text:
Prealgebra for Two-Year Colleges/Appendix (procedures)/Multiplying whole numbers with carrying

Multiplication
Multiplication is denoted by an asterisk (*), $$\times$$, or $$\cdot$$ sign. However, the X sign is normally not used in algebra, and is instead limited to very basic elementary math, as it can easily be confused with an "x" variable. The generic multiplication operator will take any two numbers, called factors, as operands. The result is called the product of the two numbers. If the multiplicants are not both written as numbers, the multiplication sign can be left out. Thus, the following example expressions are equivalent:

$$3*\!(a*b) = 3 \times \!(a \times b) = 3\! \cdot \!(a \!\cdot \! b) = 3 \!\cdot \!(ab) = 3(a \!\cdot \! b) = 3(ab) = 3ab$$

Multiplication is a form of repeated addition. For example $$ 3 \times 5 $$ means

$$ 3 + 3 + 3 + 3 + 3 \quad \operatorname{or} \quad 5 + 5 + 5 $$

Multiplication is also commutative. This means that the multiplication of two numbers (factors) will give the same product regardless of the order in which the numbers are multiplied together.

Numbers with exponents that are whole numbers larger than 1 indicate the number of factors to be multiplied, thus that number is multiplied by itself as many times as the exponent shows. Numbers with an exponent of 1 have only one factor, and therefore are equal to the number. Any number with an exponent of 0 has no factors at all, and the result is 1. Examples:

$$5^3 = 5 \times 5 \times 5 \qquad\qquad 5^1 = 5 \qquad\qquad 5^0 = 1$$

Long Multiplication
Long Multiplication is the multiplication of numbers more than 12, but usually only the facts from 1 through 9 are used. Before you attempt long multiplication, please make sure you know the facts 1 through 9. The others are optional, but makes long division a bit easier. The steps for the vertical multiplication method are:

1. Write the numbers down.

52  19  -- 2. Multiply 9 times 2. If there is a tens place for the answer, regroup. Multiply 9 times 5, add the regroup, and write the numbers down. You should have:

1  52   19 --  468

3. Multiply 1 times 2, and 1 times 5, regrouping if needed, but this time shift the answer one space to the left. If you want to, you can put a zero under the 8 instead. You should now have:

52  19 --  468  520 4. Now add.

52 19 - 468 520 - 988

If you are multiplying decimals, then multiply without the decimal point. Count the number of decimal places in both numbers, and add the number of decimal places. In the answer, count that number of spaces to the left. Put the decimal point there.

5. In a summary:

Write the problem down, vertically. Multiply the last digit of the second number to the last digit of the first number. Regroup if there is a tens place in the answer. Multiply it by the second to last number, and ADD the regroup. Repeat the process for the second to last digit of the second number, but put a 0 at the end of the line under the number you got before, if a third line put 2 0s, and repeat until the problem is done.

Fast Multiplication
Fast Multiplication is the method in which you can reasonably multiply numbers more than 10 and reasonably less than 1000 by simply multiplying by the method of "10's." This is done by recognizing how many digits there are in the numbers. Here are some steps which are useful for multiplying numbers really fast

1. See if you can recognize any zeros on the end and simply "add them" to your answer.

45,300 x 5 The easy way to do this is "taking away" the 2 zeros for now and reserving them for later. the number is now 453 x 5, which is much too mind boggling to do. Now here comes the interesting part of the method of "10's"

2. Break down the number into its "10's" parts What this means is basically breakup the number by its place value.

453 = 4 (hundreds place) + 5 (tens place) + 3 (ones place)

Knowing that, this become 400, 50 and 3.

3. Multiply and apply the "10's part" okay now simply multiply:

So, here is a step where we essentially take out the "0's" out for a bit and put it back in when were done.

so, its now 5 x 400. in order to make it easier, "take out" the zeroes for now and multiply 4 x 5 = 20. Now heres the magic. Since you magically took away the 2 zeroes, you will now suddenly make the 2 zeroes reappear! 20 + "00" = 2000! AMAZING! (the quotes means they're magic zeroes, and simply not the value   of zero!)

50 is done the same away. Take away the "0" and multiply 5 x 5 = 25. Now add it back, 25 + "0" = 250

Simply 3 x 5 = 15

4. Now abracadabra, Add them all together!

2000    250     15   -    2265!

5. Now take the two zeroes you reserved in the beginning (from the original 45,300), and tack them onto the end to get your answer: 226,500.

Step 2: Multiplying numbers that are not zero friendly

2102 x 52

Using the step from before recognize that:

2102 = 2 (thousand) + 1(hundred) + 0 (tens) + 2 (ones) 52 = 5 (tens) + 2 (ones)

Now, to make it easier on yourself, circle the number 2 of "52" and put it in your magic hat. (2)

Now the problem becomes 2102 x 50. Look familiar? First of all, take out the magic "0" and put it in the hat, too. Since we recognized that 50 is basically 5 with an added magical "0" to it, we now see the problem as

2102 x 5!

Now break down the bigger, uglier number and start multiplying: 2000 x 5 (take away the magic zeroes) = 2 x 5 = 10 + "000"(now put them back!) = 10,000 (notice it has 4 zeroes) 100 x 5 (take away the magic zeroes) = 1 x 5 = 5 + "00" (now put them back!) =   500 (2 zeroes) 2 x 5 (sadly, no magical zeroes)  = 2 x 5 = 10                             =    10 (1 zero)

Remember, after every step, be sure to put your friendly magical "0" back in: 10,000 + "0" =100,000      500 + "0" =  5,000      10 + "0" =    100    (notice how the number of zeroes on the left side equal the number of zeroes on the right    side)

Now add them all together:

100000   5000     100  -  105100....... That's not all yet folks! Do you remember the 2 in your magic hat? Lets get it to work:

2 x 2102 = 2000 x 2 = 2 x 2 = 4 = 4000 100 x 2 = 1 x 2 = 2 = 200 2 x 2            =    4 total: 4204

So the answer should be   105,100 4,204   ---    109,304! Wow!

Division
Division uses the ÷ sign. It may also be signified by the slash, /, :, or the fraction bar. The generic division operator will take any two numbers as operands. The number before the ÷ sign is called the dividend and the number after the ÷ sign is called the divisor. The result is called the quotient of the two numbers.

Division is not a commutative operation. Switching the dividend and the divisor will likely give a different quotient (but sometimes not). The division with a divisor of 0 is not defined. There is no answer for it.

Example:
 * $$4 / 2 = 2$$ and $$2 / 4 = 0.5$$

Long Division
In arithmetic, long division is an algorithm for division of two real numbers. It requires only the means to write the numbers down, and is simple to perform even for large dividends because the algorithm separates a complex division problem into smaller problems. However, the procedure requires various numbers to be divided by the divisor: this is simple with single-digit divisors, but becomes harder with larger ones.

A more generalized version of this method is used for dividing polynomials (sometimes using a shorthand version called synthetic division).

In long division notation, 500 ÷ 4 = 125 is denoted as follows:

\begin{matrix} \quad 125\\ 4\overline{)500}\\ \end{matrix} $$

The method involves several steps:

1. Write the dividend and divisor in this form:

 * $$4\overline{)500}$$

In this example, 500 is the dividend and 4 is the divisor.

2. Consider the leftmost digit of the dividend (5). Find the largest multiple of the divisor that is less than the leftmost digit: in other words, mentally perform "5 divided by 4". If this digit is too small, consider the first two digits.

In this case, the largest multiple of 4 that is less than 5 is 4. Write this number under the leftmost digit of the dividend. Write the multiple divided by the divisor (4 divided by 4 = 1) above the line over the leftmost digit of the dividend.

\begin{matrix} 1\\ 4\overline{)500}\\ 4 \end{matrix} $$

3. Subtract the digit under the dividend from the digit used in the dividend. Write the result (remainder) (5 − 4 = 1) under the bottom digit, then drop the zero (the second digit) to the right of it.

\begin{matrix} 1\\ 4\overline{)500}\\ \underline{4}\\ \;\,10 \end{matrix} $$

4. Repeat steps 2 and 3, except use the number you just created to divide by, and write above and under the second digit.

\begin{matrix} \,\,12\\ 4\overline{)500}\\ \underline{4}\\ \;\,10\\ \quad\underline{8}\\ \quad\;\,20 \end{matrix} $$

5. Repeat step 4 until there are no digits remaining in the dividend. The number written above the bar is the quotient, and the last remainder calculated is the remainder for the entire problem.

\begin{matrix} \quad 125\\ 4\overline{)500}\\ \underline{4}\\ \;\,10\\ \quad\underline{8}\\ \quad\;\,20\\ \quad\;\,\underline{20}\\ \qquad0 \end{matrix} $$


text:
Prealgebra for Two-Year Colleges/Appendix (procedures)/Dividing whole numbers by long division

Multiplication
Multiplication is denoted by an asterisk (*), $$\times$$, or $$\cdot$$ sign. However, the X sign is normally not used in algebra, and is instead limited to very basic elementary math, as it can easily be confused with an "x" variable. The generic multiplication operator will take any two numbers, called factors, as operands. The result is called the product of the two numbers. If the multiplicants are not both written as numbers, the multiplication sign can be left out. Thus, the following example expressions are equivalent:

$$3*\!(a*b) = 3 \times \!(a \times b) = 3\! \cdot \!(a \!\cdot \! b) = 3 \!\cdot \!(ab) = 3(a \!\cdot \! b) = 3(ab) = 3ab$$

Multiplication is a form of repeated addition. For example $$ 3 \times 5 $$ means

$$ 3 + 3 + 3 + 3 + 3 \quad \operatorname{or} \quad 5 + 5 + 5 $$

Multiplication is also commutative. This means that the multiplication of two numbers (factors) will give the same product regardless of the order in which the numbers are multiplied together.

Numbers with exponents that are whole numbers larger than 1 indicate the number of factors to be multiplied, thus that number is multiplied by itself as many times as the exponent shows. Numbers with an exponent of 1 have only one factor, and therefore are equal to the number. Any number with an exponent of 0 has no factors at all, and the result is 1. Examples:

$$5^3 = 5 \times 5 \times 5 \qquad\qquad 5^1 = 5 \qquad\qquad 5^0 = 1$$

Long Multiplication
Long Multiplication is the multiplication of numbers more than 12, but usually only the facts from 1 through 9 are used. Before you attempt long multiplication, please make sure you know the facts 1 through 9. The others are optional, but makes long division a bit easier. The steps for the vertical multiplication method are:

1. Write the numbers down.

52  19  -- 2. Multiply 9 times 2. If there is a tens place for the answer, regroup. Multiply 9 times 5, add the regroup, and write the numbers down. You should have:

1  52   19 --  468

3. Multiply 1 times 2, and 1 times 5, regrouping if needed, but this time shift the answer one space to the left. If you want to, you can put a zero under the 8 instead. You should now have:

52  19 --  468  520 4. Now add.

52 19 - 468 520 - 988

If you are multiplying decimals, then multiply without the decimal point. Count the number of decimal places in both numbers, and add the number of decimal places. In the answer, count that number of spaces to the left. Put the decimal point there.

5. In a summary:

Write the problem down, vertically. Multiply the last digit of the second number to the last digit of the first number. Regroup if there is a tens place in the answer. Multiply it by the second to last number, and ADD the regroup. Repeat the process for the second to last digit of the second number, but put a 0 at the end of the line under the number you got before, if a third line put 2 0s, and repeat until the problem is done.

Fast Multiplication
Fast Multiplication is the method in which you can reasonably multiply numbers more than 10 and reasonably less than 1000 by simply multiplying by the method of "10's." This is done by recognizing how many digits there are in the numbers. Here are some steps which are useful for multiplying numbers really fast

1. See if you can recognize any zeros on the end and simply "add them" to your answer.

45,300 x 5 The easy way to do this is "taking away" the 2 zeros for now and reserving them for later. the number is now 453 x 5, which is much too mind boggling to do. Now here comes the interesting part of the method of "10's"

2. Break down the number into its "10's" parts What this means is basically breakup the number by its place value.

453 = 4 (hundreds place) + 5 (tens place) + 3 (ones place)

Knowing that, this become 400, 50 and 3.

3. Multiply and apply the "10's part" okay now simply multiply:

So, here is a step where we essentially take out the "0's" out for a bit and put it back in when were done.

so, its now 5 x 400. in order to make it easier, "take out" the zeroes for now and multiply 4 x 5 = 20. Now heres the magic. Since you magically took away the 2 zeroes, you will now suddenly make the 2 zeroes reappear! 20 + "00" = 2000! AMAZING! (the quotes means they're magic zeroes, and simply not the value   of zero!)

50 is done the same away. Take away the "0" and multiply 5 x 5 = 25. Now add it back, 25 + "0" = 250

Simply 3 x 5 = 15

4. Now abracadabra, Add them all together!

2000    250     15   -    2265!

5. Now take the two zeroes you reserved in the beginning (from the original 45,300), and tack them onto the end to get your answer: 226,500.

Step 2: Multiplying numbers that are not zero friendly

2102 x 52

Using the step from before recognize that:

2102 = 2 (thousand) + 1(hundred) + 0 (tens) + 2 (ones) 52 = 5 (tens) + 2 (ones)

Now, to make it easier on yourself, circle the number 2 of "52" and put it in your magic hat. (2)

Now the problem becomes 2102 x 50. Look familiar? First of all, take out the magic "0" and put it in the hat, too. Since we recognized that 50 is basically 5 with an added magical "0" to it, we now see the problem as

2102 x 5!

Now break down the bigger, uglier number and start multiplying: 2000 x 5 (take away the magic zeroes) = 2 x 5 = 10 + "000"(now put them back!) = 10,000 (notice it has 4 zeroes) 100 x 5 (take away the magic zeroes) = 1 x 5 = 5 + "00" (now put them back!) =   500 (2 zeroes) 2 x 5 (sadly, no magical zeroes)  = 2 x 5 = 10                             =    10 (1 zero)

Remember, after every step, be sure to put your friendly magical "0" back in: 10,000 + "0" =100,000      500 + "0" =  5,000      10 + "0" =    100    (notice how the number of zeroes on the left side equal the number of zeroes on the right    side)

Now add them all together:

100000   5000     100  -  105100....... That's not all yet folks! Do you remember the 2 in your magic hat? Lets get it to work:

2 x 2102 = 2000 x 2 = 2 x 2 = 4 = 4000 100 x 2 = 1 x 2 = 2 = 200 2 x 2            =    4 total: 4204

So the answer should be   105,100 4,204   ---    109,304! Wow!

Division
Division uses the ÷ sign. It may also be signified by the slash, /, :, or the fraction bar. The generic division operator will take any two numbers as operands. The number before the ÷ sign is called the dividend and the number after the ÷ sign is called the divisor. The result is called the quotient of the two numbers.

Division is not a commutative operation. Switching the dividend and the divisor will likely give a different quotient (but sometimes not). The division with a divisor of 0 is not defined. There is no answer for it.

Example:
 * $$4 / 2 = 2$$ and $$2 / 4 = 0.5$$

Long Division
In arithmetic, long division is an algorithm for division of two real numbers. It requires only the means to write the numbers down, and is simple to perform even for large dividends because the algorithm separates a complex division problem into smaller problems. However, the procedure requires various numbers to be divided by the divisor: this is simple with single-digit divisors, but becomes harder with larger ones.

A more generalized version of this method is used for dividing polynomials (sometimes using a shorthand version called synthetic division).

In long division notation, 500 ÷ 4 = 125 is denoted as follows:

\begin{matrix} \quad 125\\ 4\overline{)500}\\ \end{matrix} $$

The method involves several steps:

1. Write the dividend and divisor in this form:

 * $$4\overline{)500}$$

In this example, 500 is the dividend and 4 is the divisor.

2. Consider the leftmost digit of the dividend (5). Find the largest multiple of the divisor that is less than the leftmost digit: in other words, mentally perform "5 divided by 4". If this digit is too small, consider the first two digits.

In this case, the largest multiple of 4 that is less than 5 is 4. Write this number under the leftmost digit of the dividend. Write the multiple divided by the divisor (4 divided by 4 = 1) above the line over the leftmost digit of the dividend.

\begin{matrix} 1\\ 4\overline{)500}\\ 4 \end{matrix} $$

3. Subtract the digit under the dividend from the digit used in the dividend. Write the result (remainder) (5 − 4 = 1) under the bottom digit, then drop the zero (the second digit) to the right of it.

\begin{matrix} 1\\ 4\overline{)500}\\ \underline{4}\\ \;\,10 \end{matrix} $$

4. Repeat steps 2 and 3, except use the number you just created to divide by, and write above and under the second digit.

\begin{matrix} \,\,12\\ 4\overline{)500}\\ \underline{4}\\ \;\,10\\ \quad\underline{8}\\ \quad\;\,20 \end{matrix} $$

5. Repeat step 4 until there are no digits remaining in the dividend. The number written above the bar is the quotient, and the last remainder calculated is the remainder for the entire problem.

\begin{matrix} \quad 125\\ 4\overline{)500}\\ \underline{4}\\ \;\,10\\ \quad\underline{8}\\ \quad\;\,20\\ \quad\;\,\underline{20}\\ \qquad0 \end{matrix} $$

Mathematics for grade 7 and English grade 7
Long division

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/


text: User talk:76.176.94.120

title: User talk:76.176.94.120


text:
Prealgebra for Two-Year Colleges/Appendix (procedures)/Order of operations

Order of operations
The order of operations is the order in which all algebraic expressions should be simplified. Oftentimes, the meaning of a complex expression changes depending upon the order in which it is calculated. The order of operations is:

 * Parentheses
 * Exponents (and Roots)
 * Multiplication & Division
 * Addition & Subtraction

This means that expressions within parentheses are evaluated first, then exponents (including roots, i.e. radicals), then multiplication and division (at the same level), and finally addition and subtraction (at the same level). If there are multiple operations at the same level on the order of operations, move from left to right.

There are a number of different abbreviations for memorizing the order. PEMDAS, BEDMAS and BODMAS (B is Brackets) are common. Another common way to remember the order is the mnemonic

 * "Please Excuse My Dear Aunt Sally,"

with the beginning letters standing for each operation. Whichever mnemonic you use, be aware that multiplication does not always come before division, and addition does not always come before subtraction. For example:

If you have an expression like

 * 3 × 3 - 5 + 2

you work like this: First notice that, there are no Parentheses or Exponents, so we move to Multiplication and Division. There's only the one multiplication, so we do that first and end up with 9 - 5 + 2. Now we move to Addition and Subtraction, working left to right. So firstly we do the subtraction to get 4 + 2, and finally the addition to give 6. If we had blindly done the addition first, we would have got the answer 2, which is wrong!

The rationale for the grouping (apart from parentheses, which are obviously first) is that multiplication is repeated addition and exponentiation is repeated multiplication. Also, division is multiplication by the reciprocal and subtraction is addition of the negative, so these operations are equivalent. In fact PEMA would be a better phrase ("Please Excuse My Aunt"), but in lower arithmetic courses MDAS is often taught without explaining reciprocals.

Parentheses are curved symbols, ( and ), that are put around part of an expression in order to convey that the expressions inside them should be evaluated first. Within a set of parentheses, the order of operations should be followed. Square brackets, [ and ], are sometimes used around parentheses to avoid confusion: [(3+5)×2]2 means the same as ((3+5)×2)2. The fraction bar and radical bar (often called a vinculum) groups expressions like parentheses.

For example, the expression 2×(6+7)-82 should be solved in the following order:

If the desired order for solving the expression were different (based on the initial problem), parentheses would be positioned differently, or even omitted.

The meaning of the fraction and radical bars must be deciphered carefully. The part of the expression directly below or above the bar is to be treated as parenthesised. (Care must be taken in writing expressions with a bar.)

The expression $$\sqrt{a + b} \times c$$ means c times the root of a + b, not the root of a + b × c or even the root of c times the sum a + b, since the bar is above the a and the b, but not the c.

The expression $$\frac{4 + 5}{1 + 2}$$ could be written in one line as (4+5)/(1+2) = 9/3 = 3, not as 4+5/1+2 = 4+5+2 = 11. As you see, the expression above the bar is evaluated, as is the expression below the bar. Finally we can divide.

Because of order of operations -22 = -(22) = -4, not (-2)2 = +4: the negative sign can be considered to have an implicit 0 in front, making the expression 0 - 22.

When it comes to distributing a power, use the raise a power to a power rule. Example: (xy^2)to the fourth power (^4) =(x)^4 (y^2)^4 =x^4 &times y^8 (originally it was y^6; this would only be true for y^2 * y^4, where you add the exponents)

The order of operations is very important, and you must remember the order when using simple calculators. Expressions such as 2+3×5 vary on the order used. Entering [2] [+] [3] [×] [5] on most calculators would result in adding three to two and then multiplying by five, resulting in 25; the proper evaluation sequence would be 2+(3×5), multiplying three and five and then adding that to two to get 17. Some scientific calculators and most graphing calculators use the proper order of operations, but four-function calculators typically use "left-to-right" evaluation, which can return incorrect results.


text:
Prealgebra for Two-Year Colleges/Appendix (procedures)/Rounding

Rounding
To round a number, you need to look at the numbers right of the decimal. If the first number right of the decimal is bigger than or equal to 5 we add 1 to the number left of the decimal this is known as rounding up. If the first number right of the decimal place is less than 5 we simply rewrite the number again excluding all the numbers right of the decimal place; this is known as rounding down. Note that after the number is rounded up or down all of the numbers right of the decimal are discarded and only the numbers left of the decimal are written.

Examples
1.49999999 rounds to 1 1.50000000 rounds to 2 1234.56 rounds to 1235


text:
Prealgebra for Two-Year Colleges/Appendix (procedures)/Lowest common multiple

To find the Lowest Common Multiple (LCM) of several numbers, we first express each number as a product of its prime factors.

For example, if we wish to find the LCM of 60, 12 and 102 we write

$$ \begin{matrix} 60=2^2 \cdot 3 \cdot 5 \\ 12=2^2 \cdot 3 \\ 102=2 \cdot 3 \cdot 17 \end{matrix} $$

The product of the highest power of each different factor appearing is the LCM.

For example in this case, $$2^2\cdot3\cdot5\cdot17=1020$$. You can see that 1020 is a multiple of 12, 60 and 102 ... the lowest common multiple of all three numbers.

Another example: What is the LCM of 36, 45, and 27?

Solution: Factorise each of the numbers

$$ \begin{matrix} 36=2^2\cdot3^2\\ 45=5\cdot3^2 \\ 27=3^3 \end{matrix} $$

The product of the highest power of each different factor appearing is the LCM, i.e;

$$2^2\cdot5\cdot3^3=540$$

Properties of the LCM
If the LCM of the numbers is found and 1 is subtracted from the LCM then the remainder when divided by each of the numbers whose LCM is found would have a remainder that is 1 less than the divisor. For example if the LCM of 2 numbers 10 and 9 is 90. Then 90-1=89 and 89 divided by 10 leaves a remainder of 9 and the same number divided by 9 leaves a remainder of 8.


text:
Prealgebra for Two-Year Colleges/Appendix (procedures)/Greatest common factor

The greatest common factor (GCF) can be found in a similar fashion. In this case, the product of the prime factors common to all the numbers gives the HCF.

For example, if we wish to find the HCF of 60, 12 and 102 we write

$$ \begin{matrix} 60=2^2 \cdot 3 \cdot 5 \\ 12=2^2 \cdot 3 \\ 102=2 \cdot 3 \cdot 17 \end{matrix} $$

Now the GCF is $$2 \cdot 3=6$$.

Two numbers which have an HCF of 1, for example 12 and 5, are said to be coprime - they have no factors in common (except 1).


text:
Prealgebra for Two-Year Colleges/Appendix (procedures)/Absolute value

Absolute value
The absolute value of a number is found by applying a simple rule: If you see a negative sign in front of the number, change it to a plus sign. If you see a plus sign, leave it alone. So, for example, the absolute value of -17 is +17. The absolute value of +36 is +36.

Another way to understand the absolute value of a number is to think about the number line:

The absolute value of a number is the distance from zero to that number on the number line.

The absolute value of x is usually written as |x|. On calculators and computers it is sometimes written as abs (x).

Questions:

1. Calculate the absolute value of the following numbers:
 * a. -5
 * b. 9
 * c. -3.8
 * d. -139,462
 * e. 5/8

2. What is the absolute value of zero? Explain.

3. Calculate the following:

 * a. |27|
 * b. |-1.9|
 * c. |3 - 7|
 * d. |3 - 0.5|
 * e. abs (-6)

4. Draw a graph of abs(x) from -5 to +5. Can abs(x) ever be less than zero? How can you see that from your graph?

Answers:

1. a. 5 b. 9 c. 3.8 d. 139,462 e. 5/8

2. Zero, because zero is exactly zero away from zero on the number line.

3. a. 27 b. 1.9 c. |3-7| = |-4| = 4 d. |3-0.5| = |2.5| = 2.5 e. 6

4. [[Media:Absolute value.svg|Image of the absolute value of x]]


text:
Prealgebra for Two-Year Colleges/Appendix (procedures)/Exponents

Exponents
Exponents or 'powers' are a process of repeated multiplication, in much the same way that multiplication is a process of repeated addition.

Exponents are normally written in the form $$a^b$$, where $$a$$ is the base and $$b$$ is the exponent. In contexts where superscripts are not available, such as in many contexts in computers, $$a^b$$ is commonly written as "a^b" or less often as "a**b". If you're not familiar with algebra, you can just imagine the letters a and b as representing numbers. We pronounce $$a^b$$ as a to the power of b, a to the b or a exponent b.

Integer exponents
When the exponent is a positive integer, then it is just a simple case of multiplying the base by itself a certain number of times. For example,

$$3^4 = 3*3*3*3 = 81\,$$

Here, 3 is the base, 4 is the exponent (written as a superscript), and 81 is 3 raised to the 4th power. Notice that the base 3 appears 4 times in the repeated multiplication, because the exponent is 4.

Some more examples:

$$ \begin{matrix} 12^2=12\times12=144\\ 2^8=2\times2\times2\times2\times2\times2\times2\times2=256\\ 1^5=1\times1\times1\times1\times1=1 \end{matrix} $$

Multiplying exponents
If you have two or more exponents with the same base, then multiplying them has the same effect as adding their exponents.

For instance $$(a^b)*(a^c)\,$$ is the same as $$a^{b+c}\,$$. For example,

$$(3^4)*(3^2)=(3*3*3*3)*(3*3)=(3*3*3*3*3*3)=(3^6)=(3^{4+2})\,$$

Dividing exponents
If you have two or more exponents with the same base, then dividing them has the same effect as subtracting their exponents.

For instance $$(a^b)/(a^c)\,$$ is the same as $$a^{b-c}\,$$. For example,

$$(3^6)/(3^2)=(3*3*3*3*3*3)/(3*3)=(3*3*3*3)=(3^4)=(3^{6-2})\,$$

Exercises

 * 1) What are $$4^3\,$$, $$3^4\,$$, $$1^{250}\,$$, $${250}^1\,$$
 * 2) Write these numbers as powers of 2: $$128, 8, 1024\,$$
 * 3) What is $$(2^3)*(2^2)\,$$
 * 4) What is $$(2^6)/(2^2)\,$$
 * 5) Harder: Why does $$3^0=1\,$$ (clue: think about $${3^2}/{3^2}\,$$, for example)

Click here for Answers.

Negative exponents
Negative exponents work slightly differently. Let's say you want to calculate $$3^{-2}$$. To do that, you take $$1/3^2$$ to get your answer. We do the exponent first, see Order of Operations

$$3^{-2}=\frac{1}{3^2}=\frac{1}{9}$$

The commutative property doesn't apply in exponents. See for yourself! Try to calculate 23, and then see if it is the same as 32 (answer here). The distributive and associative properties don't apply either.

Exponents do, however, have their own set of axioms that they consistantly follow. Consistent with the preceding examples, one can state generally that:

(a^b)\times(a^c)=a^{b+c} \,\,\,\,$$ and

(a^b)/(a^c)=a^{b-c} \,\,\,\,$$

It's also easy to see that $$ \begin{matrix} (a^b)^c=a^{b\times c} \end{matrix} $$

Fractional exponents
So far, we have only seen exponents as whole numbers, but exponents can be fractional as well. With a fractional exponent, the numerator acts as a normal whole-number exponent, while the denominator acts as a root.

In general, $$a^{p/q}=\sqrt[q]{a^p}\,\,\,\,$$ for any real number $$q$$ ≠ 0.

Let's look at $$8^{2/3}$$ as an example. First, we raise 8 to the power of the numerator, 2. Then, since the denominator is 3, we take the third root of this number. The expression is read as the cubed root of eight squared, and written as:

$$8^{2/3}=\sqrt[3]{8^2}=\sqrt[3]{64}=4$$

It should then be evident that when the numerator of the fractional exponent is 1, the expression is a simple root. That is, $$1/2$$ is a square root, $$1/3$$ is a cubed root, $$1/4$$ is a fourth root, etc.

For example, $$9^{1/2}$$ would be read as the square root of nine, and written as:

$$9^{1/2}=\sqrt[2]{9^1}=\sqrt{9}=3$$

See also: Radicals


text:
User talk:Balloonguy/Japanese

I like your layout. If you wanted to create a definite structure to everything I'd be glad to contribute. Personally I think dialects should be an appendix rather than something taught. Standard should be emphasized (even though I love kansai.)

Kanji is lacking alot, but it's quite the task to add them.

I agree with pretty much all you say but getting people together to do this is very difficult. I believe that if we get a group of people together and make a skeleton outline, we can actually make a decent reference.

One thing I think should be implemented is the use of span texts for mouseovers so you can use kanji and still have understanding, such as:

母 の顔 が変 です.

Basic structure is:

word

example:

word

title: User talk:Balloonguy/Japanese


text:
User talk:Kadamczy

Welcome, Kadamczy!

Come introduce yourself at the new users page. If you have any questions, you can ask there or contact me personally. Be sure to sign your comments with four tildes ( ~ ). Νεοπτόλεμος ( talk | email | contribs ) 02:39, 21 March 2008 (UTC)

Japanese Wikibook discussion
Hi Kadamczy,

I just added a comment on Talk:Japanese which I wanted to draw your attention to as you've been involved in this book. --Swift (talk) 05:22, 10 June 2008 (UTC)

title: User talk:Kadamczy


text:
User talk:Archmage flash

Welcome message
Welcome, Archmage flash!

Come introduce yourself at the new users page. If you have any questions, you can ask there or contact me personally. Webaware talk 04:56, 21 March 2008 (UTC)

title: User talk:Archmage flash


text:
Embedded Control Systems Design/A design example 2

This chapter illustrates the various steps in the design of an embedded system by means of a concrete example: an automated People Mover.

Introduction

In order to understand what is involved in the design of embedded control systems, it is useful to elaborate an example of such a system. The chosen example comes from a commonly known application domain, so that all readers can quickly grasp the complexity and the required features of the design. At the same time, the example is sufficiently realistic to cover all relevant aspects (economical, technical, human resources, etc.) that show up (during the various phases) in the design and the lifecycle of an embedded control system. The example of a (automated) people mover meets these requirements. This Chapter is conceived as the story of how a fictitious company (People Move Company) deals with the decision to design a new people mover; this story approach allows to illustrate the different design criteria that become relevant in different phases of the product's design and life cycle.

A people mover is an unmanned train. The first people mover became operational in 1967, as an attraction in Tomorrowland in Disneyland, California. (See the Wikipedia article for more details about the history and deployment of people movers.) Nowadays, people movers are used as transport systems all around the world.

As the staff of People Move Company knows, designing an embedded control system is quite a job. A lot of people have to work together to attain an operating whole. From the moment on that People Move Company decided to take part in the call for bids for a new people mover project, the company tries to operate in a structured way. It subdivides the design job into three parts.

 * In a first step, the requirements analysis, the chief engineer of the company, together with (a representative of) the CEO, make a product requirements document, in which the company expresses its views on the requirements of the client. In case the Board of the People Move Company decides the company can compete realistically in the bid —and this decision is in itself not an easy issue!— the requirements list will be used to draw up a quotation.
 * Secondly, and if the company is allowed to participate in the bid, it internally enters the high level design phase, during which it completes the list of client and product requirements with requirements the company imposes for the sake of internal interest. Design criteria as component reuse, service after sale, and maintenance get the attention during this phase, since these can make a large financial difference for the company. Moreover, the responsibles of the project make a plan of which of their employees can work on which parts of the job.
 * Then all involved employees can implement the detailed design of the people mover.

After these design tasks, the peoplemover has to be produced, serviced and maintained. The production process will not be discussed in detail here though.

Requirements analysis
People Move Company wants to design a fully automated vehicle for public transport. The responsibles for the project make up a list of technical, commercial and practical requirements the system has to meet in order to comply to the client's call for bids. This result is the product requirements document that is written in plain language. As the document expresses the requirements of the client, People Move Company wants to draw up the product requirements in dialogue with the client. In one such meeting, the client explains that he already is in the possession of rails on which the people mover will ride. People Move Company gets the specifications of these rails, as well as the standards for people transport that apply in the client's country. This brings in some design constraints, for example with respect to maximum velocities, acceleration profiles, weight, energy consumption, etc. In turn, People Move Company profits from the interactions with the client to explain the company's competitive advantages (technically as well as economically) as well as some product constraints that the company will want to comply to, for internal technical and economical reasons. Making these new constraints clear, in both directions, can indeed be of great importance when it comes to a lawsuit between client and company.

The interaction with the client brings People Move Company to its final product requirements document, that will be submitted to the call for bids. Internally, the same document will serve as the first input for the company's own design team. While it cannot be intended to specify all requirements in full technical detail, it described the following list of important requirements that cannot be omitted. Where exactly the border lies between a detail and a high-level requirement, and between important and lower-priority requirements, is very hard to explain or teach in general; especially because it involves so many company-specific aspects: available man power and expertise, existing products whose design can be re-used (partially) for the new product, etc. The economic success of a company will to a large extent depend on how good the design requirements team is in its job to create a realistic and competitive requirements document.

Because of already existing products in its portfolio, the company decides to design a people mover that consists of a locomotive and one or more passenger carriages. The vehicle is driven by an electric motor and receives its power from the rails. The people mover can only run on special railroads that are equipped with beacons. These beacons are placed along the railroad and indicate to the people mover's control system a desired change in speed of the people mover. A map of the railroad together with the meaning of each beacon has to be loaded into the memory of the people mover controller before the vehicle can drive on this railroad. A sensor on the vehicle detects the beacons and reacts as indicated on the map. The possible reactions are:

 * speeding up with a certain amount. Remark: velocity and acceleration are limited to a specified maximum, according to a specified smooth profile, in order to realize a user friendly peoplemover. Such profiles are part of many already existing railroad standards.
 * slowing down with a certain amount. Remark: deceleration is limited to a certain maximum.
 * stopping with a specified deceleration.

As an example, a beacon just before a bend in the tracks indicates that the peoplemover has to slow down to a specified velocity. On the places where a terminal is indicated on the tracks, the vehicle stops. The place where the people mover stands still is specified by the client to differ with a maximum of one meter from the place indicated on the map. The doors open at least 0.1 and at most 0.5 seconds after halting. Thirty seconds after the opening of the doors a warning signal sounds for one second and the doors close. At least 1 and at most 2 seconds after the warning signal the vehicle continues its travel.

When it is getting dark, the headlights of the people mover and the internal illumination in the carriages must switch on on. The lights have a minimal intensity, specified by the client.

The company is not responsible when the railroad is blocked by any obstacle.

Although the people mover is fully automated, a control tower is able to decide on the actions of the people mover. The speed and position of the vehicle (to a certain precision) can be seen on a display in the control tower.

High-level design
Hooray, People Mover Company was among the winners of the call for bids, and now gets some funding from the client to make a more detailed high-level design, in which all the technical details that are relevant to the client will have to be explained. The list of requirements mentioned above meets the overal needs of the client, but is not specific enough for the company's designers to make a complete analysis of the feasibility and the costs of the project. These people also have to take into account the requirements that are in the People Move Company's own interest, regardless of the client's wishes. For example, as People Move Company wants to accept only jobs that fit into its long term product portfolio vision and that it can support for decades, the designers have to think of features that facilitate component reuse and maintenance of the people mover product. These efforts will lower the cost of service after sale, and of later similar products that the company wants to bring to the market. To complete the list of requirements, the company's design team has a list of design criteria that its designers always have to have in mind. For the people mover these design criteria are:
 * Functionality: the people mover's first functionality is its ability to transport people in a fully automated way.
 * Safety: has to be incorporated at all levels of the design. Including failure modes and adding redundancy are common safety precautions.
 * Robustness: the design should have some margin to survive in conditions that are somewhat beyond what is asked by the client.
 * Cost: the price still has to be competitive.
 * Energy consumption: the company wants to improve its image of the least energy consuming vehicles.
 * Insensitivity for varying environmental conditions (humidity, temperature and vibrations)
 * Controllability: the people mover must be controllable from a control room.
 * Space and weight constraints
 * Producibility
 * Certification and standardization
 * User interface
 * Life time
 * Maintenance
 * Service after sale

Now all design criteria have to be translated into requirements that can be added to the ''product requirements. ''The extra requirements thus ensure that all design criteria will be incorporated in the final peoplemover. Some examples illustrate how this can be done.

As already stated above People Move Company can save money by including features that facilitate service after sale. This could be implemented with a "maintenance from distance" system.

Another way to save money and time is reusing infrastructure and components of previous models of the peoplemover. The safety requirement was already partially included in the product requirements by specifying that a control tower has to be able to take over all commands. But this is not enough, as safety has to be incorporated in all design levels. For example, while designing the cruise control function, the designer has to make sure that the current in the motor never exceeds a limit value. This could occur when the vehicle tries to go too fast on a slope.

Some design criteria interfere with each other. User interface, for example, has to do with safety. The system design has to anticipate inattention of the users. It could happen that someone enters the peoplemover while the doors are already closing. A sensor detecting this dangerous act, can prevent accidents. Also the design criteria life time, robustness and cost can interact. A designer has to have an idea of how long the system has to be in use. A long life time asks for robust components of higher quality. Of course this has an influence on the price.

In high level design the requirements are translated in specifications, which means that vague requirements as cost, life time, maintenance, service after sale and energy consumption are expressed more precisely. Expected life time, cost and energy consumption are defined by a certain number.

After fathoming the design criteria People Move Company has a more extended version of the product requirements in hand. With this the system engineers or system architects create a more formal set of functional specifications. They make up the global system architecture. This scheme divides the system into three main parts:

 * a traffic command system
 * a control command system
 * a vehicle command system

Each part has a couple of inputs and outputs, represented by arrows in figure 1. The traffic command system for example gets information from beacons placed along the railroad and from the slopes themselves. These beacons indicate a change in speed of the peoplemover. A map of the railroad together with the meaning of each beacon has to be loaded into the memory of the peoplemover before the vehicle can drive on the railroad. A sensor on the vehicle detects the beacons and reacts as indicated on the map. The slopes result in a certain load torque. The inputs and outputs of the other parts are indicated on figure 1. Figure 2 describes the content of each part in more detail.

Now the top of People Move Company decides who will work on which part of the job. In some cases it is useful to work with subcontractors. The company can also hire an expert with relevant experience. By dividing the system in subsystems, different teams can work concurrently to realize these different subsystems. This is called concurrent engineering. Good inter-team communication, e.g. in daily or weekly meetings, makes it possible to attune to each other. The optimal division in subsystems minimizes subsystem-communication and maximizes subsystem-independency. Afterwards the subsystems are assembled into a fully operational system. Applying the division strategy on the automated peoplemover results in seven subsystems. Figure 3 illustrates the subsystems for the vehicle system of the automated peoplemover. The last two subsystems are omitted in this illustration as they are not really related with the embedded aspect of the design. Therefore they will not be treated in the rest of this example.

 * A "Wireless communication" subsystem implements the transfer of information from the control tower to the train and vice versa.
 * Subsystem "CPU" functions as the brain of the train. In this subsystem all information is centralized and afterwards divided over the different subsystems.
 * Subsystem "Power" is responsible for building in the motor.
 * Subsystem "Control loop" makes the control loop of the drive.
 * Of course the peoplemover needs information on the environment. Subsystem "Sensors" gathers information on bends in the railway, terminals, light intensity outside and detects the radio signal of the control room.
 * As the peoplemover is a physical object, a subsystem has to take care of the "Mechanical design".
 * Appearances also count, especially if People Move Company wants to impress the public at large. Therefore a subsystem is in charge of the "Aesthetic design".

Of course these subsystems cannot work on their own islands. Interaction between the different subsystems is inevitable. People Move Company designates an 'integrator' who will coordinate the discussions on how transfer of information between subteams will be implemented. Figure 4 shows which subsystems need to communicate with each other. The mechanical and aesthetic design teams are not drawn as they have to be in contact with all other teams. For example, "CPU" needs to know when the control room wants to take over control. This information can be wirelessly transferred to the peoplemover. "Sensors" can provide information from beacons that indicate a desired velocity. This is a necessary input for the "Control loop" subsystem.

Detailed design
Hooray again! The People Mover Company won the competition, and has a contract to deliver 100 people movers within 36 months from now. So, the design team sits together with the human resources manager, the CTO and the CFO, to turn the high-level design document into a detailed design document that will be used to plan and guide all activities in this new project.

Architecture
The starting point for the detailed design is the global architecture with the division in subsystems. In a first step each subsystem has to find out which components he needs. Brainstorm sessions can be of great help here. The result of this process is the architecture of the peoplemover, illustrated in figure 5.

Furthermore the subsystems agree on the form in which the information will be communicated. This can be in an analog or a digital signal. The specific number of bits or bytes is specified together with the exact meaning of each signal. If any of these conventions turns out to be unnatural in a later stadium of the process, the involved subsystems have to sit together again to modify the agreements. As already mentioned an 'integrator' coordinates these meetings.

Now the different subteams can be set to work. The system architects avoid to (re)design every component though. Components that already exist are part of the Intellectual Property (IP) and can be used without having to design them again. Here the system architect makes a "make-or-buy decision.

Wireless communication
The control tower wirelessly sends commands to the peoplemover. This set of commands is finite. Each command corresponds with a sequence of ones and zeros, a binary codeword. A huffman code is one way to assign a binary codeword to each command. The sequence of ones and zeros now has to be transferred to the peoplemover using a modulated electromagnetic wave.

A device that performs modulation and the inverse operation demodulation is known as a modem. The peoplemover must be with equipped with an antenna and a modem. More information on communication can be found in the wikibook on Communication Systems and the wikibook on Data Coding Theory. The system designer's task is to choose a modulation method, a code and hardware (antennas and modems).

Sensors
From the product requirements document it is already clear that sensors will be needed in the peoplemover. The system designer thus has to choose suitable sensors, decide where to place them and make sure that the sensor information ends up at the right address. For each type of sensor the designer has to look for different requirements.

In every fact the peoplemover needs sensors that can detect when it is getting dark outside. An obvious solution is a light sensor with fotovoltaic cells. Of course a lot of sensors of this kind exist. It is up to the designer which final sensor will be used. It is also important to decide where this sensor will be placed. The best place for the light sensors is probably on the roof of the vehicle.

Also the communication with the control room asks for a sensor. Here one can think of sensors for radio detection. The environment in which the peoplemover will be used is of great importance here, as the control tower always has to be able to make contact with the peoplemover, even when the vehicle is in a tunnel.

The designer has to choose how the beacons will work. At first sight optical sensors seem to be very interesting. The beacons could even send out coded signals. This choice will certainly lead to difficulties though, because the environment is very 'hostile'. A leaf or other natural object can easily disturb the signal. A more robust solution is thus needed. An option is to make the beacons magnetic. A sensor in the peoplemover can then detect this magnetic field.

Furthermore the peoplemover needs to be equipped with a velocity sensor and weight sensors to detect overload. The outputs of this subteam is sent to the CPU and will result in an action.

In general sensors are quite expensive. The cost design criterion has to be considered as a consequence when choosing the right sensors.

CPU
The central processing unit (CPU) handles the incoming information. In first instance this information comes from the magnetic beacons, but the control room can take over. The possible commands are:
 * Desired speed = maximum speed
 * Desired speed = 0.8*maximum speed
 * Desired speed = 0.6*maximum speed
 * Desired speed = 0.4*maximum speed
 * Desired speed = 0.2*maximum speed
 * Stop
 * Desired speed = 0.2*maximum speed in the other direction
 * Desired speed = 0.4*maximum speed in the other direction
 * Desired speed = 0.6*maximum speed in the other direction
 * Desired speed = 0.8*maximum speed in the other direction
 * Desired speed = maximum speed in the other direction
 * Reset counter
 * Lights on
 * Lights off

Each time the wheel senses a magnetic signal from the railway, a counter is incremented with one unity. In this way every beacon has a corresponding counter's output. A logic circuit decodes this counter output. A programmable logic controller (PLC) provides a robust implementation. Figure 6 shows a possible implementation of the CPU. For a trajectory with sixteen beacons a 4-bit counter is sufficient. The decoder contains the necessary information to give the right command each time a beacon signal arrives. A speed signal goes to the control loop subsystem or a signal is sent to the light controller.

Control loop
One of the teams is in charge of the drive's control loop. The process this team implements is illustrated in figure 7.

This subsystem receives the desired speed of the CPU team and the measured speed of the Power Team. These are digital signals. To convert them into analog signals the inputs are sent through Digital to Analog Converters. With the design criterion of robustness in mind it is wise to include low pass filters here. This increases the signal-to-noise ratio. A comparison of the resulting signals has to lead up to a voltage signal that steers the motor. The associated controller is the PI velocity controller.

As already indicated in the high level design the current in the motor has to be limited for safety reasons. A clipper can be used to achieve this.

A second controller is the current controller, which allows quick motor control. In fact the two PI-controllers are in a master-slave configuration. The master PI controls the speed by adjusting the requested current, while the slave controls the current manipulating the motor voltage.

After all discussed manipulations the control loop gives a voltage signal in the output. This signal goes to the Power Team.

The conceptual scheme of figure 7 thus consists of Digital to Analog Converters (DAC1 and DAC2), low pass filters (LPF), PI-controllers (PI vel and PI curr), amplifiers (A) and a clipper. These elements can be realized with a collection of resistors, capacities and other simple electronic components.

Power

An automated peoplemover is driven by an electric motor. Different implementations are used in practice.

Most trains nowadays are driven by a variable frequency drive. A variable-frequency drive (VFD) is a system for controlling the rotational speed of an alternating current (AC) electric motor by controlling the frequency of the electrical power supplied to the motor. The motor used in a VFD system is usually a three-phase induction motor. AC input power is first converted to DC intermediate power using a rectifier bridge. The DC intermediate power is then converted to quasi-sinusoidal AC power using an inverter switching circuit. The VFD steering circuit is pictured in figure 8.

Since the control loop subsystem steers the power subsystem, these two subsystems will have to communicate. The control loop's steer signal (named PWM) carries information in its frequency. This signal is compared with a triangle wave with a much higher frequency. The comparator's output steers the 2 switches of one phase. As a result each phase has a quasi-sinusoidal waveform with the same frequency as the control loop's steer signal. The power subsystem also has other functions to fulfill such as measuring the current through the motor and measuring the speed of the motor. Again multiple implementations are possible. Just one of those is described here.

The control loop subsystem limits the desired acceleration to avoid a large current through the motor. Therefore the control loop subsystem needs an input signal proportional to the motor current. The current can be measured by connecting a small resistance in series with the motor. The voltage across this small resistance is proportional to the motor current. A differential amplifier is used to become a voltage signal with a range of [-5V, 5V].

In the power subsystem itself, a current limiter is also built in. In all kinds of applications redundancy is used to increase safety. This is very important to become a robust and safe system.

The actual speed of the peoplemover is necessary information for subsystem control loop to control the speed. The speed can be measured by a rotary encoder placed on the wheel axis. The output of the encoder serves as input of a microcontroller that sends out a 8-bit digital signal to subsystem CPU and subsystem control loop.

Another important functionality that has to be implemented in the power subsystem is the emergency brake. When subsystem CPU sets the stop signal high (5V) the motor has to be stopped immediately. This functionality can be implemented in many ways. The most obvious way is to OR the Stop signal with the pulses from the pulse width modulation.

Testing
An indispensable part of the design is testing. All subteams have to test their own system. Moreover the different subsystems have to be tested in smaller groups. Of course the final step is to test the whole. For example, more on software testing can be found on software testing.

Implementation and production
When all design steps are passed through, People Move Company can start with the implementation and production of the peoplemover. These tasks again ask for a structured approach. They are not discussed in this Wikibook though.


text:
Embedded Control Systems Design/Aviation

This Applications chapter discusses the entire system that enables planes to fly and communicate with each other. Taking-off and landing multiple planes at major airports for instance, is a difficult task and requires a lot of subsystems. The main functionalities of the entire flight management system are navigation, flight planning and aircraft control.

There are three main levels in the full flight management system:

 * Air Traffic Control

ATC contains everything that enables planes to communicate directly with each other, or through a control tower. This level is essential to create a global coordinated system that transports people or equipment.

 * Flight Control System

FCS is the entire system that enables one plane to take-off, stay in the air, and land. A lot of an airplane's embedded control systems are linked to this functionality and are a part of the Flight Control System as a component.

 * Components

Each component has a specific task in an airplane. Most of them have a very straightforward functionality and are easy to describe as an embedded system. For example: door opening mechanism, engine control, inflight entertainment system, Integrated Modular Avionics (IMA), black box, ...

The goal of this applications chapter is to point out the requirements of the different system levels in aviation and their impact on the design. There are many links to other sites for detailed functionality information.

Air Traffic Control
Of all different system levels, ATC is the top level. This is a special level because it doesn't exist in other applications like automotive, where there is no global communication between vehicles. The primary goal of ATC is collision prevention which requires communication and radar systems. To provide redundancy most aircrafts are equipped with TCAS ( Traffic alert and Collision Avoidance Systems).

Discussion
...

Introduction
There are a lot of embedded control systems in an airplane. One of them is the flight control system which controls the flight trajectory and the stability of the airplane. The actuators of this system are the engines and the movable devices of the main wing and tail. They are actuated by the pilot and by the embedded control system of the airplane.

Different kinds of FCS exist. They are categorized according to the number of axes they control. A one-axis FCS prevents the plane of rolling. A two-axis FCS also gives the possibility to control the direction of flight. The three-axis FCS is even more complex by providing the ability of automatic climbing and descending.

The figure above is a schematic overview of the flight control system. The flight control system (FCS) controls the flight trajectory and the stability of the airplane. The actuators of this system are the engines and the movable devices of the main wing and tail. They are actuated by the pilot and by the embedded control system of the airplane. This includes stability control, automatic pilot and engine control. The control outputs are calculated from sensor inputs, like navigation systems, weather radar, velocity measuring, altitude measuring, sensors in the engine...

Technical requirements
Airplane systems have specific requirements that are not found or are not as critical in other systems:

Airplanes are made using strong but light materials. The aviation industry was the first to use aluminium on a large scale, and modern materials such as carbon fibre are beginning to make their entry. The weight requirement is also one of the reasons for the evolution from a mechanical linkage to a fly-by-wire system.
 * Weight

Material quality must be high, less variation on material properties is allowed for aviation. Even though the black box is useless considering the safety of a single flight, it provides a crucial feedback loop in the entire design and maintenance process.
 * Safety

A lot of systems are made in redundancy in order to increase the reliability. Airplanes are also maintained frequently. Dependability is very important regarding safety, the flight control system must be available at all times, unlike automotive where simply stopping is an option.
 * Dependability (Reliability, Maintainability, Availability)

Larger planes are always equipped with an auto-pilot of some sort. This control system can work in different ways. In most planes, a control system is optional and is used to simplify the pilot's task. Other planes, usually military, need a control system because they are inherently unstable and no human could control them without a control system. More recently, stability control also exists in automotive, where systems such as ABS and ESP are making their entry.
 * Stability control

Other more common requirements are size, response time, cost, energy consumption, noise control, fun factor, ... (see how they are dealt with in automotive)

Legislation requirements
Because of the high safety and dependability requirements, regulation for aircraft systems is very strict. Software(DO-178B), hardware(DO-254) and network(AFDX) design are all standardized. Therefore, the certification of a new aircraft design and the paper trail that comes with it are very complex and time-consuming. The whole procedure can be found on the site of the Federal Aviation Administration: http://www.faa.gov/licenses_certificates/ Not only the airplane but also the pilot is subjected to high demands. Pilot training, testing and certification procedures can also be found on the site.

Optimization
As one can imagine, there are a lot of possible system designs. There is also the difficult task of combining their strengths in order to be able to meet the imposed requirements. Due to the extra dependability requirements some back up systems must be preserved, which makes everything even more complex. The design of an airplane and more specific of this control system, is an iterative design. The concept is already invented in the early days of aviation. Each airplane is the product of an iteration step in order to approximate more and more the 'ideal' airplane. In the early days design was mainly based on the experience of the designer but since complexity grows, a more systematic method is needed. An example and discussion of such a computer aided design method can be found in the following paper by Airbus. http://www.mip.ups-tlse.fr/publis/files/06.26.pdf

Pilot
The pilot interacts with the airplane systems and therefore can be considered as part of the flight control system

Training
The pilot must be well trained and experienced so he can react appropriately in case of unexpected situations or system failures. The plane design is based on this assumption, unlike automotive, where far less training is required.

HMI
To make interaction possible between the pilot and the airplane systems the airplane has a HMI (human machine interface). A good HMI design can greatly reduce the amount of effort needed to control the aircraft.

Redundancy
In larger airplanes with an auto-pilot, the pilot can also be seen as a redundant system to fly the plane if the auto-pilot fails. Even the pilots themselves can replace each other. Even though they are supposed to do different tasks, all pilots can fly the plane in case of an emergency. In the early days, up to four pilots were needed to control an airplane, now only two are required for large airplanes.

Sensors
The following sensors are commonly used by FCS. The object is mainly to navigate the aircraft autonomously. Therefore, an inertial navigation system is used. An aircraft has six independent degrees of freedom, three rotational and three translational. If they are all measured, it is possible to navigate and control an aircraft. The rotational movements are measured by three gyroscopes. The translational movement is measured by accelerometers. Because of the rotational movement of an aircraft, it is necessary to place the accelerometers on a frame which is referenced parallel to the Earth's surface. This is done by measurements of the gyro's. In military aircrafts, all measurements are performed by accelerometers. These are attached rigidly to the aircraft. Three of these are then used to measure the angular acceleration. By integration, the angles (roll, pitch and yaw) are calculated and used to derive the translation measured by the other accelerometers.

Control System
The autopilot is a system that serves to diminish the workload of the pilot. After long hours in the air, the pilot must be concentrated enough to be able to land safely. Another reason to use a control system in larger planes is the fact that the control surfaces of these planes require large control forces, which are impossible for a human pilot. Further it is to notice that a computer controlled system is more accurate.

Sensors will produce a signal according to the direction the aircraft is flying. The control system will compare this information with the desired direction. If there is a difference between both it will try to correct the present situation by controlling the actuator. In the more advanced, three-axis systems more signals will be used than in the one-axis system. The control system will control both the desired orientation of the plane and the desired course of the aircraft (in direction and altitude). Some aircrafts, especially the military fighters, are designed to fly unstable. This means that the center of lift lies before the center of gravity. In that case a small deflection from the balanced situation will cause the aircraft to become unstable and uncontrollable. This behaviour is desired in military fighters to be able to make short turns and react quickly. The response time will indeed be shorter when flying an unstable aircraft. Because a human won't be able to control such an aircraft, the control system is very important to keep the aircraft virtually stable.

Dependability requirement for control systems
Now an important aspect, dependability requirements, is explained. This is applicable to any embedded control system where the dependability requirement is important. The design of a control system is an iterative process. Several systems have been used and replaced and nowadays fly-by-wire is hot topic. Regarding the design criteria and especially the dependability requirement, the designers have chosen to use two identical computers for the Flight Control (see image). The exact software implementation is not now, but some question can be posed:

 * How do you know which signal of which computer is the correct one?
 * What if one computer breaks down?
 * Is there enough redundancy?

Maybe some modifications to this configuration can be made. The first possible configuration uses two computers. It can be useful to run different algorithms, who are developed separately, by different companies. But these different algorithms are doing the same calculations. The results can be compared and there can be a voting for one final output signal. An other possibility is to increase the number of computers. Nowadays as many as 5 computers can be used.

Fly-by-wire
In order to explain why there is the quest for fully fly by wire, a short overview of the systems used in the early days and nowadays is given. Basically the pilot action consists of moving the stick. The stick can be directly coupled to the movable wing devices, there can be a servo mechanism or the action of the pilot can be measured and an electric signal is generated. The direct mechanical coupling between the pilot and the movable wing devices is removed. This is fly by wire. Pro's and con's of fly by wire

 * (+) The size of the airplane or weather conditions are not important. The force that has to be delivered by the pilot can be perfectly controlled.
 * (+) Using a haptic interface, the 'feel' of all airplanes can be generalized. This can reduce the required amount of pilot training or experience and thus decreases the cost of the 'pilot' component.
 * (+) The action of the pilot can be processed as an electrical signal together with outputs of sensors, in order to obtain optimal control, which can be safer, less energy consumption and fun to fly for the passengers.
 * (+) Electrical systems have less weight.
 * (-) Because the direct coupling between pilot and actuator are vanished and the system is more complex, a failure of one of the signal processing or transmitting elements can be fatal. Much attention must be paid to dependability.
 * (-) Since electric systems depend fully on electricity, a power breakdown is fatal.

Architectures
Another aspect of fly-by-wire systems, is the architecture used. In 'Principles of Avionics' (see references), other names are giving to the different possible architectures then here in this book. The two names will be given.

Central architecture -- centralized hardware/centralized software

 * One computer is used for, let say, maybe 10 subsystems. As all hardware is centralized, the environment can be controlled very well. Also the maintenance of these systems is easy.
 * If you want to meet the dependability requirement, maybe three computers can be used, what still is less than 10 computers for 10 subsystems.
 * All calculations are centralized, so that analog sensor signal must be transported over al long range, making them sensitive to noise.

Distributed architecture -- distributed hardware/distributed software

 * Calculations are finding place in the sensors. Only results are transmitted.
 * There is no central system, so the subsystems must be capable of communicating with each other.
 * If every sensor has software for its own calibrations, then changing the sensor is easy, because no modifications must be made to a central system.

Federated architecture -- distributed hardware/centralized software

 * It is a compromise between central and distributed architecture.
 * There are less subsystems than in the case of distributed hardware, but more than in the case of centralized hardware.
 * Because the whole system is divided in functional block, e.g. a system for engine control and a system for communications, the failure of one of those will not influence the others

Additional information about fly-by-wire is already extensively covered on Wikipedia in Aircraft flight control systems.

Actuators
The control surfaces of aircraft are actuated by servo motors. Important in the design of the actuating chain, is the following. If the servo motor would fail, the system should fall back to manual control. Otherwise the pilot would lose control over whole the plane. In larger planes, control surfaces are always steered by the use of motors. The forces acting on these planes are too high for a human to bear. Still a mechanical connecting exists. The system is called “servo assisted”. In the more advanced aircraft, a system called “fly by wire” is used. Here there is no mechanical connection anymore, and the aircraft is fully controlled by servo motors. The pilot only uses a small sidestick, which generates the controlling signals.

Discussion
The example of the flight control system applied to a Boeing 737 was removed due to its length and relevance regarding the flight control system described on a system level.

...

Components
There are many embedded control systems in an airplane on the component level. Only some of the more interesting ones will be explained in further detail.

Black Box
The FDR (flight data recorder) or black box is a special component in the way that it interacts with aircraft systems. It is designed to record specific aircraft data, often used for crash investigation. The design is subjected to internationally recognized standards. The requirements and design are described in the link above.

Discussion
...

Integrated Modular Avionics
IMA represent standardized computer network systems which simplify the development of avionics software. It is a concept which proposes an integrated system architecture analogue to the AUTOSAR architecture in automotive applications. Apart from the easier software development it also simplifies hardware and software integration and maintenance.

Discussion
...

= References =

 * Flight-control system architecture optimization for fly-by-wire airliners by Christophe BAUER, Kristen LAGADEC†, Christian BES and Marcel MONGEAU§.
 * Redundancy Management Technique for Space Shuttle Computers by J. R. Sklaroff.
 * Principles of Avionics by Albert Helfrick


text:
Embedded Control Systems Design/Robotics

This Application chapter discusses embedded robot control systems that control industrial robots, machine tools, and autonomous robots (such as RoboCup soccer playing robots).

The robotics domain has a couple of specific general design requirements: each robot system requires the close integration of planning, sensing, control and modelling, and the robot must take into account the interactions between itself, its task and its environment. The more intelligent a robot system must be, the more it has to be robust (see Design Criteria for more information on robustness) against deviations that can occur in what it expects from its task and its environment. In other words: each such robot is a system-of-systems, where many of the systems are not under direct control of the robot itself, because both task and environment can contain agents that have their own behaviour.

 * A team of RoboCup robots


text:
General Chemistry/Atomic Structure/History of Atomic Structure

Why Is The History Of The Atom So Important?
It is fundamental to the understanding of science that science is understood to be a process of trial and improvement and represents the best known at the time, not an unerring oracle of truth. Development of an idea and refinement through testing is shown more in the understanding of atomic structure.

The Greek Theorists
The earliest known proponent of anything resembling modern atomic theory was the ancient Greek thinker Democritus. He proposed the existence of indivisible atoms as a response to the arguments of Parmenides and the paradoxes of Zeno.

Parmenides argued against the possibility of movement, change, and plurality on the premise that something cannot come from nothing. Zeno attempted to prove Parmenides' point by a series of paradoxes based on difficulties with infinite divisibility.

In response to these ideas, Democritus posited the existence of indestructible atoms that exist in a void. Their indestructibility provided a retort to Zeno, and the void allowed him to account for plurality, change, and movement. It remained for him to account for the properties of atoms, and how they related to our experiences of objects in the world.

Democritus proposed that atoms possessed few actual properties, with size, shape, and mass being the most important. All other properties, he argued, could be explained in terms of the three primary properties. A smooth substance, for instance, might be composed of primarily smooth atoms, while a rough substance is composed of sharp ones. Solid substances might be composed of atoms with numerous hooks, by which they connect to each other, while the atoms of liquid substances possess far fewer points of connection.

Democritus proposed 5 points to his theory of atoms. These are:

 * 1) All matter is composed of atoms, which are bits of matter too small to be seen. These atoms CANNOT be further split into smaller portions.
 * 2) There is a void, which is empty space between atoms.
 * 3) Atoms are completely solid.
 * 4) Atoms are homogeneous, with no internal structure.
 * 5) Atoms are different in their size, their shape, and their weight.

Alchemy
Empedocles proposed that there were four elements, air, earth, water, and fire and that everything else was a mixture of these. This belief was very popular in the medieval ages and introduced the science of alchemy. Alchemy was based on the belief that since everything was made of only four elements, you could transmute a mixture into another mixture of the same type. For example, it was believed that lead could be made into gold.

Alchemy's problem was exposed by Antoine Lavoisier when he heated metallic tin in a sealed flask. A grayish ash appeared on the surface of the melting tin, which Lavoisier heated until no more ash formed. After the flask cooled, he inverted it and opened it underwater. He discovered the water rose one-fifth of the way into the glass, leading Lavoisier to conclude that air itself is a mixture, with one-fifth of it having combined with the tin, yet the other four-fifths did not, showing that air was not an element.

Lavoisier repeated the experiment again, substituting mercury for tin, and found that the same happened. Yet after heating gently, he found that the ash released the air, showing that the experiment could be reversed. He concluded that the ash was a compound of the metal and oxygen, which he proved by weighing the metal and the ash, and showing that their combined weight was greater than that of the original metal.

Lavoisier then stated that combustion was not an element, but instead was a chemical reaction of a fuel and oxygen.

John Dalton
Modern atomic theory was born with Dalton when he published his theories in 1803. His theory consists of five important points, which are considered to be mostly true today: (from Wikipedia)

 * Elements are composed of tiny particles called atoms.
 * All atoms of a given element are identical.
 * The atoms of a given element are different from those of any other element; the atoms of different elements can be distinguished from one another by their respective relative weights.
 * Atoms of one element can combine with atoms of other elements to form chemical compounds; a given compound always has the same relative numbers of types of atoms.
 * Atoms cannot be created, divided into smaller particles, nor destroyed in the chemical process; a chemical reaction simply changes the way atoms are grouped together.

We now know that elements have different isotopes, which have slightly different weights. Also, nuclear reactions can divide atoms into smaller parts (but nuclear reactions aren't really considered chemical reactions). Otherwise, his theory still stands today.

Dmitri Mendeleev
In the late 1800's, Russian scientist Dmitri Mendeleev was credited with creating one of the first organized periodic tables. Organizing each element by atomic weight, he cataloged each of the 56 elements discovered at the time. Aside from atomic weight, he also organized his table to group the elements according to known properties.

While writing a series of textbooks, Mendeleev realized he was running out of space to treat each element individually. He began to regularly "linewrap" the elements onto the next line, and create what is now called the periodic table of the elements. Using his table, he predicted the existence of later-discovered elements, such as "eka-aluminum" and "eka-silicon" (gallium and germanium) according to patterns found earlier. His predictions were a success, proving his table to be exceptionally accurate. Later theories, those of the electrons around the atom, explain why elements in the same period, or group, have similar chemical properties. Chemists would later organise each element by atomic number, not atomic weight, giving rise to the modern Periodic Table of Elements.

Discovery of the Electron
In the year 1889 the British physicist J.J. Thomson discovered the electron. Thomson conducted a number of experiments using cathode ray tube. Cathode rays are constructed by sealing two electrodes in a glass tube connected to a voltage supplier and a vacuum pump removing the air from it. When the electrodes are attached to high voltage of about 15000v and low pressure, a beam of radiation is emitted from the negative electrode(cathode)moving towards the positive electrode(anode). These beams are said to be green rays called cathode rays. (1) The ray is said to be taken towards Faraday's tube(gold leaf electroscope) and was charged by induction and deflect positively charged gold leaf electroscope (2) a freely moving paddle wheel was placed on the path of the rays and it was able to move it suggesting that it has momentum (3) The ray was placed in a magnetic and electric field moving towards the north and positive pole respectively Thomson discovered that cathode rays travel in straight lines except, when they are bent by electric or magnetic fields. The cathode rays bent away from a negatively charged plate, Thomson concluded that these rays are made of negatively charged particles; today we call them electrons. Thomson found that he could produce cathode rays using electrodes of various materials. He then concluded that electrons were found in all atoms and are over a thousand times smaller than protons. Thompson used an apparatus to determine the charge to mass ratio(e/m)after the ray emerge he placed a magnetic field of a known magnetic influence and the rays bent towards the north to a particular position then he added an electric field to return the ray back to it original position and recorded the charges used by the electric field so he divided them to get the ratio of about -1.7*10^8kg

The "Plum Pudding" Atomic Model
Soon after the discovery of the electron, Thomson began speculating on the nature of the atom. He suggested a "plum pudding" model. In this model the bits of "plum" were the electrons which were floating around in a "pudding" of positive charge to match that of the electrons and make an electrically neutral atom. A modern illustration of this idea would be a chocolate chip cookie, with the chips representing negatively charged electrons and the dough representing positive charge.

Rutherford
Ernest Rutherford is known for his famous gold foil experiment in 1911. Alpha particles, which are heavy and positively charged (actually, helium nuclei, but that's beside the point), were fired at a very thin layer of gold. Most of the alpha particles passed straight through, as expected. According to the plum pudding model all of the particles should have slowed as they passed through the "pudding", but none should have been deflected. Surprisingly, a few alpha particles were deflected back the way they came. He stated that it was "as if you fired a 15-inch shell at a piece of tissue paper and it came back and hit you."

The result of the experiment allowed Rutherford to conclude that the plum pudding model is wrong.
 * Atoms have a nucleus, very small and dense, containing the positive charge and most of the atom's mass.
 * The atom consists of mostly empty space.
 * The electrons are attracted to the nucleus, but remain far outside it.

Niels Bohr

Bohr created his own model of the atom, improving on Rutherford's. Bohr used an equation developed by Rydberg that provided a mathematical relationship between the visible spectral lines of the hydrogen atom. The equation requires that the wavelengths emitted from the hydrogen atom be related to the difference of two integers. Bohr theorized that these integers represent "shells" or "orbits" in which electrons travel around the nucleus, each with a certain energy level. The energy of an orbit is proportional to its distance from the nucleus. An atom will absorb and release photons that have a specific amount of energy. The energy is the result of an electron jumping to a different shell. Starting with Rydberg's equation, along with Planck and Einstein's work on the relationship between light and energy, Bohr was able to derive an equation to calculate the energy of each orbit in the hydrogen atom. The Bohr model depicts the atom as a nucleus with electrons orbiting around it at specific distances. This model is also referred to as the Planetary Model.

Robert Millikan
Robert Millikan is credited for the "Oil Drop Experiment", in which the value of the electron charge was determined. He created a mechanism where he could spray oil drops that would settle into a beam of X rays. The beam of X rays caused the oil drops to become charged with electrons. The oil droplets were in between a positively charged plate and a negatively charged plate which, when proper electric voltage was applied, caused the oil droplet to remain still. Robert Millikan measured the diameter of each individual oil drop using a microscope.

Millikan was able to calculate the mass of each oil droplet because he knew the density of the oil ($$volume * density = mass$$). Using the mass of each oil droplet and the equation for the force of gravitational attraction (which he rearranged from $$Force = mG$$ to $$m_{droplet}G = Ee$$, where $$m_{droplet}$$ is the mass of each individual oil droplet, $$G$$ is the acceleration due to gravity, and $$Ee$$ is the electrical force which equals force in the first equation), Millikan was able to find the value of the charge of the electron, $$e$$.

The X rays, however, did not always produce an oil drop with only one negative charge. Thus, the values Millikan obtained may have looked like this: Millikan found that these values all had a common divisor: $$-1.6*10^{-19}$$ coulomb. He concluded that different values occurred because the droplets acquired charges of -5, -4, -3, and -2, as in this example. Thus, he stated that the common divisor, $$-1.6*10^{-19}$$ coulomb, was the charge of the electron.
 * $$-8.0*10^{-19}$$ coulomb
 * $$-6.4*10^{-19}$$ coulomb
 * $$-4.8*10^{-19}$$ coulomb
 * $$-3.2*10^{-19}$$ coulomb


text:
User:OoEtMePasty

Home-made, home-grown and locally-reared Cornish Cuisine a speciality. Other stuff from the Celtic/western edges of these rain-lashed rocks (especially fish and traditional) liked too. When leaving the Atlantic behind, I go for Middle Eastern and Japanese (when I can get it) scoff. Strong supporter of the Slow Food movement. Never eaten a cook/chill in me life though I think some of them (Marks and Spencers) seem fine. Would rather haddock and chips or a shish kebab than Maccy D's any day. If I need to go to a supermarket I go to the Co-op. Boycott Tescos for their contributions to the ruination of British High Streets.

title: User:OoEtMePasty


text:
User:Rundmc81

I am many things to many people. I am a rapper, author, researcher, computer software programmer, podcaster, etc. My name is David. My 'rap name' is David Chamberz or 'DC'. Most people just call me 'DC'. I plan to edit and write many articles here in Wikipedia. I have had many websites over my lifetime (as well as many email addresses), but my current pages and contact information is:

http://www.myspace.com/huntsvillerapstar http://davidchamberz.blogspot.com http://rundmc81.livejournal.com

rundmc81@gmail.com rundmc81@yahoo.com [IM only]

title: User:Rundmc81


text: Wikibooks:Requests for deletion/Easy Guide to Famous Artists


text:
User talk:Srhat

Welcome to Wikibooks, Srhat! It's for me from me :) Srhat (talk) 14:02, 5 April 2008 (UTC)

Please leave your messages bottom. Don't forget to put four tildes at the end of your message.

 * As an active member of the Wikibooks community, we'd like your input on the CheckUser nomination for Thenub314, which can be found here. – Adrignola talk 19:45, 19 August 2010 (UTC)

Language books
Hi Srhat

Are you still active here? I am the author of the Dutch wikibook and I have been trying to update that book to make it more attractive and interactive. I think the authors of language books could all benefit if we share our tricks a bit better, e.g. I am now making icons to illustrate Dutch prepositions see e.g. Dutch/Vocabulary/Prepositions for a few examples. These icons are very easy to adapt for another language. You can even do that in notepad.

I wonder if we couldn't start a language book user group or portal to exchange that sort of things better. I feel that I am reinventing many wheels, probably square ones too.

Jcwf (discuss • contribs) 18:23, 16 May 2015 (UTC)

Useful tricks page
Hi

I have talked to some of you about the need to better pool our resources and ideas about writing good foreign language wikibooks. I noticed that some information was available under Authoring Foreign Language Textbooks, but not so much on what kind of tricks you can use to make your boom more interactive. So I added a page "Useful tricks to it and gave a description of some of the tricks I am using @ Dutch. Hopefully we can use the Useful tricks page and its discussion page for a better exchange of ideas. I still think we need a better venue for that, but it's a start. Jcwf (discuss • contribs) 00:39, 18 May 2015 (UTC)

title: User talk:Srhat


text:
User:Ttv~enwikibooks

hi! every comment is welcome:

title: User:Ttv~enwikibooks


text:
User talk:207.112.85.207

Join Us!
-- – Mike.lifeguard  | talk 21:26, 21 March 2008 (UTC)

title: User talk:207.112.85.207


text: User talk:159.191.12.25

title: User talk:159.191.12.25