts to solving a type of equations called Diophantine equations. A Diophantine equation is an equation of the form
 * ax + by = d

where x and y are unknown.

As an example, we should try to find the inverse of 216 in mod 811. Let the inverse of 216 be x, we can write
 * $$216x \equiv 1 \pmod{811} \!$$

we can rewrite the above in every day arithmetic
 * $$216x + 811y = 1 \!$$

which is in the form of a Diophantine equation.

Now we are going to do the inelegant method of solving the above problem, and then the elegant method (using Magic Tables).

Both methods mentioned above uses the Euclid's algorithm for finding the gcd of two numbers. In fact, the gcd is closely related to the idea of an inverse. Let's apply the Euclid's algorithm on the two numbers 216 and 811. This time, however, we should store more details; more specifically, we want to set up an additional column called PQ which stands for partial quotient. The partial quotient is just a technical term for "how many n goes into m" e.g. The partial quotient of 3 and 19 is 6, the partial quotient of 4 and 21 is 5 and one last example the partial quotient of 7 and 49 is 7.

The tables says three 216s goes into 811 with remainder 163, or symbollically:
 * 811 = 3×216 + 163.

Let's continue:

Reading off the table, we can form the following expressions
 * 811 = 3× 216 + 163
 * 216 = 1× 163 + 53
 * 163 = 3× 53 + 4
 * 53 =13× 4 + 1

Now that we can work out the inverse of 216 by working the results backwards
 * 1 = 53 - 13×4
 * 1 = 53 - 13×(163 - 3×53)
 * 1 = 40×53 - 13×163
 * 1 = 40×(216 - 163) - 13×163
 * 1 = 40×216 - 53×163
 * 1 = 40×216 - 53×(811 - 3×216)
 * 1 = 199×216 - 53×811

Now look at the equation mod 811, we will see the inverse of 216 is 199.

Magic Table
The Magic Table is a more elegant way to do the above calculations, let us use the table we form from Euclid's algorithm

Now we set up the so-called "magic table" which looks like this initially

Now we write the partial quotient on the first row:

We produce the table according to the following rule:
 * Multiply a partial quotient one space to the left of it in a different row, add the product to the number two space to the left on the same row and put the sum in the corresponding row.

It sounds more complicated then it should. Let's illustrate by producing a column:

We put a 3 in the second row because 3 = 3×1 + 0. We put a 1 in the third row because 1 = 3×0 + 1.

We shall now produce the whole table without disruption:

We can check that
 * 199×216 - 811×53| = 1

In fact, if the magic table is contructed properly, and we cross multiplied and subtracted the last two column correctly, then we will always get 1 or -1, provided the two numbers we started with were coprimes. The magic table is just a cleaner way of doing the mathematics.

Exercises
1. Find the smallest positive x:
 * $$216x \equiv 1 \ \ \mbox{(mod 816)}$$

2. Find the smallest positive x:
 * $$42x \equiv 7 \ \ \mbox{(mod 217)}$$

3.

(a) Produce the magic table for 33a = 1 (mod 101)

(b) Evaluate and express in the form p/q
 * $$3 + {1 \over 16 + {1\over 2}}$$

What do you notice?

4.

(a) Produce the magic table for 17a = 1 (mod 317)

(b) Evaluate and express in the form p/q

18 + {1 \over 1 + {1\over {1 + {1\over 1 + {1\over 5}}}}} $$ What do you notice?

Chinese remainder theorem
The Chinese remainder theorem is known in China as Han Xing Dian Bing, which in its most naive translation means Han Xing counts his soldiers. The original problem goes like this:
 * There exists a number x, when divided by 3 leaves remainder 2, when divided by 5 leaves remainder 3 and when divided by 7 leaves remainder 2. Find the smallest x.

We translate the question into symbolic form:

\begin{matrix} x&\equiv &2 \pmod{3}\\ x&\equiv &3 \pmod{5}\\ x&\equiv &2 \pmod{7}\\ \end{matrix} $$

How do we go about finding such a x? We shall use a familiar method and it is best illustrated by example:

Looking at x = 2 (mod 3), we make the jump into ordinary mathematics

\begin{matrix} x&\equiv &2 \pmod{3}\\ x &=& 2 + 3a \qquad (1) \end{matrix} $$

Now we look at the equation modulo 5

Substitute into (1) to get the following
 * $$2 + \!$$
 * $$3a \!$$
 * $$\equiv 3 \pmod{5} \!$$
 * $$3a \!$$
 * $$\equiv 1 \pmod{5} \! $$
 * $$a \!$$
 * $$\equiv 2 \pmod{5} \!$$
 * $$ a \!$$
 * $$= 2 + 5b \!$$
 * }
 * $$a \!$$
 * $$\equiv 2 \pmod{5} \!$$
 * $$ a \!$$
 * $$= 2 + 5b \!$$
 * }
 * $$= 2 + 5b \!$$
 * }

Now look at the above modulo 7
 * $$x = 8 + 15b \equiv 2 \pmod{7} \!$$

we get
 * $$b \equiv 1 \pmod{7} \!$$

We choose b = 1 to minimize x, therefore x = 23. And a simple check (to be performed by the reader) should confirm that x = 23 is a solution. A good question to ask is what is the next smallest x that satisfies the three congruences? The answer is x = 128, and the next is 233 and the next is 338, and they differ by 105, the product of 3, 5 and 7.

We will illustrate the method of solving a system of congruences further by the following examples:

Example 1 Find the smallest x that satisfies:
 * $$ x \equiv 1 \pmod{3} \!$$
 * $$ x \equiv 2 \pmod{5} \!$$
 * $$ x \equiv 3 \pmod{7} \!$$

Solution

now substitute back into the first equation, we get
 * $$ x = \!$$
 * $$ 1 + 3 \!$$
 * $$a \!$$
 * $$\equiv \!$$
 * $$ 2 \pmod{5} \!$$
 * $$a \!$$
 * $$2 + 5b \!$$
 * }
 * $$a \!$$
 * $$2 + 5b \!$$
 * }
 * }

we obtain
 * $$x \!$$
 * $$= 1 + 3(2 + 5b) \!$$
 * $$= 7 + 15b \!$$
 * $$\equiv 3 \pmod{7} \!$$
 * }
 * $$\equiv 3 \pmod{7} \!$$
 * }
 * $$\equiv 3 \pmod{7} \!$$
 * }
 * $$b \equiv 3 \pmod{7} \!$$
 * $$b = 3 + 7c \!$$

again substituting back Therefore 52 is the smallest x that satisfies the congruences.

Example 2
Find the smallest x that satisfies:
 * $$ x \equiv 5 \pmod{11} \!$$
 * $$ x \equiv 3 \pmod{7} \!$$
 * $$ x \equiv 8 \pmod{9} \!$$

 Solution

substituting back
 * $$x \!$$
 * $$ = 5 + 11 \!$$
 * $$a \equiv 3 \pmod{7} \!$$
 * $$ a \equiv 3 \pmod{7} \!$$
 * $$ a = 3 + 7b \!$$
 * }
 * $$ a \equiv 3 \pmod{7} \!$$
 * $$ a = 3 + 7b \!$$
 * }
 * $$ a = 3 + 7b \!$$
 * }
 * }

now solve for b
 * align="right" | $$x = \!$$   || $$5 + 11(3 + 7b) \!$$
 * align="right" | $$ = \!$$    || $$38 + 11\times 7b  \!$$
 * align="right" | $$\equiv \!$$ || $$8 \pmod{9} \!$$
 * }
 * align="right" | $$\equiv \!$$ || $$8 \pmod{9} \!$$
 * }

again, substitue back
 * align="right" | $$2 + 2\times 7b \!$$ || $$\equiv 8 \pmod{9} \!$$
 * align="right" | $$            b \!$$ || $$\equiv 3 \pmod{9} \!$$
 * align="right" | $$ b            \!$$ || $$= 3 + 9c \!$$
 * }
 * align="right" | $$ b            \!$$ || $$= 3 + 9c \!$$
 * }

 * $$x \!$$ || $$ = \!$$ || $$ 38 + 11\times 7(3 + 9c) \!$$
 * || $$ = \!$$ || $$269 + 11\times 7\times 9c \!$$
 * }
 * }

Therefore 269 is the smallest x that satisfies the congruences.

Exercises
1. Solve for x

\begin{matrix} 3x &\equiv & 5 \pmod{14} \\ 2x &\equiv & -3 \pmod{17} \\ x &\equiv &6 \pmod{15} \\ \end{matrix} $$

2. Solve for x

\begin{matrix} 3x &\equiv & 5 \pmod{19} \\ 7x &\equiv & -3 \pmod{17} \\ x &\equiv &6 \pmod{11} \\ \end{matrix} $$

*Existence of a solution*
The exercises above all have a solution. So does there exist a system of congruences such that no solution could be found? It certainly is possible, consider:
 * x ≡ 5 (mod 15)
 * x ≡ 10 (mod 21)

a cheekier example is:
 * x ≡ 1 (mod 2)
 * x ≡ 0 (mod 2)

but we won't consider silly examples like that.

Back to the first example, we can try to solve it by doing:

\begin{matrix} x & = & 5 + 15k      & \equiv & 10          & \pmod{21} \\ &  & 15k           & \equiv & 5           & \\ &  & 3k            & \equiv & 1           & \\ \end{matrix} $$ the above equation has no solution because 3 does not have an inverse modulo 21!

One may be quick to conclude that if two modulo systems share a common factor then there is no solution. But this is not true! Consider:
 * $$x \equiv 4 \pmod{15} \!$$
 * $$x \equiv 7 \pmod{21} \!$$

we can find a solution

we now multiply both sides by the inverse of 5 (which is 17), we obtain
 * $$x = \!$$ || align=right|$$     4 + 15k \!$$||$$ \equiv 7 \pmod{21} \!$$
 * ||align=right | $$       15k \!$$||$$ \equiv 3 \pmod{21} \!$$
 * ||align=right | $$5\times 3k \!$$||$$ \equiv 3 \pmod{21} \!$$
 * }
 * ||align=right | $$5\times 3k \!$$||$$ \equiv 3 \pmod{21} \!$$
 * }
 * $$3k \equiv 9 \!$$

obviously, k = 3 is a solution, and the two modulo systems are the same as the first example (i.e. 15 and 21).

So what determines whether a system of congruences has a solution or not? Let's consider the general case:
 * $$x \equiv a \pmod{m} \!$$
 * $$x \equiv b \pmod{n} \!$$

we have
 * $$x = a + km \!$$
 * $$x = b + ln \!$$

essentially, the problem asks us to find k and l such that the above equations are satisfied.

We can approach the problem as follows
 * $$0 = (a - b) + (km - ln)\,\! $$
 * $$(ln - km) = (a - b) \,\!$$

now suppose m and n have gcd(m,n) = d, and m = dmo, n = dno. We have
 * $$dln_o - dkm_o = (a - b)\,\!$$
 * $$ln_o - km_o = (a - b)/d\,\!$$

if (a - b)/d is an integer then we can read the equation mod mo, we have:
 * $$ln_o \equiv (a - b)/d \pmod{m_o}\,\! $$

Again, the above only makes sense if (a - b)/d is integeral. Also if (a - b)/d is an integer, then there is a solution, as mo and no are coprimes!

In summary: for a system of two congruent equations
 * $$x \equiv a \pmod{m}\,\!$$
 * $$x \equiv b \pmod{n}\,\!$$

there is a solution if and only if
 * d = gcd(m,n) divides (a - b)

And the above generalises well into more than 2 congruences. For a system of n congruences:
 * $$x \equiv a_1 \pmod{ m_1}\,\!$$
 * $$x \equiv a_2 \pmod{ m_2}\,\!$$
 * $$x \equiv a_n \pmod{ m_n}\,\!$$
 * $$x \equiv a_n \pmod{ m_n}\,\!$$

for a solution to exist, we require that if i ≠ j
 * gcd(mi,mj) divides (ai - aj)

Exercises
Decide whether a solution exists for each of the congruences. Explain why.

1.
 * x ≡ 7 (mod 25)
 * x ≡ 22 (mod 45)

2.
 * x ≡ 7 (mod 23)
 * x ≡ 3 (mod 11)
 * x ≡ 3 (mod 13)

3.
 * x ≡ 7 (mod 25)
 * x ≡ 22 (mod 45)
 * x ≡ 7 (mod 11)

4.
 * x ≡ 4 (mod 28)
 * x ≡ 28 (mod 52)
 * x ≡ 24 (mod 32)

To go further
This chapter has been a gentle introduction to number theory, a profoundly beautiful branch of mathematics. It is gentle in the sense that it is mathematically light and overall quite easy. If you enjoyed the material in this chapter, you would also enjoy Further Modular Arithmetic, which is a harder and more rigorous treatment of the subject.

Also, if you feel like a challenge you may like to try out the Problem Set we have prepared for you. On the other hand, the project asks you to take a more investigative approach to work through some of the finer implications of the Chinese Remainder Theorem.

Acknowledgement
Acknowledgement: This chapter of the textbook owes much of its inspiration to Terry Gagen, Emeritus Associate Professor of Mathematics at the University of Sydney, and his lecture notes on "Number Theory and Algebra". Terry is a much loved figure among his students and is renowned for his entertaining style of teaching.

Reference
1. The Largest Known Primes--A Summary


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Jakarta EE Programming

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 * JACC (Java Authorization service provider contract for containers)
 * Web Services
 * Jakarta XML Web Services (JAX-WS; formerly Java API for XML Web Services)
 * Jakarta RESTful Web Services (JAX-RS; formerly Java API for RESTful Web Services)
 * Jakarta EE Management
 * Java Management Extensions (JMX)
 * Managed Objects
 * Events
 * State Management
 * Performance Monitoring
 * Index

Related Concepts

 * IDE
 * Database
 * Web Engine
 * Attribute Oriented Programming
 * JavaBeans

title: Jakarta EE Programming


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File:Trapizoidruleexample.jpg

This is an artists impression of the trapizod rule used in summing up in this case y = - x^2.

title: File:Trapizoidruleexample.jpg


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Jakarta EE Programming/EJB-QL

EJB Query Language is used to define what a finder method of an Entity bean should return. The EJB 1.1 specification offered no way to query entity beans. It resulted in different query languages for different application servers. When Sun published the EJB 2.0 specification it added a query language to find entity beans more easily.

Basic Syntax
SELECT_CLAUSE FROM_CLAUSE [WHERE_CLAUSE]

Both SELECT_CLAUSE and FROM_CLAUSE must be used but WHERE_CLAUSE is optional.

Finder methods
Finder methods are used to return a single or multiple Entities from the storage. The finder method is defined in entity's home interface. The return value of a finder is the beans remote interface or a collection of entity beans implementing the remote interface which were found.

Basic finder method
The simplest possible EJB-Query would be one which return all Entities in a given table.

@ejb.finder signature="Collection findAll" query="SELECT DISTINCT OBJECT(p) FROM Player p"

As you can see, the EJB-Query looks nearly the same as a standard SQL query, although it's not the name of the tables but the name of the EJB classes and it's not the name of the columns but the name of the EJB properties. The only difference is the terminal object(o) which is equivalent to: take all columns of a given row and transform the result into an entity bean instance.

Finder method with WHERE clause
Look at the example above. It differs from the finder method above only that a conditional for the players which should be returned is added.

@ejb.finder signature="Collection findByPosition(java.lang.String position)" query="SELECT DISTINCT OBJECT(p) FROM Player p WHERE p.position = ?1"

This finder method returns all Players which play at the position which is entered by parameter one. If you have more than one parameter you can use them in your query with ?2, ?3 and so on.

Finder method which operates on relation
EJB Query differs from SQL especially if you want to use different relational tables.

@ejb.finder signature="Collection findByCity(java.lang.String city)" query="SELECT DISTINCT OBJECT(p) FROM Player p, IN (p.teams) AS t WHERE t.city = ?1"

This query return all players whose team plays in the city given with parameter one. In other words: return all Players who play in the team which is situated in a given city.

Select methods
In opposite to finder methods select methods might return a persistence field or other entity beans. So it's possible to define a select method which returns a primitive data type instead of a complete object.

title: Jakarta EE Programming/EJB-QL


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User:Moewe~enwikibooks

I am interested in J2EE and web technologies. Since I started learning J2EE some months ago I will try to save my knowledge in this book J2EE.

title: User:Moewe~enwikibooks


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Electronics/Noise in electronic circuits

 * Electrical Noise: any unwanted form of energy tending to interfere with the proper and easy reception and reproduction of wanted signals.

Based on Origin

 * 1) External noise
 * 2) Atmospheric
 * 3) Extraterrestrial
 * 4) solar
 * 5) Cosmic
 * 6) Industrial
 * 7) Internal noise
 * 8) Thermal Agitation Noise
 * 9) Shot Noise
 * 10) Transit Time Noise
 * 11) Flicker Noise
 * 12) Miscellaneous Sources

Thermal noise
$P_n \propto\ T\;\delta\!f = k\,T\;\delta\!f$ where k = Boltzmann's constant = 1.38x10-23J/K
 * Thermal Agitation Noise: Also known as Johnson noise or White noise.
 * T = absolute temperature, K = 273 + °C
 * δ f = bandwidth of interest
 * Pn = maximum noise power output of a resistor

$P_n = \frac{V^2}{R_L} = \frac{\left(\frac{V_n}{2}\right)^2}{R} = \frac{V_n^2}{4R}$ $V_n^2 = 4RP_n = 4RkT\;\delta\!f$ $V_n = \sqrt{4kT\;\delta\!f\;R}$

Shot Noise
$i_n = \sqrt{2ei_p\;\delta\!f}$ where in = r.m.s. shot-noise current
 * e = charge of an electron = 1.6x10-19C
 * ip = direct diode current
 * δ f = bandwidth of system

Addition due to several sources
noise voltages: $V_n1 = \sqrt{4kT\,\delta\!f\,R_1}$, $V_n2 = \sqrt{4kT\,\delta\!f\,R_2}$...and so on, then $V_{n,tot} = \sqrt{{V_n1^2}+{V_n2^2}+...} = \sqrt{4kT\,\delta\!f\,R_{tot}}$ where Rtot = R1+R2+...

Addition due to Cascaded Amplifier stages
Req = R1+R'2 $$R_{eq} = R_1 + \frac{R_2}{A_1^2} + \frac{R_3}$$

Differential signaling
Differential signaling is a method of transmitting information electrically by means of two complementary signals sent on two separate wires. The technique can be used for both analogue signaling, as in some audio systems, and digital signaling, as in RS-422, RS-485, PCI Express and USB.

Good grounding
An ideal signal ground maintains zero voltage regardless of how much electrical current flows into ground or out of ground.

When low-level signals travel near high currents, their return currents shouldn't be allowed to flow in the same conductor. Otherwise, noise such as AC ripple on the high current will modulate the low-level signal.


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Written Chinese/Cantonese pronunciation

The Cantonese characters pronunciation consists of three elements: syllable onset(聲母), syllable nucleus(韻母) and tone(聲調). A character consists of syllable onset(though some rare characters does not have this element), following by a syllable nucleus, with a tone for each character.

The concept of syllable onset and syllable nucleus can be explained by examing the english word "sing". When you pronounce this word, try to divide it to "s-" and "-ing". Note that the "s-" determines the shape of your mouth, and the "-ing" is the actual voiced part. That's the concept of syllable onset and syllable nucleus respectively. The tone is a special feature in Chinese language. There is a total of nine tones, and each character belongs to one of them. Note that there is no concept of stress in Chinese words - each word has only one syllable.

Syllable onsets
There are totally 19 syllable onsets in Cantonese. The following table shows their pronunciation guide:

title: Written Chinese/Cantonese pronunciation


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A-level Mathematics/Advanced

The objective of this book is to provision the reader with a more in depth knowledge of the material covered at A-Level in mathematics. It includes a focus on the definition and proof of concepts rather than the understanding of these concepts, which the reader is assumed to be somewhat familiar with. It is organised by concept and is neither comprehensive nor syllabus specific. Instead it is intended to allow the keen student the opportunity to gain a greater understanding of the subjects covered at A-Level and hopefully perform better in the field of mathematics as a consequence. It should not therefore be used as a sole reference for examinations and many of the explanations will contain more depth than the average student is expected to reproduce in an exam.

Subject Index

 * Basic Mechanics
 * Advanced Mechanics
 * Basic Statistics
 * Advanced Statistics
 * Basic Pure Maths
 * Advanced Pure Maths

title: A-level Mathematics/Advanced


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Developing A Universal Religion/Consciousness And Conscience

What is the “me” that makes one think in a manner that is peculiar to only oneself?

The total “me” is easy to imagine; it must be the accumulation of events and understandings that one has experienced during one’s life, added to the genetically inherited abilities and aptitudes present in one’s brain. As we have noted, molecular memories and the understandings they represent are stored as linked paths and networks of greater or lesser significance through everyone’s brain—the whole constituting the “mind,” just part of it forming the “me” concept. This collection, together with the biochemical activities and emissions of the cells of one’s body, makes the “me” think and act the particular way that one does. As Descartes said, cogito, ergo sum (“I think, therefore, I am”).

Second-level thinking, i.e., when animals analyze situations and recognize their implications, then act upon (or dismiss) what they have understood, may or may not involve consciousness of self. If the situation is totally independent of their own individuality (for example, for an antelope when a lion walks nearby) then an awareness of their own unique identity is not called for (only the need to include the knowledge of such things as their proximity to the lion, wind direction, etc., in their analysis). But, if, in order to correctly assess a situation, an animal needs to separate its identity from that of others (for example, when in a family or grooming group, where knowing one’s social standing, and how others act and react toward one’s presence and actions), then a degree of consciousness of self must be present.

The recognition of personal identity, a separate self or me would have occurred very early in the development of third-level thinking ability. Cassirer knew this when he stated, "it is language that makes his existence in a community possible; and only in society, in relation to a 'Thee', can his subjectivity assert itself as a 'Me'."

Third-level thinking, using words and languages, provides the consciousness we are familiar with, where thoughts can be consciously directed and where moral questions are formulated. This is the detached self that can examine (with some difficulty) what is happening at the second level.

As we have discussed, everything we “know” is built, held and maintained as second-level constructs—developed by second-level thinking that builds neural networks which form and link the memories. These give us the mental images of objects, events and ideas that we carry in our mind’s eye; they are our own Platonic cave-wall shadows. It is the mental construct of one’s own body that one “sees” when experiencing an out-of-body sensation (that of looking down upon oneself). Out-of-body sensations occur as the conscious third level of thought is (semi-consciously) drawn by prevailing circumstances to picture mind images of one’s body as though they were separate and distinct (i.e., disconnected) from the mental networks that denote self.

It was third-level thinking that made some early scientists postulate that there was an imp, or homunculus, directing mental traffic within the brain; the imp turns out to be the mind’s second-level activities.

Consciousness, then, amounts to an awareness of the existence of an assemblage of thoughts and memories within the brain, and of the particular significance that these have to the possessor. The awareness occurs at the third level, and it is the presence of mental constructs that creates the sense of permanency to one’s concept of self. Consciousness is aware of second-level activities, but their rapidity and subconscious independence make them hard to analyze. Second and third-level activities block ready access to first-level consciousness; training and practice aimed at decreasing third and second-level thinking activities (meditation, for instance) may occasionally allow first-level awareness to make itself known (as an experience, not as a detailed representation of the external environment).

Research demonstrates that subconscious biochemical and electrical flows occur before we become consciously aware of them. (We should expect this because mental images must first form subconsciously to be recognized and analyzed for relative significance; only then can those of importance be selected and fed to our conscious third-level thinking where, finally, they may be put into words.) This is why the semi-consciousness we are occasionally aware of seems to have a life of its own. It does. Thoughts at the second level run their own course before we become aware of them. This effectively detaches them from our third-level thinking, and makes them seem to exist as an independent body of thought within our minds.

Conscience is an entirely different issue. In essence, exercising one’s conscience amounts to expressing one’s concept of truth. This, therefore, represents both the highest and the most fundamental level of life’s activities: the highest, because life survives by determining the true nature of its environment; the most fundamental, because life does this to most effectively exploit its resources in order to live. Unfortunately, the true nature of things is readily distorted; by one’s sensors and one’s understanding of signals received by them, and by the words and mental constructs we and others use. Truth, to the extent that it exists, is often costly to obtain; finding it requires openness to the widest possible range of experiences, facts and ideas, then a constant debate over their meaning, with oneself and with others.

Conscience is often associated with morality—knowing “right” from “wrong,” and behaving accordingly. In that morality is always relative to its time and circumstances, it is a lesser concept than the concept of conscience. Most theologies recognize this, some going as far as saying that one’s conscience is God-given and must be followed, even if it contradicts religious teachings.

Most biologists dismiss any discussion of an animal conscience. For example, Hauser, in Wild Minds, states categorically (page 253) that animals have no moral system. But I think that advanced animals may possess a conscience of some degree because they do seek to understand the true nature of their environment, and they can separate a knowledge of “self” from that of another. Animal altruistic behaviour, which has not infrequently been observed, may demonstrate the operation of a conscience and of animal morality.


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Developing A Universal Religion/Purpose And Meaning

There is an important difference between asking, “what is life’s purpose?” and asking, “what is life’s meaning?”

The first question is by far the most important, for we seek an answer that must be universally and permanently true. All of life, wherever it is and in whatever form it exists, is expected to be subsumed within the answer to the question about life’s purpose. “To do God’s will” might be the reply of many to this question, giving an answer that, they would claim, applies to all things and all creatures for all time. The point to note is that, although the chosen “life’s purpose” might vary from one person to another, every choice selected must meet the criteria of universality and timelessness.

The second question, asking life’s meaning, is clearly of less significance, because subjective and multiple answers are acceptable and even expected. Everyone is quite willing to accept different replies from the same person for we fully recognize that “life’s meaning” can change from day to day. “Life’s purpose” has no such freedom.

The answer to the question of life’s purpose turns out to be the key that unlocks the puzzle of life’s meaning. We can be sure of this, because whenever someone says that life has meaning for them, we always find that they are expressing a feeling that stems from acting to achieve one or more purposes they deem to be important.

Many do not recognize that they live their daily lives happily striving to attain a multitude of purposes. The desire to live comfortably, to provide for a family, to be without pain, to be emotionally satisfied, to enjoy life; all these and thousands of similar phrases are statements of purpose, all more or less distant from conscious thought, but all significant to our minds as they go about their task of making the decisions that guide our daily activities.

We sometimes consciously chose one or two purposes to have particular significance for us, and their achievement may then take primacy over others. Getting a degree, saving money to buy a house, or helping charitable organizations might be examples, and many of us spend much time and effort supporting the attainment of goals such as these. However, whether or not we recognize the fact, every one of our activities is directed toward the achievement of one purpose or another.

Of course, we often react to emotions and feelings as well. But these actions are taken to satisfy or alleviate the emotions or feelings that prompted them; thus acting to satisfy our emotions is also acting to achieve a purpose. The difference is that these are not purposes directed by conscious thoughts, they are responses to body chemicals. Thus, they are usually more primitive or animal-like in nature (although emotional responses to music probably pertain to relatively recent evolutionary developments).

The happy feeling that we are living a meaningful life, or that life possesses meaning for us, is a by-product of a mind that is doing its job well by directing its support system (the body) to meet a multitude of purposes. The feeling stems from the mind being able to work relatively stress-free, both consciously and subconsciously, because the tasks we perform, the thoughts we arrange, the decisions we make, are directed toward some worthwhile, i.e. purposeful, end.

It is pointless to directly seek the meaning of life. Feeling that life is meaningful is the normal state—a feeling of well-being, when the bloodstream is relatively free of stress-causing chemicals, because the mind is working efficiently and effectively, making purpose-directed decisions. Such a mind has no need to instruct the release of anxiety-causing chemicals.

There is no physical or biological requirement to feel that life has meaning. Living entities can eat, survive and reproduce, without any such feeling, as the daily lives of bacteria, plants and insects presumably affirm. In organisms capable of conscious thought, however, there is a definite requirement for such thought to be purpose-directed. Solving problems and making decisions rationally requires a desire to achieve some purpose—this systematizes conscious thought. Working rationally is the activity that makes the mind valuable to survival, thus ensuring its own survival.

People who lack valued purposes are susceptible to depression, when nothing seems worthwhile and life can feel meaningless. The cure is not to directly seek meaning, but to find a purpose worthy of being valued and sought, then use this purpose to make decisions and guide actions.


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Developing A Universal Religion/Rationality In Science And Religion

Science and mathematics reign supreme in our understanding of the universe because, as has been elsewhere noted, each scientific fact and theory, each mathematical statement, has been exhaustively tested for inconsistencies and illogic before being given membership among the hierarchy of theories and facts that make up the totality of these disciplines. Our underlying belief in the universe’s causality requires this rigour, as any discrepancy, until resolved, threatens to demolish our whole understanding of the cosmos.

The mark of a true scientist is that he or she willingly investigates discrepancies. The hallmarks of scientific method are that its findings are repeatable, measurable, and universal in implication and relationship, yet simple in concept once understood. Although humans have uncovered a great deal about the universe, we have much to learn, and the meticulousness of the scientific method is the only sound approach.

Most religions, in contrast, do not tolerate such a questioning attitude and, consequently, many have become less and less in tune with reality.

Religions fail us when they ignore our need to be rational. Of course, anyone can hold any belief or statement to be true. There is no necessity (other than the mind’s own operational need for rationality) to demand a logical relationship between subordinate statements and apex belief in religion. Each statement could be taken at face value, true and absolute in itself. Indeed, many people (particularly those who follow their religion’s fundamental text conscientiously) seem able to accept most, and sometimes all, of their religion’s statements as absolutes, and require no inter-connectivity between them. But this treats us all as though we were automaton, incapable of thought, simply being required to believe and to act as we are told.

(Preachers do not usually ignore the inter-relationships within their religion. In my limited experience, preachers spend much time attempting to show links between a belief in God and subordinate statements [such as the existence of Heaven and Hell], or between actions and consequences. Possibly this is because preachers are more rational than most of us, and have been drawn to religion because of the need to find an authority for the moral decisions their mental constructs require them to address. If this is even partially so, it is truly ironic to realize that it is the universe’s very own rationality expressing itself within their minds that has induced them to become leaders in an arena where rationality has become subordinated to faith!)

As earlier stated, we can believe anything we choose. Our beliefs will cause us no harm, as long as they do not cause us to act in a way that reality will punish. Thus, we can believe, to use an extreme example, that the moon is made of green cheese, that the man-in-the-moon is our real father, and that we will all go to be with him after death. We can construct a subsystem of subordinate beliefs, all guiding us in our daily decision making, and all assuring us that we will be rewarded with an afterlife amid green cheese. And we can live more or less happily within this belief system and die content.

We can sustain these beliefs forever, provided we somehow filter and modify any incoming stimuli that fails to support them. For instance, we could believe in a cheesy moon as long as we did not spectrally analyze moon light, and we could continue to believe in moon heaven for as long as we did not physically visit the moon. Others could go, but they could not return to report its true nature and so challenge our beliefs, for it is clear that our green-cheese belief would become untenable after the return from a moon landing. (Of course, this is exactly why a belief in heaven or paradise can exist.)

All belief systems run the risk of unravelling when forced to confront reality. The only belief systems that survive close encounters with the real world are those that are based upon the rationality of the real world. The less our belief system agrees with reality, the more difficult following that belief becomes. We can enjoy our beliefs for as long as we like, but reality will compel us to revise such beliefs as soon as they create situations which threaten our existence.

We stand at this juncture in our current religions. Many of our traditional beliefs have become less convincing. Reality keeps pulling at the tangled skein of religious thought, attempting to correct false assumptions and misunderstandings. More and more, we become obligated to ask if current religions really are the best source of guidance in contemporary issues. Moral decisions are in danger of becoming little more than political trade-offs at the parish level. Matters relating to population control or gene manipulation, for example, that need a global consensus if such decisions are ever to amount to anything of significance, cannot even be raised at the international level for fear of the religious conflict this would create. Surely a global civilization can never be established until conditions like these are corrected.


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Developing A Universal Religion/Creativity, Free Will, And A Revelation

Creativity
To be creative, our thoughts must stop following the well worn neural paths they are accustomed to travelling. Thus, we start the creative process when we (mentally) say “no.” By saying “no” to any particular thought, we force our minds to conceive alternatives, to search for different neural routes among our existing store of memories, to look for new links. Let me provide a couple of examples.

Consider an artist who paints a picture that is appealing and sells well. He or she might churn out several paintings of like vein without thinking and live well. But, other than the first painting, this is being productive, not creative. To be creative, the artist must first say “no” to some aspect of his or her work. Some large or small part of it must not be produced by rote; it has to be produced as the result of a conscious effort to discover and present something new. A novel way of using titanium white, an insight into the nature of anything, a new way of presenting an emotional experience, a rational discovery, anything at all, as long as it produces an informing entity that is unique. The refusal to repeat what has been done before, because there must be a better way, is the act that initiates creativity.

Alternatively, consider a manager, heading a division within a factory producing widgets. There are opportunities galore to be creative; all such a person need do is think, “No, there must be a better way of doing this,” then work out what that might be. On the other hand, the manager may not be faulted for thinking little and creating nothing, being paid for his efforts, and going home knowing that he has done his job, content. (But, somehow, perhaps, feeling a little incomplete, for he has simply managed an operation, not led. To lead others one must first be creative.)

Everyone can be creative. And we all are, some of the time. We have all created many things, suggested a different approach, changed a routine, come up with a new thought, improved upon the past. Each time, in a moment prior to each occasion, we would have, in effect, said “no,” and this would have stopped us from repeating what we may have done many times before.

Routine living, simply repeating with minor, externally induced, variations what has been done before, is not being creative. No improvements—no increases in life’s quality—are produced, and life stagnates. It is the creative acts of millions who have lived before that has given us everything we have that animals do not.

When we refuse to allow our thoughts to follow their usual pathways, the search for new neural associations begins. Finding them completes the act of creativity; new links are made, new memories laid down, our mind’s store of constructs enlarges, and our future abilities increase. Creativity adds to oneself as well as gives to others.

Free Will
Thinking And The Universe noted that we discover how the universe and its contents operate by examining the cause-and-effect relationships that tie events together. It was emphasized that nothing happens in isolation; every object and every event is linked through a chain of causality to everything that existed before and to everything that exists currently. Not one iota can change without being brought about by some prior event happening.

But, if no change can occur without being caused by some prior event, which itself must have had a prior cause, and so on and so on, backwards in time to the universe’s beginning, then how can free will exist? How can any individual have any thought that hasn’t been pre-programmed into the cause-and-effect network that exists throughout the universe?

Moreover, if free will does not exist, how can anyone be responsible for their actions? And if no one can be held accountable, then no one should be disciplined for anything they do.

This problem has agitated theologians and philosophers for centuries, and has never been satisfactorily resolved.

Intuitively we feel that we do have free will. We all think that we are free to make any decision we like, and most of us expect to be held accountable for the decisions we make and the actions we carry out as a result of those decisions. So there is some explaining to be done—where does causality give way to allow free will?

Possibly the answer is simple. It may be that causality does not apply to single particle events. What follows is an elaboration of what is meant by this statement, and some evidence supporting the premise.

We think of particles as discrete objects, having mass and able to move from place to place (factors that we can easily measure) because this is our direct experience of the (large and small) objects that we encounter everyday. However, if particles are small enough (about the size of an atom) scientists find that their true nature is not that simple.

One of the peculiarities we find, is that we can never know simultaneously the position and momentum of single particles such as an atom or an electron. This is because any time we try to measure either of these properties, the act of measuring one changes the other. For example, light photons reflecting from a particle that, once detected, should tell us the information we seek, actually give the particle a backward kick as they bounce off. This shifts the particle’s position in an unknown way. It’s a bit like trying to find out what a falling feather is doing by poking at it with a stick. Heisenberg developed the Uncertainty Principle as a description of this problem in 1927. (It precipitated much free-will discussion among scientists and philosophers at the time.)

Another unexpected feature of particle behaviour is that, if we send a beam of them (or even shoot them, one at a time) through two thin parallel slits to make an image on a screen, rather than the expected two lines appearing on the screen, we see several parallel lines of varying density (called an interference pattern). Moreover, these lines are being drawn by a succession of many single hits. Now, on our everyday scale, objects like bullets can’t produce interference patterns. But waves can, so our explanation of such occurrences is that particles show “duality”; they move as if they were waves, but hit targets as if they were bullets.

Quantum mechanics helps physicists solve conundrums like these. This discipline has determined that energy exists as waves of interacting energy fields and that matter is actually bunched-up packets of waves (or photons). (You will remember that Einstein showed that matter and energy were different forms of the same thing, so quantum mechanics and the Theory of Relativity support each other’s explanations.) Since waves are spread-out entities, the position of the particle they represent can only be calculated as a probability of being at any particular place. (For instance, a scientist might say, “If a photon were to hit this screen, it is 60% likely to hit at this spot.”) This may be the first clue in resolving our free will enigma.

The existence and behaviour of “virtual particles” might be another clue that causality does not apply to particles. Myriads of virtual particles constantly flicker into and out of existence everywhere (obtaining the energy to do so from the “vacuum energy” of space, see The Expanding Universe). They are called “virtual” because they cannot be directly detected. Their occurrence was predicted by Heisenberg, and they are found by looking for the real particles and antiparticles they create. (These created particles and antiparticles immediately destroy one another; it is the leftovers from this destruction that scientists observe.) The comings and goings of the unseen originating virtual particles are completely unpredictable from our point of view within the universe. Since the behaviour of these particles is unpredictable, the results they produce are also unpredictable. Causality seems not to come into play on very small events in our universe.

Observations such as these suggest that tiny particles, virtual or real, possess properties that are somewhat different from those exhibited when the same particles form large conglomerations. The observations suggest that, at the grassroots level, only probabilities exist. These probabilities build to become statistical certainties as the particle groupings becomes bigger. Once large enough, their behaviours exhibit the causality upon which the laws of physics depend, and that allows scientists to explain so much. In short, predictable causality is not possible (and absolute certainty does not exist) for events involving objects of small dimensions.

So much for causality. Now, for free will.

Thinking, as we have noted, can actually be observed occurring in the brain. Imaging scans reveal that electrically charged ions travel along neural axons and prompt chemical transmissions across synapses to neighbouring neurons. But ions, we remember from our school science class, are atoms or tiny groups of atoms that have lost or gained electrons. That is, they are very small particles.

Now we have all we need to allow free will to exist. We exercise free will when we first say “no.” (As stated in the above postscript “Creativity,” this is exactly what we do to start the creative process.) Subsequent, conscious, third-level thought, capped by a decision, completes the act. This process consciously overrules, or at least re-thinks, any decision that may have been made by prior thinking, including that done at the second-level subconscious in our mind. We are able to do this, because causality does not force small particles (particularly chemical neurotransmitters travelling between synaptic gaps) to trace previously determined paths.

There we have it! We possess free will because small particles are not causally constrained; they obey probability laws which only become certainties when large numbers are involved. We can say “no,” and be responsible for creating an original decision.

(For what it’s worth, we might note that we exhibit no free will at all when we just “go with the flow.” All flexibility and freedom disappears when multiple entities merge, be they particles, photons, ions travelling along existing neural pathways, or mobs of people.)

A Revelation
A “revelation” can be experienced that takes a person from from simply thinking that an idea could be true, to believing that it is true. Its sudden, totally unexpected arrival; its mental fireworks and fascinating light trails; its prolonged accompanying feelings of certainty, exultation and joy; and its ability to direct my actions even now, are all a matter of natural biochemistry. With the loss of its original wonder, and the expectation of never again experiencing its magic, comes a degree of sadness. But, it also brings the knowledge that something similar, something fully explainable and rational, must have happened to many people, many times in the past. Each one of history’s prophets and mystics who claimed to have received divine intervention, with its accompanying brightness and light, surely experienced a similar phenomenon. Scientists, perceiving a sudden solution to a problem that had long occupied their thoughts, have reported similar happenings, sometimes adding that it was as though the universe had spoken to them. These feelings, as noted in The Source Of Revelations, do not emanate from some external consciousness, some universal spirit; they come from the individual’s own second-level subconscious mental activities.

Is it sad to say goodbye to the notion that these exotic experiences are proof that a higher Being exists, that some divine force is slowly but certainly bringing enlightenment to us? I don’t hold so, for this imposes a dependency and subordination that harks back to the days when superstition and fear ruled the human mind. The idea of being manipulated by a god apportioning ideas in this manner is quite distasteful to me. I do not like to think that any god worthy of its title would do this to any of its creations.

Is it distressing to know that our thinking is not being supported or even directed by stimuli received from a god—to know that we are acting all alone? No; not at all. To the contrary, to know that life, starting from scratch and with absolutely no outside help, is actually on a journey towards comprehending the entire universe is far from troubling—it is magnificent. It is inspiring to think that any single one of us might add another piece to the elucidation of the puzzle. It is exhilarating to know that, as infinitesimal as humans may be on the grand scale of things, they are nevertheless slowly unravelling the nature of the universe, and, through the understanding this brings, gaining some measure of power over fragments of the cosmos itself.

To think that some distant descendant of ours might someday control all! None of the religions I have read about give life and living such an overwhelming sense of purpose and destiny to me. They may have done so, to some people at one time, and they may still do so, to other people today. But not to me. What makes my life meaningful to me is the thought that some of my actions might, through the later efforts of others, contribute morsels toward the eventual evolution of oB.


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A-level Mathematics/Advanced/Basic Mechanics

Mechanics is a broad subject, relating to the behaviour and interaction of physical objects. A more specific definition is that this transcends pointlessness because mechanics is one of those broad abstract ideas that one acquires a definition of by studying its specifics rather than by summarising in a single sentence. With that in mind, we consider its basic concepts.

Concepts

 * Basic Concepts in Mechanics
 * Space and Time
 * Vector Algebra
 * Concepts in Motion
 * Momentum and Impulse
 * Newton's Laws of Motion
 * Newtonian Gravity
 * Energy


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Developing A Universal Religion/Gödel’s Theorem, General Systems Theory, and The Conservation Laws

Gödel’s Theorem
Kurt Gödel, in a paper published in 1931, proved that any mathematical system that includes the natural numbers (1, 2, 3, and so on) contains questions whose answers can neither be proved nor disproved using the axioms to be found within that system. This is now known as Gödel’s Incompleteness Theorem. It implies that there are many mathematical truths that can never be proved, and, by extension, that any system will contain questions that cannot be answered from within that system. Since any meaningful questions that we might ask about the nature of any possible supersystem will inevitably involve use of the natural numbers, there are questions we will never be able to answer. Asking if our universe was “designed to meet some kind of purpose,” is just one such question.

Another way of using Gödel’s theorem to address why one is unable to understand everything from within a system, is as follows. A system is complete when all statements (or their negations) can be proved from within that system. Systems can be proved to be consistent, i.e. free from contradictions, but only by involving a larger frame of reference, which then requires an even larger system to prove its consistency, and so on. In short, we require information that is only available from outside of our universe to determine the accuracy of any answer to any question that can be raised within the universe. In other words, we can never know the answers to questions like, “does God exist?” or, “what is the universe’s purpose in existing?” because we cannot obtain information about what exists beyond the boundaries of our universe.

In effect, Gödel is saying that we can never know anything fully and completely. Thus, even the very best of our scientific understanding is ultimately unverifiable.

Some might say that if this is true, then what is the point of doing anything. Why bother to develop a universal religion, for instance? The answer, of course, is that we need one just as much as we need an understanding of how things work, because both improve the quality of our lives. Time enough to stop doing our best after we are dead!

General Systems Theory
The basic concepts underlying General Systems Theory are simple, once the terms employed are recognized and their meaning understood. The theory’s power stems from its generalisability, for all systems (whether living or non-living, small or large) demonstrate the same principles.

General Systems Theory can be summarized as follows.

A system is a processing complex of interrelated parts that acquires supplies and turns them into something else. Thus, a person can be called a system, for each individual takes in oxygen, nutrients and water, processes them, and turns out movement, growth, and bodily waste. A factory can be called a system, for it takes in raw materials and energy, carries out operations, and turns out manufactured goods and waste.

Physical and biological systems exist everywhere; they range in scale and form from primitive Archaean life to galactic clusters, and the same systems terminology applies to all.

No system within our universe is “closed.” In other words, there is no system within the universe whose boundary is impermeable to everything. For example, the Earth receives and processes energy from the Sun; the Sun was formed from earlier Milky Way galactic dust, and radiates energy that interacts with the galaxy’s particles; all galaxies exert gravitational pulls upon one another (so their motions are interdependent), and so on. An alternative way of expressing this property is to state that all systems are “open.”

Thus, all systems (except, possibly, the universe itself) are subsystems of larger supersystems, and, in particular, the biological system is a subset of the physical system. In other words, life is a subsystem of the universe.

We cannot tell if our universe is a subsystem of a larger universe. If there is no linkage to anything external, if the universe is entirely self-contained, and neither takes in nor gives out any form of material, then it is closed. If our universe is somehow related to (i.e., exchanges information, energy and/or matter with) a larger Universe, then it is open.

Systems are dependent upon, and thus controlled by, their various supersystems. This fact becomes readily apparent when a system can no longer obtain needed resources from its environment (its supersystem) and shuts down. It is also demonstrated when a system’s outputs are so excessive or aggravating that its supersystem (its environment) can accept no more, and the resulting back-pressure (a.k.a. feedback) shuts down the system’s operations.

The criteria that determine what we can do, or what we are able to produce, are all to be found in the way our supersystems react to our behaviour. Our supersystems can reject (partially or fully) or accept (partially or fully) our outputs. (For example, a rejection by family, friends, employer or society can soon effect our welfare.) Conversely, supersystem acceptance creates the demand for more of the same output and thus encourages more of the same processing activity.

General Systems Theory terminology can be used to increase our understanding of the problem-solving and decision-making processes described in ../Solving Problems/ and ../Making Decisions/. Thus, the supersystem (earlier termed the environment) provides the criteria that determine the success or failure of its subsystems’ behaviours. Physically existing supersystems exhibit and enforce many real criteria, and we make practical decisions successfully by knowing and respecting these. Similarly, mentally existing supersystems (i.e. major constructs) exhibit and enforce many abstract criteria, and we make moral decisions successfully by knowing and respecting these.

The Conservation Laws
As the Conservation Laws have been referred to several times in the text, it might be useful to say a little more about them.

Conservation Laws state that, amid all the changes that occur throughout the universe, certain quantities and qualities (for instance, the total amount of energy/mass, momentum, charge, spin, parity, etc.) are always conserved. The value of each (although not necessarily the form) after an interaction is always the same as the value of each before. These laws explain why, for instance, a perpetual motion machine cannot be built. (Interacting system-parts generate heat which is lost to the surrounding environment. Thus the machine loses energy and eventually stops.) Conservation laws explain why there is a property we call inertia. (We feel a force termed inertia when, for instance, we push an object to start it moving. Accelerating an object in this way changes its velocity, and this means we have added to its momentum. Since momentum must be conserved, it must be taken from somewhere; in this case from our hand and body, and, ultimately, from the Earth, reducing its spin (i.e., its angular momentum) a tiny fraction. The inertial force we feel is our body’s reaction [Newton’s Third Law of Motion] to the force that transfers momentum from the Earth to the object.)

The several Conservation Laws are likely to be sub-manifestations of one comprehensive law that we have not yet discovered. Superstring theory may soon be able to tell us more, not only about the Conservation Laws, but also about why certain physical constants are just right for our universe to exist and to create and nourish life. Superstring theory, in Witten and Townsend’s M-theory version, can now account for the existence of the known forces (gravity, electromagnetic, strong and weak nuclear), showing that they may all be derivatives of minute vibrating strings 10-35 meters or less in length, and existing in either 10- or 26-dimensional hyperspace. Superstring theory exercises the minds of many physicists and cosmologists (the first group, because it may be the elusive TOE/GUT [Theory Of Everything or Grand Unified Theory] long searched for, and cosmologists, because it predicts and allows for the existence of other universes.)

Some seemingly inexplicable phenomena (such as the behaviour of virtual particles, or the instantaneous transmission of quantum states, as well as time-travel and teleportation [both recently demonstrated to exist ]) might represent a window through which we may glean a little knowledge about the possible existence of any such super-universes.

Our universe may be just an adjunct of a larger Universe, with the larger Universe retaining ultimate control. Control by a super-Universe could be rigid, creating nothing more than a fully deterministic sub-universe if the connections between the two were entirely inflexible. However, this seems not to be the case. Wave/particle duality and the laws of conservation allow minuscule events unlimited freedom to act, as long as conglomerate activities obey the conservation constraints (see also the earlier discussion on free will).

Alternatively, our universe’s existence could be simply a manifestation of nothing, just as branches of mathematics can be created from definitions rather from actuality. All that is needed is for the whole to sum to zero, and that sum to be maintained regardless of how its parts manifest or become manipulated (a condition maintained by causality and described by the Conservation Laws).

 * Postscript 2: Origin Theory Modifications


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Developing A Universal Religion/Origin Theory Modifications

Theories related to life’s creation and its evolution are still being proposed. In the 1960’s Francis Crick, Leslie Orgel and Carl Woese independently proposed that RNA preceded proteins in evolution, and introduced the idea of an early “RNA world.” In 1983, Thomas Cech and Sidney Altman found evidence that RNA can act as a catalyst, so supporting the RNA world theory.

Current thinking posits that early RNA exchanges occurred laterally (within a commune of different cells), rather than generationally (from parents to offspring), as DNA now replicates. The former may have facilitated many primitive evolutionary developments, and has been found to occur in bacteria.

An alternative theory, put forward by Graham Cairns-Smith, suggests that an inorganic genetic system is likely to have existed before the RNA world, possibly one involving the irregular distribution of cations since found in clays.

Ideas and findings such as these may help us to replicate possible early life forms in the laboratory. That we will create life from scratch one day is not much doubted by any biologist (although some prefer that we never make the attempt). However, the methodology we use to succeed may not be the way that life started (on this planet or elsewhere in the universe), because there are likely to be many ways that life could begin. We will probably use water as its basic ingredient, and we already know the other elements and molecules from which Earthly life is constructed. However, it is not a matter of which ingredients to use, the problem is how to assemble these to form a self-contained processing system.

A web search for articles relating to the creation of life or the discovery of extraterrestrial life forms will yield many other interesting details. ../Life And Exploiting/ adds a further note to this discussion of life’s beginning.

 * Return to Postscript 1. Gödel’s Theorem, General Systems Theory, and The Conservation Laws


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Developing A Universal Religion/Multi-year Targets

Anything and everything is possible when fantasizing about the future. Everyone has their own views, and the great majority, we know, will turn out to be wrong. For what little they might be worth, here are some conjectures concerning the possible development of a universal religion.

25-year achievements: A suitable meta-purpose is envisioned, and its legal definition as a universal purpose is agreed upon. Public awareness programs are developed and initiated. Political, financial, and religious support is being sought. National and international groups are being formed. Media and public relations units are active. Possible administration, legal, accountability, and other organizational structures necessary for the continuance of the universal purpose project are being discussed.

50-year achievements: The universal purpose is used by 20% of the world’s governments to guide their decision making in such areas as law, genetics, population, pollution, world aid, trade, and exploitation-control.

100-year achievements: The universal purpose has been adopted by 25% of the world’s population, and varieties of a universal religion, headed by this purpose, are being used by them to make moral decisions in arenas where their personal religion provides no answers.

250-year achievements: Self-sustaining, off-world colonies have been built on several of the sun’s planets or their moons and are supplying Earth with rare minerals; the continuation of any indigenous life forms found there is protected by the tenets of the (now commonly accepted) universal religion. Colony ships are on their way to several hospitable exoplanets.

500-year achievements: Information exchanges between humans and intelligent alien life are becoming common occurrences. The possible need to reformulate our meta-purpose, universal purpose and universal religion (and its moral codes) is debated in light of what we learn.

1000-year achievements: Interchanges between numerous other-world species are facilitated by the adoption of a universal meta-purpose that guides the manner by which life’s continuing evolution is supported.

Clearly many other targets have to be met, if any of this is to come to pass. But, look back the same number of years into our past: what changes have humans wrought during that short span. Then look ahead. All might become reality—if we take courage and act upon our dreams.


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Wikijunior talk:Ancient Civilizations/Vedic

203.200.10.35, I don't know if you'll see this, but this is what I took from the Vedic Civilization article on Wikipedia:

The main deities of the Vedic pantheon were Indra, Agni (fire), and Soma. Other deities were Varuna, Surya (the Sun), Mitra, Vayu (the wind). Goddesses included Ushas (the dawn), Prithvi (the Earth) and Aditi. Rivers, especially Sarasvati, were also considered goddesses.

I noticed you added Brahma, Vishnu, and the other one whose name I can't spell :) as the most important Gods. I thought they didn't become important until much later, after the Vedic civilization.--Shanel 15:06, 25 August 2005 (UTC)

As far as I know (and I am sure being a part of Vedic Civilization / Hindu) that the followings were not part of Vedic Civilization. They were however a part of Indus Valley Civilization but not the vedic.

The Ashvamedha or horse sacrifice The Purushamedha or human sacrifice. Rituals concerning demonology and magic.

Vedic rituals never included any magic, sacrifice etc.

--Gurudatt 14:44, 17 December 2005 (IST)

Indus Valley Civilization?
This section was added by the anon above; it sounds like the Indus Valley Civilization, not the Vedic. I'll remove it for now until someone can verify it:

"The cities of the Vedic people were divided into two parts, both covered by walls. The inner region of cities contained the administrative block along with the town centre, king's palace, main temple and living quarters of important persons of the city. The outer regions were inhabited by the general public, and contained buildings such as markets. Cities had roads running from north to south and east to west, dividing the city into squares. Their cities also had drainage systems, much like the cities of today."--Shanel 05:45, 28 August 2005 (UTC)

Sanskrit originals
I can't read the original sanskrit in my browser (konqueror), does anyone know how to make them work?


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A-level Mathematics/Advanced/Basic Mechanics/Space and Time

I will ignore special relativity for the time being and consider Newtonian space and time.

Space
For our purposes Space is the background in which mechanics takes place. Before we can say more than this we require a better understanding on the concepts of co-ordinates.

Co-ordinates
Co-ordinates are simply a collection of labels. As an example consider telephone numbers in an office building. I can label all of the rooms in the building (assuming they all have phones) by the phone number of the room. This would then be a set of co-ordinates. I could also label the rooms by a room number, this is another set of co-ordinates. Co-ordinates are useful in that they let me specify something by a label (like room number) but they are in no way fundamental, at least from the perspective of modern mechanics. Notice that with the case of rooms in an office there are many ways one could form co-ordinates of the office. If one had a directory of room numbers and phone numbers then one could convert from one set of co-ordinates to another if one should so choose. This is useful because in mechanics sometimes it is better to operate in one set of co-ordinates rather than another. In the case of the office if you are looking for Dave in Room 112 then one wouldn't use the telephone number of 112 to go looking for it. Rooms generally have room numbers on their doors, not their phone numbers. But if you were going to phone Dave you would find his room number rather useless unless you converted to the phone number (via the phone directory). In mechanics we specify position via co-ordinates. That is we assign to each location of interest a label, almost always a number. NEED SOME EXAMPLES HERE. Just as when looking for someone in a building, in mechanics one must pick sensible co-ordinates to get something done. When is one co-ordinate system better than another? This is something you have to learn or intuit via problems, which is why just looking at a text book is a waste of time in mathematics, you will need experience if you are to solve problems in mechanics. Experience will usually allow you to pick, based on the 'I've seen one like this before' principle. We could have more than one number to specify a location via co-ordinates. For example consider a building with many floors. The rooms might be numbered 1, 2, 3 etc. on each floor. If I say to you to go to room 43, you wont be able to unless I tell you what floor it is on. If I say go to room 43 on floor 4 you will be able to find the room in question. We now have two co-ordinates, the room number and the floor number. This is the way most people first meet co-ordinates - on a graph, where the two values indicate the location.

Dimensions
Dimensionality is difficult to pin down. One simple way to think of dimensionality is to define it as the number of numbers needed to give a label to every point in some small region in a space. In our previous example we could if we wanted just use phone numbers to classify all the rooms, so the space could be thought of as one dimensional (the space in this case is the list of rooms, not the spatial distributions of the rooms, which is something different). Ordinary space is three dimensional, I can classify every position in space using three numbers. An example of a set of three numbers I could use would be my distance from three (distinct, non-colinear) points. I could also classify my position using cartesian co-ordinates, or by my distance from some point and the angle the line from said point to my position to two fidicule directions. In this sense ordinary space is 3 dimensional. Sometimes we examine problems in less dimensions because additional dimensions are superfluous or we can generalise from special cases. This is another clever trick one must learn to become truly proficient at mechanics, know when you can reduce the number of dimensions in a problem, you will correspondingly reduce the complexity of the problem and make a solution much easier to find.

Time
If one wanted, an entire book on what time is could be written. But we want a simple definition. In short time is defined to make motion look simple, an in a desperate effort to correspond to psychological time. Another way to think about time is to define it empirically. Time is something that is measured by clocks. For the purposes of mechanics we define time to be a parameter which always increases, and which a set of devices known as clocks will agree upon at some given configuration of our system (an instant). Put another more simplistic way, time is something that all good clocks will agree upon at any one instant.


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User:Meatcleaver

Real Name Chris Leaver

Age 27 been married 2 years :-)

Interests Cinema, Cycling, Linux, Photography

Musical Taste See FishMonkey@LastFM

title: User:Meatcleaver


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User talk:SV Resolution/sandbox

Wait. Isn't this the talk page for my sandbox?
 * Indeed it is! 217.205.244.157 20:19, 5 November 2005 (UTC)

title: User talk:SV Resolution/sandbox


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Talk:XML - Managing Data Exchange/Introduction/Answers

 Gustav Sielmann 2028 Development gsielmann@Dot.com 6045 Arnold Rummer 2037 Development</Department> <Email>arummer@Dot.com</Email> <Tel.>6085</Tel.> <Employees Name>Johann Neumeier</Employees Name> <ID>5042</ID> <Department>Marketing</Department> <Email>jneumeier@Dot.com</Email> <Tel.>6093</Tel.> <Employees Name>Max Muller</Employees Name> <ID>5048</ID> <Department>Marketing</Department> <Email>mmueller@Dot.com</Email> <Tel.></Tel.> <Employees Name>Karl Thaler</Employees Name> <ID>7123</ID> <Department>IT</Department> <Email>kthaler@Dot.com</Email> <Tel.>6152</Tel.>


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Cellular Automata/Index

0–9

 * 1D
 * 2D
 * 3D
 * 4D

A

 * age
 * attributes

B

 * birthday
 * boundary
 * cyclic boundary (cyclic lattice)

C

 * cellular automaton
 * configuration

E

 * evolution

L

 * lattice

N

 * neighborhood

P

 * pattern
 * preimage
 * preimage diagram
 * preimage matrix
 * preimage network
 * preimage vector

S

 * sequence

title: Cellular Automata/Index


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Diagnostic Radiology/Chest Imaging

Chest imaging is a medical imaging technique that focuses on visualizing the structures within the chest, including the heart, lungs, blood vessels, and surrounding tissues. It is essential for diagnosing and monitoring a wide range of chest-related conditions and diseases. Several imaging modalities are commonly used in chest imaging:

 * Chest X-ray (CXR):
 * Description: A chest X-ray is one of the most common imaging studies used to examine the chest. It provides a two-dimensional image of the chest, including the heart, lungs, ribs, and major blood vessels.
 * Applications: Chest X-rays are used to detect and assess various conditions, such as pneumonia, lung cancer, rib fractures, and heart abnormalities. They are also used for preoperative assessments and monitoring medical conditions.

 * Computed Tomography (CT) of the Chest:
 * Description: Chest CT involves the use of X-rays to create detailed cross-sectional images of the chest. It can provide high-resolution images with multiple viewing angles.
 * Applications: CT scans of the chest are used to diagnose and evaluate a wide range of conditions, including lung cancer, pulmonary embolism, infections, trauma, and chest wall abnormalities. They are also valuable for assessing complex lung and mediastinal lesions.

 * Magnetic Resonance Imaging (MRI) of the Chest:
 * Description: Chest MRI uses powerful magnets and radio waves to create detailed images of the chest's soft tissues, including the heart and blood vessels.
 * Applications: MRI is particularly useful for assessing the heart and vascular structures. It is used to evaluate congenital heart diseases, aortic diseases, cardiac tumors, and soft tissue chest abnormalities.

 * Positron Emission Tomography-Computed Tomography (PET-CT):
 * Description: PET-CT combines functional and anatomical imaging. It involves injecting a small amount of radioactive material (tracer) into the body, which accumulates in areas with increased metabolic activity.
 * Applications: PET-CT is commonly used for staging and monitoring cancer, particularly lung cancer and lymphomas. It can identify areas of abnormal metabolic activity, helping determine the extent of disease.

 * Ultrasound of the Chest:
 * Description: Chest ultrasound uses high-frequency sound waves to create images of the chest's internal structures, such as the heart and pleura (lining of the chest cavity).
 * Applications: It is often used in pediatric medicine to assess congenital heart abnormalities and in critical care settings to evaluate pleural effusions and pneumothorax.

 * Fluoroscopy:
 * Description: Fluoroscopy is a real-time imaging technique that uses X-rays to visualize dynamic processes in the chest, such as swallowing or the movement of the diaphragm during breathing.
 * Applications: It is used for various diagnostic and interventional procedures, including barium swallow studies, assessment of swallowing disorders, and guided interventions like bronchoscopy.

Chest imaging is essential for diagnosing and monitoring conditions such as lung diseases, heart diseases, trauma, infections, tumors, and vascular disorders. Radiologists, pulmonologists, cardiologists, and other healthcare professionals rely on these imaging techniques to provide accurate diagnoses and guide treatment decisions in chest-related medical cases.

Chest Radiology

 * 1) Signs in Chest Radiology
 * 2) Specific Indications of Chest Radiology
 * 3) Interstitial Lung Disease
 * 4) Alveolar Lung Disease
 * 5) Atelectasis, Airways and Obstructive Lung Disease
 * 6) Mediastinal Masses and Mediastinal/Hilar Lymph Node Enlargement
 * 7) Solitary and Multiple Pulmonary Nodules
 * 8) Benign and Malignant Neoplasms of the Lung
 * 9) Chest Trauma
 * 10) Chest Wall, Pleura and Diaphragm
 * 11) Infection (Immunocompetent, Immunocompromised and Post-transplant Patients
 * 12) Unilaterla Hyperlucent Lung (or hemthorax)
 * 13) Congenital Lung Disease
 * 14) Pulmonary Vascular
 * 15) Thoracic Aorta and Great Vessels
 * 16) Ischemic Heart Disease
 * 17) Myocardial Disease
 * 18) Cardiac Valvular Disease
 * 19) Pericardial Disease
 * 20) Congenital Heart Disease in the Adult
 * 21) Monitoring and support devices -- "tubes and lines"
 * 22) Postoperative Chest

title: Diagnostic Radiology/Chest Imaging


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Diagnostic Radiology/Chest Anatomy

Chest anatomy refers to the structures and components located in the thoracic region of the body, commonly known as the chest. The chest houses vital organs and structures, including the heart, lungs, major blood vessels, and the thoracic cage (ribcage). Here is an overview of the key components of chest anatomy:

 * Thoracic Cage (Ribcage):
 * Ribs: The ribcage consists of 12 pairs of ribs, each attached to the thoracic vertebrae in the spine. Ribs provide protection to the organs within the chest, particularly the heart and lungs.
 * Sternum: The sternum, or breastbone, is a flat, elongated bone located in the center of the chest. It consists of three parts: the manubrium, body, and xiphoid process. The sternum serves as an attachment point for several ribs and supports the chest's structure.

 * Lungs:
 * Lungs: The lungs are a pair of respiratory organs situated on either side of the chest. They are responsible for the exchange of oxygen and carbon dioxide during breathing. The right lung has three lobes, while the left lung has two lobes, allowing space for the heart.
 * Bronchi and Bronchioles: The trachea (windpipe) divides into the right and left main bronchi, which further branch into smaller bronchi and bronchioles within the lungs. These airways transport air to and from the alveoli (air sacs) for gas exchange.

 * Heart and Great Vessels:
 * Heart: The heart is a muscular organ located in the chest, slightly to the left of the midline. It pumps blood throughout the body, supplying oxygen and nutrients to tissues. The heart consists of four chambers: two atria (left and right) and two ventricles (left and right).
 * Aorta: The aorta is the body's largest artery and originates from the left ventricle of the heart. It carries oxygenated blood from the heart to the systemic circulation, delivering it to the body's tissues.
 * Superior and Inferior Vena Cava: These large veins return deoxygenated blood from the upper (superior) and lower (inferior) parts of the body to the right atrium of the heart.

 * Mediastinum:
 * Mediastinum: The mediastinum is the central compartment of the chest that houses the heart, major blood vessels, esophagus, and other structures. It extends from the sternum to the vertebral column.

 * Esophagus:
 * Esophagus: The esophagus is a muscular tube that carries food and liquids from the mouth to the stomach through a process called peristalsis. It passes through the chest as it travels to the abdominal cavity.

 * Diaphragm:
 * Diaphragm: The diaphragm is a dome-shaped muscle that separates the chest (thoracic) cavity from the abdominal cavity. It plays a crucial role in the breathing process, contracting and relaxing to change the volume of the thoracic cavity and facilitate breathing.

 * Major Blood Vessels:
 * Pulmonary Arteries and Veins: Pulmonary arteries carry deoxygenated blood from the right side of the heart to the lungs for oxygenation, while pulmonary veins return oxygenated blood from the lungs to the left side of the heart.
 * Coronary Arteries: Coronary arteries supply the heart muscle with oxygenated blood, ensuring its proper function.

Understanding chest anatomy is essential for healthcare professionals, especially those in cardiology, pulmonology, thoracic surgery, and emergency medicine. It enables the diagnosis and treatment of conditions related to the heart, lungs, and other thoracic organs.

Normal Chest Anatomy
Name and define the three zones of the airways

Define a secondary pulmonary lobule

Define an acinus

List the lobar and segmental bronchi of both lungs

 * Identify the following structures on the posteroanterior (PA) chest radiograph:
 * Lungs -- right, left, right upper, middle and lower lobes, left upper and lower lobes, lingula
 * Fissures -- minor, superior accessory, inferior accessory, azygous
 * Airway - trachea, carina, main bronchi
 * Heart - right atrium, left atrial appendage, left ventricle, location of the four cardiac valves
 * Pulmonary arteries -- main, right, left, interlobar
 * Aorta -- ascending, arch, descending
 * Veins -- superior vena cava, azygous, left superior intercostal ("aortic nipple")
 * Bones -- spine, ribs, clavicles, scapulae, humerus
 * Right paratracheal stripe
 * Junction lines -- anterior, posterior
 * Aortopulmonary window
 * Azygoesophageal recess
 * Paraspinal lines
 * Left subclavian artery

Identify the following structures on the lateral chest radiograph:
 * Lungs -- right, left, right upper, middle and lower lobes, left upper and lower lobes, lingula
 * Fissures n-- major, minor, superior accessory
 * Airway -- trachea, upper lobe bronchi, posterior wall of bronchus intermedius
 * Heart -- right ventricle, right ventricular outflow stripe, left atrium, left ventricle, the location of the four cardiac valves
 * Pulmonary arteries -- right, left
 * Aorta --ascending, arch, descending
 * Veins -- SVC, IVC, left brachiocephalic (innominate), pulmonary vein confluence
 * Bones -- spine, ribs, scapulae, humerus
 * Retrosternal line
 * Posterior tracheal stripe
 * Right and left hemidiaphragms
 * Raiderís triangle
 * Brachiocephalic (innominate) artery

title: Diagnostic Radiology/Chest Anatomy