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High School Mathematics Extensions/Supplementary/Complex Numbers

Introduction
Although the real numbers can, in some sense, represent any natural quantity, they are in another sense incomplete. We can write certain types of equations with real number coefficients which we desire to solve, but which have no real number solutions. The simplest example of this is the equation:

\begin{matrix} x^2 + 1 &=& 0 \\ x^2 &=& -1 \\ x &=& \sqrt {-1} \end{matrix} $$ Your high school math teacher may have told you that there is no solution to the above equation. He/she may have even emphasised that there is no real solution. But we can, in fact, extend our system of numbers to include the complex numbers by declaring the solution to that equation to exist, and giving it a name: the imaginary unit, $$i$$.

Let's imagine for this chapter that $$i = \sqrt{-1}$$ exists. Hence x = i is a solution to the above question, and $$i^2 = -1 $$.

A valid question that one may ask is "Why?". Why is it important that we be able to solve these quadratics with this seemingly artificial construction? It is interesting delve a little further into the reason why this imaginary number was introduced in the first place - it turns out that there was a valid reason why mathematicians realized that such a construct was useful, and could provide deeper insight.

The answer to the question lies not in the solution of quadratics, but rather in the solution of the intersection of a cubic and a line. The mathematician Cardano managed to come up with an ingenious method of solving cubics - much like the quadratic formula, there is also a formula that gives us the roots of cubic equations, although it is far more complicated. Essentially, we can express the solution of a cubic $$x^3 = 3px + 2q$$ in the form

$$x=\sqrt[3]{q + \sqrt{q^2 - p^3}} + \sqrt[3]{q - \sqrt{q^2-p^3}}$$

An unsightly expression, indeed!

You should be able to convince yourself that the line $$y = 3px + 2q$$ must always hit the cubic $$y=x^3$$. But try solving some equation where $$q^2 < p^3$$, and you run into a problem - the problem is that we are forced to deal with the square root of a negative number. But, we know that in fact there is a solution for x; for example, $$x^3 = 15x + 4$$ has the solution x = 4.

It became apparent to the mathematician Bombelli that there was some piece of the puzzle that was missing - something that explained how this seemingly perverse operation of taking a square root of a negative number would somehow simplify to a simple answer like 4. This was in fact the motivation for considering imaginary numbers, and opened up a fascinating area of mathematics.

The topic of Complex numbers is very much concerned with this number i. Since this number doesn't exist in this real world, and only lives in our imagination, we call it the imaginary unit. (Note that $$i$$ is not typically chosen as a variable name for this reason.)

The imaginary unit
As mentioned above

\begin{matrix} i^2 & = & -1 \end{matrix} $$.

Let's compute a few more powers of i:

\begin{matrix} i^1 & = & i \\ i^2 & = & -1 \\ i^3 & = & -i\\ i^4 & = & 1\\ i^5 & = & i\\ i^6 & = & -1\\ &\mbox{...}& \end{matrix} $$

As you may see, there is a pattern to be found in this.

Exercises

 * 1) Compute $$i^{25}$$
 * 2) Compute $$i^{100}$$
 * 3) Compute $$i^{1000}$$
 * Exercise Solutions

Complex numbers as solutions to quadratic equations
Consider the quadratic equation:

\begin{matrix} x^2 - 6x + 13 & =& 0 \\ x & = & \frac{6 \pm \sqrt{36 - 4 \times 13}}{2} \\ x & = & \frac{6 \pm \sqrt{-16}}{2} \\ x & = & \frac{6 \pm \sqrt{-1}\sqrt{16}}{2} \\ x & = & \frac{6 \pm 4i}{2} \\ x & = & 3 + 2i \, \ 3 - 2i\\ \end{matrix} $$ The x we get as a solution is what we call a complex number. (To be nitpicky, the solution set of this equation actually has two complex numbers in it; either is a valid value for x.) It consists of two parts: a real part of 3 and an imaginary part of $$\pm 2$$. Let's call the real part a and the imaginary part b; then the sum $$a+bi = 3 \pm 2i$$ is a complex number.

Notice that by merely defining the square root of negative one, we have already given ourselves the ability to assign a value to a much more complicated, and previously unsolvable, quadratic equation. It turns out that 'any' polynomial equation of degree $$n$$ has exactly $$n$$ zeroes if we allow complex numbers; this is called the Fundamental Theorem of Algebra.

We denote the real part by Re. E.g.:
 * $$\mathrm{Re}(x) = 3$$

and the imaginary part by Im. E.g.:
 * $$\mathrm{Im}(x) = \pm 2$$

Let's check to see whether $$x = 3 + 2i$$ really is solution to the equation:

\begin{matrix} x & = &3 + 2i &\\ x^2 & = & (3)^2 + 2(3)(2i) + (2i)^2 \\ & = & 5 + 12i\\ x^2 - 6x + 13 &=& 5 + 12i - 6(3+2i) + 13\\ &=& 0\\ \end{matrix} $$

Exercises

 * 1) Convince yourself that x = 3 - 2i  is also a solution to the equation.
 * 2) Plot the points A(3, 2) and B(3, -2) on a XY plane. Draw a line for each point joining them to the origin.
 * 3) Compute the length of AO (the distance from point A to the Origin) and BO. Denote them by r1 and r2 respectively. What do you observe?
 * 4) Compute the angle between each line and the x-axis and denote them by $$\phi_1$$ and $$\phi_2$$. What do you observe?
 * 5) Consider the complex numbers:

\begin{matrix} z & = & r_1 \cdot (\cos{\phi_1} + i\sin{\phi_1})\\ w & = & r_2 \cdot (\cos{\phi_2} + i\sin{\phi_2})\\ \end{matrix} $$ Substitute z and w into the quadratic equation above using the values you have computed in Exercise 3 and 4. What do you observe? What conclusion can you draw from this?
 * 1) Find the complex solutions to the equation $$x^2-2x+2=0$$. Perform the same steps as above. What can you conclude here?

Addition and multiplication
Adding and multiplying two complex number together turns out to be quite straightforward. Let's illustrate with a few examples. Let x =  3 - 2i and y  =  7 + 11i, and we do addition first

and now multiplication
 * $$ x + y \!$$ || $$= \!$$ || $$(3+7) + (-2+11)i \!$$
 * || $$= \!$$ || $$ 10 + 9i \!$$
 * }
 * }

Let's summarise the results here.
 * $$ x \cdot y \!$$ || $$= \!$$ || $$(3 - 2i)\cdot(7 + 11i) \!$$
 * || $$= \!$$ || $$3\cdot 7 + 3\cdot 11i - 2i\cdot 7 - 2\cdot 11\cdot i^2\!$$
 * ||$$= \!$$ || $$43 + 19i\!$$
 * }
 * ||$$= \!$$ || $$43 + 19i\!$$
 * }
 * When adding complex numbers we add the real parts with real parts, and add the imaginary parts with imaginary parts.
 * When multiplying two complex numbers together, we use normal expansion. Whenever we see i2 we put in its place -1. We then collect like terms.

But how do we calculate:
 * $$\frac{3 + 2i} {7 - \sqrt{5}i}$$

Note that the square root is only above the 5 and not the i. This is a little bit tricky, and we shall cover it in the next section.

Exercises:

\begin{matrix} x &=& 3 - 2i \\ y &=& 3 + 2i \end{matrix} $$

Compute:
 * 1) x + y
 * 2) x - y
 * x2
 * y2
 * xy
 * 1) (x + y)(x - y)

Division
One way to calculate:

\frac{1}{2\sqrt{3}+\sqrt{2}} $$ is to rationalise the denominator:

\frac{1}{2\sqrt{3}+\sqrt{2}}= \frac{2\sqrt{3}-\sqrt{2}}{(2\sqrt{3}+\sqrt{2})(2\sqrt{3}-\sqrt{2})}= \frac{2\sqrt{3}-\sqrt{2}}{10} $$

Utilising a similar idea, to calculate

\frac{3 + 2i} {7 - \sqrt{5}i} $$ we realise the denominator.

z \ = \ \frac{3 + 2i} {7 - \sqrt{5}i} $$

z \ = \ \frac{3 + 2i} {7 - \sqrt{5}i} \times \frac{7 + \sqrt{5}i} {7 +\sqrt{5}i} $$ The denominator is the sum of two squares. We get:

z \ = \ \frac{(3 + 2i) \times (7 + \sqrt{5}i)} {49 + 5} $$

z \ = \ \frac{21 - 2\sqrt{5}}{54} + \frac {14+3\sqrt{5}} {54}i $$

If somehow we can always find a complex number whose product with the denominator is a real number, then it's easy to do divisions.

If

z \ = \ a + ib $$ and

w \ = \ a - ib $$ Then zw is a real number. This is true for any 'a' and 'b' (provided they are real numbers).

Exercises
Convince yourself that the product of zw is always a real number.

Complex Conjugate
The exercise above leads to the idea of a complex conjugate. The complex conjugate of a + ib is a - ib. For example, the conjugate of 2 + 3i is 2 - 3i. It is a simple fact that the product of a complex number and its conjugate is always a real number. If z is a complex number then its conjugate is denoted by $$\bar{z}$$. Symbolically if
 * z = a + ib

then,
 * $$\bar{z} = a - ib$$

The conjugate of 3 - 9i is 3 + 9i.

The conjugate of 100 is 100.

The conjugate of 9i - 20 is -20 - 9i.

 Conjugate laws

Here are a few simple rules regarding the complex conjugate
 * $$\overline{z + w}= \bar{z} + \bar{w}$$

and
 * $$\overline{zw}= \bar{z} \bar{w}$$

The above laws simply says that the sum of conjugates equals the conjugate of the sum; and similarly, the conjugate of the product equals the product of the conjugates.

Consider this example:
 * $$(3 + 2i) + (89 - 100i) = 92 - 98i$$

and we can see that
 * $$\overline{92 - 98i} = 92 + 98i$$

which equals to
 * $$\overline{3 + 2i} + \overline{89 - 100i} = 3 - 2i + 89 + 100i = 92 + 98i$$

This confirms the addition conjugate law.

Exercise

 * Convince yourself that the multiplication law is also true.

The complex root
Now that you are equipped with all the basics of complex numbers, you can tackle the more advanced topic of root finding.

Consider the question:

\begin{matrix} z & = & -3 + 4i \\ w & = & \sqrt{z} \end{matrix} $$

Express w in the form of a + ib.

That is easy enough.

\begin{matrix} w & = & \sqrt{z}\\ w^2 & = & z\\

w &=& a + ib\\ w^2 &=& a^2 - b^2 + 2abi\\ \\ -3 &=& a^2 - b^2 &\mbox{(1)}&\\ 4 &=& 2ab &\mbox{(2)}&\\ \end{matrix} $$ Solve (1) and (2) simultaneously to work out a and b.

Observe that if, after solving for a and b, we replace them with -a and -b respectively, then they would still satisfy the two simultaneous equations above, we can see that (as expected) if w = a + ib satisfies the equation w2 = z, then so will w = -(a + ib). With real numbers, we always take the non-negative answer and call the solution $$\sqrt{x}$$. However, since there is no notion of "greater than" or "less than" with complex numbers, there is no such choice of $$\sqrt{z}$$. In fact, which square root to take as "the" value of $$\sqrt{z}$$ depends on the circumstances, and this choice is very important to some calculations.

info -- Finding the square root
 Finding the root of a real number is a very difficult problem to start with. Most people have no hope of finding a close estimate of $$\sqrt{2}$$ without the help of a calculator. The modern method of approximating roots involves an easy to understand and ingenius piece of mathematics called the Taylor series expansion. The topic is usually covered in first year university maths as it requires an elementary understanding of an important branch of mathematics called calculus.

The Newton-Raphson method of root finding is also used extensively for this purpose.

Now consider the problem

\begin{matrix} z &=& -2 + 2i \\ w &=& z^{1/3} \end{matrix} $$ Express w in the form of "a + ib".

Using the methodology developed above we proceed as follows,

\begin{matrix} w &=& z^{1/3} \\ w^3 &=& z \\ \\ w &=& (a + ib)\\ w^3 &=& (a^2 - b^2 + 2abi) \times (a + ib)\\ z &=& (a^3 - 3ab^2) + i(3a^2b - b^3)\\ \\ -2 &=& a^3 - 3ab^2 \qquad (1)\\ 2 &=& 3a^2b - b^3 \qquad (2)\\ \end{matrix} $$

It turns out that the simultaneous equations (1) & (2) are hard to solve. Actually, there is an easy way to calculate the roots of complex numbers called the De Moivre's theorem, it allows us to calculate the nth root of any complex number with ease. But to set the method, we need understand the geometric meaning of a complex number and learn a new way to represent a complex number.

Exercises

 * 1) Find (3 + 3i)1/2
 * 2) Find (1 + 1i)1/2
 * 3) Find i1/3

Complex numbers as ordered pairs
It is worth noting, at this point, that every complex number, a + bi, can be completely specified with exactly two real numbers: the real part a, and the imaginary part b. This is true of every complex number; for example, the number 5 has real part 5 and imaginary part 0, while the number 7i has real part 0 and imaginary part 7. We can take advantage of this to adopt an alternative scheme for writing complex numbers: we can write them as ordered pairs, in the form (a, b) instead of a+bi.

\begin{matrix} \mbox{Instead of} & \mbox{We could write} \\ 5+4i&(5, 4) \\ 3i&(0, 3) \\ \frac{4+5i}{3}&(\frac{4}{3}, \frac{5}{3}) \\ 42&(42, 0) \\ \sqrt{2}+\sqrt{2}i&(\sqrt{2}, \sqrt{2}) \\ \end{matrix} $$ These should look familiar: they are exactly like the ordered pairs we use to represent points in the plane. In fact, we can use them that way; the plane which results is called the complex plane. We refer to its x axis as the real axis, and to its y axis as the imaginary axis.

The complex plane
We can see from the above that a single complex number is a point in the complex plane. We can also represent sets of complex numbers; these will form regions on the plane. For example, the set
 * $$\{a+bi | -1 \le a \le 1, -1 \le b \le 1\}$$

is a square of edge length 2 centered at the origin (See following diagram).

Complex-valued functions
Just as we can make functions which take real values and output real values, so we can create functions from complex numbers to real numbers, or from complex numbers to complex numbers. These latter functions are often referred to as complex-valued functions, because they evaluate to (output) a complex number; it is implicit that their argument (input) is complex as well.

Since complex-valued functions map complex numbers to other complex numbers, and we have already seen that complex numbers correspond to points on the complex plane, we can see that a complex-valued function can turn regions on the complex plane into other regions. A simple example: the function

$$f(z) = z + (0+1i)$$

takes a point in the complex plane and shifts it up by 1. If we apply it to the set of points making up the square above, it will move the entire square up one, so that it "rests" on the x-axis.

{To make more complicated examples, I will first have to go back and introduce the polar representation of complex numbers. Makes for much more interesting functions, :-) You can use the diagrams below or modify them to make new diagrams.  I will make links to these diagrams in other places in Wikibooks:math.  In the 2nd diagram showing the point, r=4 and theta= 50 degrees.  These types of diagrams can be used to introduce phasors, which are notations for complex numbers used in electrical engineering.}

de Moivre's Theorem

 * $$(e^{i\theta})^n = (\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta$$

For example, say we have $$z = 1 + i$$. We may now write this in polar form,

 * $$z = |z|e^{i\arg z} = \sqrt{2}e^{\frac{\pi}{4}i} = \sqrt{2}\left(\cos\left(\frac{\pi}{4}\right) + i\sin\left(\frac{\pi}{4}\right)\right)$$

Using de Moivre's theorem, we can deduce,

 * $$z^2 = 2\left(\cos\left(\frac{\pi}{2}\right) + i\sin\left(\frac{\pi}{2}\right)\right) = 2i$$
 * $$z^3 = 2\sqrt{2}\left(\cos\left(\frac{3\pi}{4}\right) + i\sin\left(\frac{3\pi}{4}\right)\right) = \sqrt{2}\left(\frac{\sqrt{2}}{2}i -\frac{\sqrt{2}}{2}\right) = 2i - 2$$
 * $$z^4 = 4 \cdot -1 = -4$$

etc.

Complex root of unity
The complex roots of unity to the nth degree is the set of solutions to the equation $$x^{n} = 1$$. Clearly they all have magnitude 1. They form a cyclic group under multiplication. For any given $$n$$, there are exactly $$n$$ many of them, and they form a regular n-gon in the complex plane over the unit circle.

A closed form solution can be given for them, by use of Euler's formula:

$$u^n = \cos(2\pi\cdot j/n)+i\sin(2\pi\cdot j/n) \quad 0 \leq j < n$$

The sum of the $$n$$th roots of unity is equal to 0, except for $$n=1$$, where it is equal to 1.

The product of the $$n$$th roots of unity alternates between -1 and 1.


text:
GCSE Science/Electricity multiple choice practice question 3 answer 2

Well done! This is the correct answer.

Applying Ohm's law to the top branch:

I=V/R

=6/4 A

=1.5A

Applying Ohm's law to the bottom branch: I=V/R

=6/(2+2) A

=1.5A

On to the next question>>

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


text:
GCSE Science/Electricity multiple choice practice question 3 answer 1

I'm sorry but this is the wrong answer. Remember that each branch has the entire voltage across it.

Applying Ohm's law to the top branch:

I=V/R

=6/4 A

=1.5A

The lower branch has a combined resistance of 4Ω because the two resistirs are in series.

Applying Ohm's law to the bottom branch: I=V/R

=6/(2+2) A

=1.5A

On to the next question>>


text:
GCSE Science/Electricity multiple choice practice question 3 answer 3

I'm sorry but this is the wrong answer. Remember that each branch has the entire voltage across it.

Applying Ohm's law to the top branch:

I=V/R

=6/4 A

=1.5A

The lower branch has a combined resistance of 4Ω because the two resistirs are in series.

Applying Ohm's law to the bottom branch: I=V/R

=6/(2+2) A

=1.5A

On to the next question>>


text:
GCSE Science/Electricity multiple choice practice question 3 answer 4

I'm sorry but this is the wrong answer. Remember that each branch has the entire voltage across it.

Applying Ohm's law to the top branch:

I=V/R

=6/4 A

=1.5A

The lower branch has a combined resistance of 4Ω because the two resistirs are in series.

Applying Ohm's law to the bottom branch: I=V/R

=6/(2+2) A

=1.5A

On to the next question>>


text:
Talk:High School Mathematics Extensions/Supplementary/Complex Numbers

Please help with new content and spelling and grammar errors. I'm sick of correcting my own mistakes already!

Don't let it get you down. We all have the same problem, there are just too few of us at the moment. The page so far look really nice. Theresa knott 14:18, 3 Nov 2003 (UTC)

I hope my edits aren't unwelcome; I realize some of them are pretty major. I've been trying to order things such that they make sense to somebody who is completely unfamiliar with the concepts involved. I didn't realize I was invading someone's turf... I just got bored studying for finals, so I started clicking 'Random Page' until I found something I felt I could contribute to. :-) Glenn Willen 02:13, 10 May 2004 (UTC)

 * Don't be daft! Wikibooks is a collaboration.Theresa knott 09:04, 10 May 2004 (UTC)

Namespacing nitpick
I can see right here one of the practical problems with book-specific namespacing (and also with "breadcrumbs" like "< High School Extensions"). This page _should_ be useful for at least one of the "standard" math books in addition to HSE (which one depending on whose curriculum standards you like), but namespacing makes that awkward. Breadcrumbs are nice, but they inhibit reuse... the quickest, easiest solution I can see is to namespace more broadly (i.e. "Math:Complex number") or not at all (just "Complex number" as this was previously named), and to provide multiple sets of breadcrumbs at the top of a multiply-used page... what do people think?

Suggested To Do list
- Do you want to borrow mine? Dysprosia 09:33, 4 Nov 2003 (UTC)
 * Argand diagrams
 * Historical controversy and the "meaning" of complex numbers
 * practical uses of complex numbers eg electronics
 * Julia and Mandelbrot sets (r3m0t stares, zooms in, stares again...)

Suggested To Do list
- Do you want to borrow mine? Dysprosia 09:33, 4 Nov 2003 (UTC)
 * Argand diagrams
 * Historical controversy and the "meaning" of complex numbers
 * practical uses of complex numbers eg electronics
 * Julia and Mandelbrot sets (r3m0t stares, zooms in, stares again...)

Links you might like to take a look at

 * http://mathforum.org/johnandbetty/frame.htm A childrens book of complex numbers.

Thanks User:Xiaodai

I'm finally doing some more serious contributing!
I'm beginning to get the warm and fuzzy Wiki spirit inside me... r3m0t 00:13, 2 Jan 2004 (UTC)

 * Keep it up r3m0t! I just want to say when i drew up those plans, they are not meant to be the headings, they are there to remind me of what I want to talk about. So you might want to get creative and use your own headings, divide the chapter your way and write about things the way you desire. Don't let those plans dictate to you what you should write, or in what order you should write it. Zhuo Jia Dai

Am I right in thinking that every integer is real? In other words, is a number such as 4 + 5i an integer? r3m0t (whoops, forgot to sign, nevermind)

yes. although 4 + 5i are sometimes called Gaussian integers, we mean integers when we say integers nothing else. Zhuo Jia Dai

Some very neat work on number system there
Congratulation on some very neat work on number system there, r3m0t. Keep it up. Zhuo Jia Dai

Set theory not suitable for this level
I dont know what is taught in high school around the world. But set theory isn't taught in Australian high schoool. So one part of this chapter has to be rewritten, slightly. Xiaodai 08:59, 10 Jul 2004 (UTC)

I learned basic set theory (Venn Diagrams, sets, intersections, unions) at the age of 9. It certainly is not too complicated to be taught at this level.--206.127.177.59 21:09, 2 Apr 2005 (UTC)

Yeah, some of the set stuff is really not hard at all. But we don't learn it at all in South African primary/secondary schools. --168.209.97.34 20:13, 23 Jun 2005 (UTC) (Taejo in English Wikipedia)

DeMoivre's theorem not suitable
The student may not have had exposure to the natural logarithms at this level. It would be acceptable to include information about polar complex plane in the form z = r cos theta + ir sin theta. So a complex number in polar coordinates is like ( r, theta ). --206.127.177.59 21:09, 2 Apr 2005 (UTC)

yeah i agree, it is too hard for me (lol im crap at math anyway)

Note on curricula: In South Africa logarithms are taught to Math "Higher Grade" students in the final year. Natural logarithms are not taught in high school. --168.209.97.34 20:13, 23 Jun 2005 (UTC) (Taejo on English Wikipedia)

SOMEONE HELP ME!!!!!!!!!!!!!!!!
alright im only in year 9 so dont get too technical. theres this question that goes: find all values of a such that there are two real solutions to the equation x^3+(a-1)x^2+(9-a)x-9=0 can someone just tell me how to do cubics???

and i dunno whats calculus my teacher said i need it DONT GET TOO HARD!!!

edit: sorry its supposed to be 1 after a (thanks for telling me)

First of all, don't use all capital letters LIKE THIS, this is considered shouting in internet culture. To your question, I think either you wrote the wrong question here or your teacher made a mistakes(or you're in a pre-university class, then). At first, I thought you could take one of these approaches a12a22 − 4a0a23 − 4a13 + 18 a0a1a2 − 27a02
 * Use the formula directly(involved matrix actually)
 * Use Vieta's Substitution to work out the solution directly.
 * Differentiate it, then the discriminant of the resulting quadratic equation should be greater than zero, and one of the solution of the quadratic should also be the solution to the cubic, use vieta again to solve it.
 * Let one of the root be k, then divide the cubic equation with x-k, the resulting quadratic should be a prefect square, that is, the discriminant of it should be zero.

Now, the trouble is that all of the method above are far too complex for a grade 9 student, and even if there is an answer, it is too clumsy..(I tried, but found out the algebra expression grows very long, I also tried using a grphic calculator, and the result seems to be an irrational number) Well, once we're at cubic things never go nicely, vieta's substitution itself are too complex already itself.--Lemontea 10:51, 14 Jun 2005 (UTC)

Edit:Ok, tried method one on a equation solver and the answer match with what I found on the graph. a= 9+2.788273389 or 9-35.48344244, now both seems to be irrational numbers...

hmm, it isnt supposed to be hard. its in this noether student problem set, for year 9 (im in australia), and all the other problems are piece of cake. can you please just give me a link to vieta's substitution? i really dont no what it is. thanks

Well, in my country Vieta's substitution isn't in the core part of the course and is in the "enrichment" part. Anyway, be perpared that it is going to be a clumsy method, as in history, it is not developed until ~1541, long after the quadratic was solved.

There are more than one approaches to Vieta, but the first step is always to eliminate the quadratic terms. Let x = y + a, where a is a constant, substitute into the equation, and find "a" such that the quadratic term's coeifficient become zero.

There're two way to go after this, the first is to let y = p + q and compare coeifficient to find out p3 + q3 and (pq)3.

The second method is to let y = z + b/z. Now find the value of b such that it reduces into quadratic, which can be solved with ease.

Hope these helps.

Opps, didn't notice your correction on the problem, now it is much easier. First, expand:

$$x^3 + ax^2 - x^2 + 9x - ax - 9 = 0$$

Then factorize:

$$x^2(x-1) + 9(x-1) + ax(x-1) = 0$$

$$(x^2 + ax + 9)(x-1) = 0$$

Now either x = 1 or x2 + ax + 9 = 0. I think you should be able to do it now. --Lemontea 08:06, 24 Jun 2005 (UTC)

yea i done it, but only found -6 and 6, never thought of when one of the roots of the binomial would be one (damn)


text:
GCSE Science/Electricity multiple choice practice question 4

< Back to Questions

A plug contains three wires. Each a different colour: Red, Blue, and green/yellow. Only two of the wires in a plug carry current under normal conditions. These are:

 * The red and the green/yellow
 * The blue and the green/yellow
 * The red and the blue


text:
GCSE Science/Electricity multiple choice practice question 4 answer 3

This is the correct answer. The red is the live wire and the blue is neutral. Both wires carry current under normal working conditions.

On to the next question>>


text:
GCSE Science/Electricity multiple choice practice question 4 answer 2

I'm sorry but this is the wrong answer. The green/yellow is the Earth wire. The Earth wire does not carry any current under normal operating conditions. It is a safety device.

On to the next question>>


text:
GCSE Science/Electricity multiple choice practice question 4 answer 1

I'm sorry but this is the wrong answer. The green/yellow is the Earth wire. The Earth wire does not carry any current under normal operating conditions. It is a safety device.

On to the next question>>


text:
GCSE Science/Electricity multiple choice practice question 5

A student rubs a polythene strip with fur and suspends it from a clamp stand. She then rubs another polythene strip with fur and brings it up to the first strip. The two strips repel each other.

Look at the following four statements: Which two of the above statements are false?
 * A) One of the strips is neutral, the other charged.
 * B) The strips are both negatively charged.
 * C) The strips are both positively charged.
 * D) The strips are both neutral.
 * A and B
 * A and C
 * A and D
 * B and C
 * B and D
 * C and D


text:
GCSE Science/Forces

 * How can we describe the way things move?
 * How do we make things speed up or slow down?
 * Acceleration (advanced)
 * Force
 * Balanced and Unbalanced Forces
 * F = ma (advanced)
 * Kinetic energy
 * Momentum
 * Gravity and Astronomy

title: GCSE Science/Forces


text:
Latin/Lesson 6-Pronouns

Personal Pronouns in English
Pronouns are nouns which are used instead of another noun ('pro', in place of 'noun', noun.)

There are three categories of pronouns which are divided up into persons: 1st, 2nd, and 3rd. In addition, pronouns can be singular or plural. They are declined like all other nouns.

Table of Personal Pronouns in all of their cases: I, thou, we, ye
Note: ' is the archaic singular of the archaic plural ' - useful for distinguishing you (singular) from you (plural)

Nota Bene: the genitive is used in certain phrases like:
 * 1) memor nostrī, mindful of us
 * 2) paucī vestrum, a few of you.

For the possessive uses (my sister, your bicycle) sometimes uses the possessive adjectives:

3rd Person Pronouns
Technically, 3rd person pronouns do not exist in Latin as they do in English. However, they do have equivalents.

Adjectives modify nouns and take the gender of the noun which they modify. However, adjectives do not necessarily need a substantive present in the sentence to modify. The substantive can be presumed. In this way, '3rd person' pronouns are formed.

Example 1
Take the masculine form of the adjective 'ille'. Literally it means 'That (masculine) thing.' However one could take it for simply meaning 'he', depending on the context. Similarly, the pronoun 'iste' means 'that (masc.) thing'. Iste and ille are declined in exactly the same way, but there are a slight difference of meaning between them: 'ille' is often used with proper names for marking dignity or worth and 'iste' conveys a contemptuous sense.

Examples:

- Annibal, ille inclytus filius Amilcaris (Hannibal, that renowned Hamilcar's son).

- Iste servus improbus ante te (this bad slave in front of you).

If no substantive is provided assume words like these: 'man', 'woman', 'thing', 'idea', 'concept', 'reason' etc. Let context be your guide.

Declension of Ille (that)
Ille is often used as a kind of pronoun. In situations with multiple phrases or sentences, however, it is syntactically different from is, ea, id (see below).

For example: "Canis puero cibum dat. Is laborat in agro." means "The dog gives food to the boy. The dog works in the field".

However: "Canis puero cibum dat. Ille laborat in agro." means "The dog gives food to the boy. The boy works in the field".

Thus, ille, unlike the other pronouns makes a previous object into the subject (and vice versa).

Declension of Is, ea, id: (personal pronouns w/ translations)
Like ille, is can be used as a form of a pronoun.

Uses of the Relative Pronoun
The relative pronoun takes on the case depending on the function it serves in the relative clause. For example, in the sentence "He sees the man who has a slave," "who" is translated as nominative because it is the subject of the clause "who has a slave." The antecedent (noun to which the pronoun refers) is usually before the relative clause.

Examples of the Usage of the Relative Pronoun

 * 1) Virum videt (he/she sees) qui servum (servant) habet (he/she has).
 * He sees the man who has a slave
 * 1) Ille est vir cujus servus est malus.
 * That's the man whose slave is bad.
 * 1) Quis eum videt?
 * Who sees him?

Declension of hic, haec, hoc (meaning "this")
N.B. Hic as an adverb that means 'here'. N.B. Hic can also be used as a pronoun.

title: Latin/Lesson 6-Pronouns


text:
GCSE Science/Electricity multiple choice practice question 5 answer 1

Im sorry but this is the wrong answer. B is quite possibly true, both rods could be negative because two like charges repel. Hit back and try again.

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GCSE Science/Electricity multiple choice practice question 5 answer 2

I'm sorry but this is the wrong answer. C is quite possibly true! two positively charged rods would repel each other. Hit back and try again.

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text:
GCSE Science/Electricity multiple choice practice question 5 answer 3

This is the correct answer. Well done!

Next question>>


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GCSE Science/Electricity multiple choice practice question 5 answer 4

I'm sorry but this is the wrong answer. If both strops are negative they repel. If ther are both positive they also repel. But the question asked which statements are false. Hit back and try again.

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text:
GCSE Science/Electricity multiple choice practice question 5 answer 5

I'm sorry but this is the wrong answer. B may very well be true as "like charges repel". Hit back and try again.

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text:
GCSE Science/Electricity multiple choice practice question 5 answer 6

I'm sorry but this is the wrong answer. C may well be true. Two positively charged strips would repel.

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United States Government/The Annotated Constitution of the United States

 * ANALYSIS AND INTERPRETATION

 * ANALYSIS OF CASES DECIDED BY THE SUPREME COURT OF THE UNITED STATES TO JUNE 28, 2002.

 * PREPARED BY THE CONGRESSIONAL RESEARCH SERVICE, LIBRARY OF CONGRESS.

 * U.S. GOVERNMENT PRINTING OFFICE
 * 77–500PS
 * WASHINGTON : 2004

Acknowlegement
For assistance in preparation of this document, special thanks is extended to
 * Randall Andrews
 * Brenda Todd
 * Andrew Mendelson
 * John Bartoli

Authorization
Resolution

Introduction
Introduction

Historical Note on Formation of the Constitution
Historical Note on Formation of the Constitution

'''[The Constitution begins with a Preamble. It is followed by Articles, which are divided into Sections, which are again divided into clauses.]'''

Preamble

 * We the people of the United States, in order to form a more perfect union, establish justice, insure domestic tranquility, provide for the common defense, promote the general welfare, and secure the blessings of liberty to ourselves and our posterity, do ordain and establish this Constitution for the United States of America.

Section 1. - Legislative Power Vested

 * All legislative powers herein granted shall be vested in a Congress of the United States, which shall consist of a Senate and House of Representatives.

Section 2. - House of Representatives

 * 1) The House of Representatives shall be composed of members chosen every second year by the people of the several states, and the electors in each state shall have the qualifications requisite for electors of the most numerous branch of the state legislature.
 * 2) No person shall be a Representative who shall not
 * 3) have attained to the age of twenty five years, and
 * 4) been seven years a citizen of the United States, and who shall not,
 * 5) when elected, be an inhabitant of that state in which he shall be chosen.
 * 6) Representatives and direct taxes shall be apportioned among the several states which may be included within this union, according to their respective numbers, which shall be determined by adding to the whole number of free persons, including those bound to service for a term of years, and excluding Indians not taxed, three fifths of all other Persons.
 * 7) The actual Enumeration shall be made within three years after the first meeting of the Congress of the United States, and within every subsequent term of ten years, in such manner as they shall by law direct.
 * 8) The number of Representatives shall not exceed one for every thirty thousand, but each state shall have at least one Representative; and
 * 9) until such enumeration shall be made, the state of New Hampshire shall be entitled to chuse three, Massachusetts eight, Rhode Island and Providence Plantations one, Connecticut five, New York six, New Jersey four, Pennsylvania eight, Delaware one, Maryland six, Virginia ten, North Carolina five, South Carolina five, and Georgia three.
 * 10) When vacancies happen in the Representation from any state, the executive authority thereof shall issue writs of election to fill such vacancies.
 * 11) The House of Representatives shall choose their speaker and other officers; and shall have the sole power of impeachment.

Section 3. - Senate Elections

 * 1) The Senate of the United States shall be composed of two Senators from each state, chosen by the legislature thereof, for six years; and each Senator shall have one vote.
 * 2) Immediately after they shall be assembled in consequence of the first election, they shall be divided as equally as may be into three classes.
 * 3) The seats of the Senators of the first class shall be vacated at the expiration of the second year, of the second class at the expiration of the fourth year, and the third class at the expiration of the sixth year, so that one third may be chosen every second year; and
 * 4) if vacancies happen by resignation, or otherwise, during the recess of the legislature of any state, the executive thereof may make temporary appointments until the next meeting of the legislature, which shall then fill such vacancies.
 * 5) No person shall be a Senator who shall not have attained to the age of thirty years, and been nine years a citizen of the United States and who shall not, when elected, be an inhabitant of that state for which he shall be chosen.
 * 6) The Vice President of the United States shall be President of the Senate, but shall have no vote, unless they be equally divided.
 * 7) The Senate shall choose their other officers, and also a President pro tempore., in the absence of the Vice President, or when he shall exercise the office of President of the United States
 * 8) (Impeachment)
 * 9) The Senate shall have the sole power to try all impeachments.
 * 10) When sitting for that purpose, they shall be on oath or affirmation.
 * 11) When the President of the United States is tried, the Chief Justice shall preside: And
 * 12) no person shall be convicted without the concurrence of two thirds of the members present.
 * 13) Judgment in cases of impeachment
 * 14) shall not extend further than to removal from office, and disqualification to hold and enjoy any office of honor, trust or profit under the United States:
 * 15) but the party convicted shall nevertheless be liable and subject to indictment, trial, judgment and punishment, according to law.

Section 4. - Elections of Senators and Representatives

 * 1) The times, places and manner of holding elections for Senators and Representatives, shall be prescribed in each state by the legislature thereof; but the Congress may at any time by law make or alter such regulations, except as to the places of choosing Senators.
 * 2) The Congress shall assemble at least once in every year, and such meeting shall be on the first Monday in December, unless they shall by law appoint a different day.

Section 5. - Rules of House and Senate

 * 1) Each House shall be the judge of the elections, returns and qualifications of its own members, and a majority of each shall constitute a quorum to do business; but a smaller number may adjourn from day to day, and may be authorized to compel the attendance of absent members, in such manner, and under such penalties as each House may provide.
 * 2) Each House may determine the rules of its proceedings, punish its members for disorderly behavior, and, with the concurrence of two thirds, expel a member.
 * 3) Each House shall keep a journal of its proceedings, and from time to time publish the same, excepting such parts as may in their judgment require secrecy; and the yeas and nays of the members of either House on any question shall, at the desire of one fifth of those present, be entered on the journal.
 * 4) Neither House, during the session of Congress, shall, without the consent of the other, adjourn for more than three days, nor to any other place than that in which the two Houses shall be sitting.

Section 6. - Compensation and Privileges of Members

 * 1) The Senators and Representatives shall receive a compensation for their services, to be ascertained by law, and paid out of the treasury of the United States. They shall in all cases, except treason, felony and breach of the peace, be privileged from arrest during their attendance at the session of their respective Houses, and in going to and returning from the same; and for any speech or debate in either House, they shall not be questioned in any other place.
 * 2) No Senator or Representative shall, during the time for which he was elected, be appointed to any civil office under the authority of the United States, which shall have been created, or the emoluments whereof shall have been increased during such time: and no person holding any office under the United States, shall be a member of either House during his continuance in office.

Section 7. - Passage of Bills

 * 1) All bills for raising revenue shall originate in the House of Representatives; but the Senate may propose or concur with amendments as on other Bills.
 * 2) Every bill which shall have passed the House of Representatives and the Senate, shall, before it become a law, be presented to the President of the United States; if he approve he shall sign it, but if not he shall return it, with his objections to that House in which it shall have originated, who shall enter the objections at large on their journal, and proceed to reconsider it. If after such reconsideration two thirds of that House shall agree to pass the bill, it shall be sent, together with the objections, to the other House, by which it shall likewise be reconsidered, and if approved by two thirds of that House, it shall become a law. But in all such cases the votes of both Houses shall be determined by yeas and nays, and the names of the persons voting for and against the bill shall be entered on the journal of each House respectively. If any bill shall not be returned by the President within ten days (Sundays excepted) after it shall have been presented to him, the same shall be a law, in like manner as if he had signed it, unless the Congress by their adjournment prevent its return, in which case it shall not be a law.
 * 3) Every order, resolution, or vote to which the concurrence of the Senate and House of Representatives may be necessary (except on a question of adjournment) shall be presented to the President of the United States; and before the same shall take effect, shall be approved by him, or being disapproved by him, shall be repassed by two thirds of the Senate and House of Representatives, according to the rules and limitations prescribed in the case of a bill.

Section 8. - Scope of Legislative Power

 * 1) The Congress shall have power
 * 2) to lay and collect taxes, duties, imposts and excises;
 * 3) to pay the debts and provide for the common defense and general welfare of the United States; but all duties, imposts and excises shall be uniform throughout the United States;
 * 4) To borrow money on the credit of the United States;
 * 5) To regulate commerce with foreign nations, and among the several states, and with the Indian tribes;
 * 6) To establish
 * 7) a uniform rule of naturalization, and
 * 8) uniform laws on the subject of bankruptcies throughout the United States;
 * 9) To
 * 10) coin money, regulate the value thereof, and of foreign coin, and
 * 11) fix the standard of weights and measures;
 * 12) To provide for the punishment of counterfeiting the securities and current coin of the United States ;
 * 13) To establish
 * 14) post offices
 * 15) and post roads;
 * 16) To promote the progress of science and useful arts, by securing for limited times to authors and inventors the exclusive right to their respective writings and discoveries;
 * 17) To constitute tribunals inferior to the Supreme Court;
 * 18) To define and punish
 * 19) piracies and felonies committed on the high seas, and
 * 20) offenses against the law of nations;
 * 21) To
 * 22) declare war,
 * 23) grant letters of marque and reprisal; and
 * 24) make rules concerning captures on land and water;
 * 25) To raise and support armies, but no appropriation of money to that use shall be for a longer term than two years;
 * 26) To provide and maintain a navy;
 * 27) To make rules for the government and regulation of the land and naval forces;
 * 28) To provide for calling forth the militia to execute the laws of the union, suppress insurrections and repel invasions;
 * 29) To provide for organizing, arming, and disciplining, the militia, and for governing such part of them as may be employed in the service of the United States, reserving to the states respectively, the appointment of the officers, and the authority of training the militia according to the discipline prescribed by Congress;
 * 30) To exercise exclusive legislation in all cases whatsoever, over such District (not exceeding ten miles square) as may, by cession of particular states, and the acceptance of Congress, become the seat of the government of the United States, and to exercise like authority over all places purchased by the consent of the legislature of the state in which the same shall be, for the erection of forts, magazines, arsenals, dockyards, and other needful buildings; And
 * 31) To make all laws which shall be necessary and proper for
 * 32) carrying into execution the foregoing powers, and
 * 33) all other powers vested by this Constitution in the government of the United States,
 * 34) or in any department or officer thereof.

Section 9. - Limits on Legislative Power

 * 1) The migration or importation of such persons as any of the states now existing shall think proper to admit, shall not be prohibited by the Congress prior to the year one thousand eight hundred and eight, but a tax or duty may be imposed on such importation, not exceeding ten dollars for each person.
 * 2) The privilege of the writ of habeas corpus shall not be suspended, unless when in cases of rebellion or invasion the public safety may require it.
 * 3) No bill of attainder or ex post facto Law shall be passed.
 * 4) No capitation, or other direct, tax shall be laid, unless in proportion to the census or enumeration herein before directed to be taken.
 * 5) No tax or duty shall be laid on articles exported from any state.
 * 6) No preference shall be given by any regulation of commerce or revenue to the ports of one state over those of another: nor shall vessels bound to, or from, one state, be obliged to enter, clear or pay duties in another.
 * 7) No money shall be drawn from the treasury, but in consequence of appropriations made by law; and a regular statement and account of receipts and expenditures of all public money shall be published from time to time.
 * 8) No title of nobility shall be granted by the United States: and no person holding any office of profit or trust under them, shall, without the consent of the Congress, accept of any present, emolument, office, or title, of any kind whatever, from any king, prince, or foreign state.

Section 10. - Limits on States

 * 1) No state shall enter into any treaty, alliance, or confederation; grant letters of marque and reprisal; coin money; emit bills of credit; make anything but gold and silver coin a tender in payment of debts; pass any bill of attainder, ex post facto law, or law impairing the obligation of contracts, or grant any title of nobility.
 * 2) No state shall, without the consent of the Congress, lay any imposts or duties on imports or exports, except what may be absolutely necessary for executing its inspection laws: and the net produce of all duties and imposts, laid by any state on imports or exports, shall be for the use of the treasury of the United States; and all such laws shall be subject to the revision and control of the Congress.
 * 3) No state shall, without the consent of Congress, lay any duty of tonnage, keep troops, or ships of war in time of peace, enter into any agreement or compact with another state, or with a foreign power, or engage in war, unless actually invaded, or in such imminent danger as will not admit of delay.

Section 1. - Election, Installation, Removal

 * 1) The executive Power shall be vested in a President of the United States of America.
 * 2) He shall hold his Office during the Term of four Years, and, together with the Vice-President chosen for the same Term, be elected, as follows:
 * 3) Each State shall appoint, in such Manner as the Legislature thereof may direct, a Number of Electors, equal to the whole Number of Senators and Representatives to which the State may be entitled in the Congress:
 * 4) but no Senator or Representative, or Person holding an Office of Trust or Profit under the United States, shall be appointed an Elector.
 * 5) The Electors shall meet in their respective States, and vote by Ballot for two persons, of whom one at least shall not lie an Inhabitant of the same State with themselves.
 * 6) And they shall make a List of all the Persons voted for, and of the Number of Votes for each;
 * 7) which List they shall sign and certify, and transmit sealed to the Seat of the Government of the United States, directed to the President of the Senate.
 * 8) The President of the Senate shall, in the Presence of the Senate and House of Representatives, open all the Certificates, and the Votes shall then be counted.
 * 9) The Person having the greatest Number of Votes shall be the President, if such Number be a Majority of the whole Number of Electors appointed;
 * 10) and if there be more than one who have such Majority, and have an equal Number of Votes, then the House of Representatives shall immediately chuse by Ballot one of them for President;
 * 11) and if no Person have a Majority, then from the five highest on the List the said House shall in like Manner chuse the President.
 * 12) But in chusing the President, the Votes shall be taken by States, the Representation from each State having one Vote;
 * 13) a quorum for this Purpose shall consist of a Member or Members from two-thirds of the States, and a Majority of all the States shall be necessary to a Choice.
 * 14) In every Case, after the Choice of the President, the Person having the greatest Number of Votes of the Electors shall be the Vice President.
 * 15) But if there should remain two or more who have equal Votes, the Senate shall chuse from them by Ballot the Vice-President.
 * 16) The Congress may determine the Time of chusing the Electors, and the Day on which they shall give their Votes;
 * 17) which Day shall be the same throughout the United States.
 * 18) No person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President;
 * 19) neither shall any Person be eligible to that Office who shall not have attained to the Age of thirty-five Years, and been fourteen Years a Resident within the United States.
 * 20) In Case of the Removal of the President from Office, or of his Death, Resignation, or Inability to discharge the Powers and Duties of the said Office, the same shall devolve on the Vice President, and the Congress may by Law provide for the Case of Removal, Death, Resignation or Inability, both of the President and Vice President, declaring what Officer shall then act as President, and such Officer shall act accordingly, until the Disability be removed, or a President shall be elected.
 * 21) The President shall, at stated Times, receive for his Services, a Compensation, which shall neither be increased nor diminished during the Period for which he shall have been elected, and he shall not receive within that Period any other Emolument from the United States, or any of them.
 * 22) Before he enter on the Execution of his Office, he shall take the following Oath or Affirmation:
 * 23) "I do solemnly swear (or affirm) that I will faithfully execute the Office of President of the United States, and will to the best of my Ability, preserve, protect and defend the Constitution of the United States."

Section 2. - Presidential Power

 * 1) The President shall be Commander in Chief of the Army and Navy of the United States, and of the Militia of the several States, when called into the actual Service of the United States;
 * 2) he may require the Opinion, in writing, of the principal Officer in each of the executive Departments, upon any subject relating to the Duties of their respective Offices,
 * 3) and he shall have Power to Grant Reprieves and Pardons for Offenses against the United States, except in Cases of Impeachment.
 * 4) He shall have Power, by and with the Advice and Consent of the Senate, to make Treaties, provided two thirds of the Senators present concur;
 * 5) and he shall nominate, and by and with the Advice and Consent of the Senate, shall appoint Ambassadors, other public Ministers and Consuls, Judges of the supreme Court, and all other Officers of the United States, whose Appointments are not herein otherwise provided for, and which shall be established by Law:
 * 6) but the Congress may by Law vest the Appointment of such inferior Officers, as they think proper, in the President alone, in the Courts of Law, or in the Heads of Departments.
 * 7) The President shall have Power to fill up all Vacancies that may happen during the Recess of the Senate, by granting Commissions which shall expire at the End of their next Session.

Section 3. - State of the Union, Receive Ambassadors, Laws Faithfully Executed, Commission Officers

 * 1) He shall from time to time give to the Congress Information of the State of the Union, and recommend to their Consideration such Measures as he shall judge necessary and expedient;
 * 2) he may, on extraordinary Occasions, convene both Houses, or either of them, and in Case of Disagreement between them, with Respect to the Time of Adjournment, he may adjourn them to such Time as he shall think proper;
 * 3) he shall receive Ambassadors and other public Ministers;
 * 4) he shall take Care that the Laws be faithfully executed,
 * 5) and shall Commission all the Officers of the United States.

Section 4. - Impeachment
The President, Vice President and all civil Officers of the United States, shall be removed from Office on Impeachment for, and Conviction of, Treason, Bribery, or other high Crimes and Misdemeanors.

Section 1. - Judicial Power Vested

 * 1) The judicial power of the United States, shall be vested in one Supreme Court, and in such inferior courts as the Congress may from time to time ordain and establish.
 * 2) The judges, both of the supreme and inferior courts, shall hold their offices during good behaviour, and shall, at stated times, receive for their services, a compensation, which shall not be diminished during their continuance in office.

Section 2. - Scope of Judicial Power

 * 1) The judicial power shall extend to all cases, in law and equity, arising under this Constitution, the laws of the United States, and treaties made, or which shall be made, under their authority;
 * 2) to all cases affecting ambassadors, other public ministers and consuls;
 * 3) to all cases of admiralty and maritime jurisdiction;
 * 4) to controversies to which the United States shall be a party;
 * 5) --to controversies between two or more states;
 * 6) between a state and citizens of another state;
 * 7) between citizens of different states;
 * 8) between citizens of the same state claiming lands under grants of different states, and between a state, or the citizens thereof, and foreign states, citizens or subjects.
 * 9) In all cases affecting ambassadors, other public ministers and consuls, and those in which a state shall be party, the Supreme Court shall have original jurisdiction.
 * 10) In all the other cases before mentioned, the Supreme Court shall have appellate jurisdiction, both as to law and fact, with such exceptions, and under such regulations as the Congress shall make.
 * 11) The trial of all crimes, except in cases of impeachment, shall be by jury;
 * 12) and such trial shall be held in the state where the said crimes shall have been committed;
 * 13) but when not committed within any state, the trial shall be at such place or places as the Congress may by law have directed.

Section 3. - Treason

 * 1) Treason against the United States, shall consist only in levying war against them, or in adhering to their enemies, giving them aid and comfort.
 * 2) No person shall be convicted of treason unless on the testimony of two witnesses to the same overt act, or on confession in open court.
 * 3) The Congress shall have power to declare the punishment of treason, but no attainder of treason shall work corruption of blood, or forfeiture except during the life of the person attainted.

Section 1. - Full Faith and Credit

 * 1) Full faith and credit shall be given in each state to the public acts, records, and judicial proceedings of every other state.
 * 2) And the Congress may by general laws prescribe the manner in which such acts, records, and proceedings shall be proved, and the effect thereof.

Section 2. - Privileges and Immunities, Extradiction, Fugitive Slaves

 * 1) The citizens of each state shall be entitled to all privileges and immunities of citizens in the several states.
 * 2) A person charged in any state with treason, felony, or other crime, who shall flee from justice, and be found in another state, shall on demand of the executive authority of the state from which he fled, be delivered up, to be removed to the state having jurisdiction of the crime.
 * 3) No person held to service or labor in one state, under the laws thereof, escaping into another, shall, in consequence of any law or regulation therein, be discharged from such service or labor, but shall be delivered up on claim of the party to whom such service or labor may be due.

Section 3. - Admission of States

 * 1) New states may be admitted by the Congress into this union;
 * 2) but no new states shall be formed or erected within the jurisdiction of any other state;
 * 3) nor any state be formed by the junction of two or more states, or parts of states, without the consent of the legislatures of the states concerned as well as of the Congress.
 * 4) The Congress shall have power to dispose of and make all needful rules and regulations respecting the territory or other property belonging to the United States; and nothing in this Constitution shall be so construed as to prejudice any claims of the United States, or of any particular state.

Section 4. - Guarantees to States

 * 1) The United States shall guarantee to every state in this union a republican form of government, and shall protect each of them against invasion;
 * 2) and on application of the legislature, or of the executive (when the legislature cannot be convened) against domestic violence.

Article Five - The Amendment Process
The Congress, whenever two thirds of both houses shall deem it necessary, shall propose amendments to this Constitution, or, on the application of the legislatures of two thirds of the several states, shall call a convention for proposing amendments, which, in either case, shall be valid to all intents and purposes, as part of this Constitution, when ratified by the legislatures of three fourths of the several states, or by conventions in three fourths thereof, as the one or the other mode of ratification may be proposed by the Congress; provided that no amendment which may be made prior to the year one thousand eight hundred and eight shall in any manner affect the first and fourth clauses in the ninth section of the first article; and that no state, without its consent, shall be deprived of its equal suffrage in the Senate.

Article Six - Debts, Treaties, Oaths of Office

 * 1) All debts contracted and engagements entered into, before the adoption of this Constitution, shall be as valid against the United States under this Constitution, as under the Confederation.
 * 2) This Constitution, and the laws of the United States which shall be made in pursuance thereof;
 * 3) and all treaties made, or which shall be made, under the authority of the United States, shall be the supreme law of the land;
 * 4) and the judges in every state shall be bound thereby, anything in the Constitution or laws of any State to the contrary notwithstanding.
 * 5) The Senators and Representatives before mentioned, and the members of the several state legislatures, and all executive and judicial officers, both of the United States and of the several states, shall be bound by oath or affirmation, to support this Constitution;
 * 6) but no religious test shall ever be required as a qualification to any office or public trust under the United States.

Article Seven - Ratificiation Process
The ratification of the conventions of nine states, shall be sufficient for the establishment of this Constitution between the states so ratifying the same.

Amendment I - Religion, Speech, Press, Assembly, Petition (1791)

 * 1) Congress shall make no law respecting an establishment of religion,
 * 2) or prohibiting the free exercise thereof;
 * 3) or abridging the freedom of speech,
 * 4) or of the press;
 * 5) or the right of the people peaceably to assemble, and to petition the government for a redress of grievances.

Amendment II - Right to Bear Arms (1791)
A well regulated militia, being necessary to the security of a free state, the right of the people to keep and bear arms, shall not be infringed.

Amendment III - Quartering of Troops (1791)

 * 1) No soldier shall, in time of peace be quartered in any house, without the consent of the owner,
 * 2) nor in time of war, but in a manner to be prescribed by law.

Amendment IV - Search and Seizure (1791)
The right of the people to be secure in their persons, houses, papers, and effects, against unreasonable searches and seizures, shall not be violated, and no warrants shall issue, but upon probable cause, supported by oath or affirmation, and particularly describing the place to be searched, and the persons or things to be seized.

Amendment V - Grand Jury, Double Jeopardy, Self-Incrimination, Due Process (1791)

 * 1) No person shall be held to answer for a capital, or otherwise infamous crime, unless on a presentment or indictment of a grand jury, except in cases arising in the land or naval forces, or in the militia, when in actual service in time of war or public danger;
 * 2) nor shall any person be subject for the same offense to be twice put in jeopardy of life or limb;
 * 3) nor shall be compelled in any criminal case to be a witness against himself,
 * 4) nor be deprived of life, liberty, or property, without due process of law;
 * 5) nor shall private property be taken for public use, without just compensation.

Amendment VI - Criminal Prosecutions - Jury Trial, Right to Confront and to Counsel (1791)

 * 1) In all criminal prosecutions, the accused shall enjoy the right to a speedy and public trial, by an impartial jury of the state and district wherein the crime shall have been committed, which district shall have been previously ascertained by law, and
 * 2) to be informed of the nature and cause of the accusation;
 * 3) to be confronted with the witnesses against him;
 * 4) to have compulsory process for obtaining witnesses in his favor, and
 * 5) to have the assistance of counsel for his defense.

Amendment VII - Common Law Suits - Jury Trial (1791)
In suits at common law, where the value in controversy shall exceed twenty dollars, the right of trial by jury shall be preserved, and no fact tried by a jury, shall be otherwise reexamined in any court of the United States, than according to the rules of the common law.

Amendment VIII - Excess Bail or Fines, Cruel and Unusual Punishment (1791)
Excessive bail shall not be required, nor excessive fines imposed, nor cruel and unusual punishments inflicted.

Amendment IX - Non-Enumerated Rights (1791)
The enumeration in the Constitution, of certain rights, shall not be construed to deny or disparage others retained by the people.

Amendment X - Rights Reserved to States (1791)
The powers not delegated to the United States by the Constitution, nor prohibited by it to the states, are reserved to the states respectively, or to the people.

Amendment XI - Suits Against a State (1795)
The judicial power of the United States shall not be construed to extend to any suit in law or equity, commenced or prosecuted against one of the United States by citizens of another state, or by citizens or subjects of any foreign state.

Amendment XII - Election of President and Vice-President (1804)

 * 1) The electors shall meet in their respective states and vote by ballot for President and Vice-President, one of whom, at least, shall not be an inhabitant of the same state with themselves;
 * 2) they shall name in their ballots the person voted for as President, and in distinct ballots the person voted for as Vice-President, and
 * 3) they shall make distinct lists of all persons voted for as President, and of all persons voted for as Vice-President, and of the number of votes for each, which lists they shall sign and certify, and transmit sealed to the seat of the government of the United States, directed to the President of the Senate;
 * 4) --The President of the Senate shall, in the presence of the Senate and House of Representatives, open all the certificates and the votes shall then be counted;
 * 5) --the person having the greatest number of votes for President, shall be the President, #if such number be a majority of the whole number of electors appointed; and
 * 6) if no person have such majority, then from the persons having the highest numbers not exceeding three on the list of those voted for as President, the House of Representatives shall choose immediately, by ballot, the President.
 * 7) But in choosing the President, the votes shall be taken by states, the representation from each state having one vote; a quorum for this purpose shall consist of a member or members from two-thirds of the states, and a majority of all the states shall be necessary to a choice.
 * 8) And if the House of Representatives shall not choose a President whenever the right of choice shall devolve upon them, before the fourth day of March next following, then the Vice-President shall act as President, as in the case of the death or other constitutional disability of the President.
 * 9) The person having the greatest number of votes as Vice-President, shall be the Vice-President, if such number be a majority of the whole number of electors appointed,
 * 10) and if no person have a majority, then from the two highest numbers on the list, the Senate shall choose the Vice-President;
 * 11) a quorum for the purpose shall consist of two-thirds of the whole number of Senators, and a majority of the whole number shall be necessary to a choice.
 * 12) But no person constitutionally ineligible to the office of President shall be eligible to that of Vice-President of the United States.

Amendment XIII - Abolition of Slavery (1865)
Section 1. Neither slavery nor involuntary servitude, except as a punishment for crime whereof the party shall have been duly convicted, shall exist within the United States, or any place subject to their jurisdiction.

Section 2. Congress shall have power to enforce this article by appropriate legislation.

Amendment XIV - Privileges and Immunities, Due Process, Equal Protection, Apportionment of Representatives, Civil War Disqualification and Debt (1868)
Section 1.
 * 1) All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the state wherein they reside.
 * 2) No state shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States;
 * 3) nor shall any state deprive any person of life, liberty, or property, without due process of law;
 * 4) nor deny to any person within its jurisdiction the equal protection of the laws.

Section 2.
 * 1) Representatives shall be apportioned among the several states according to their respective numbers, counting the whole number of persons in each state, excluding Indians not taxed.
 * 2) But when the right to vote at any election for the choice of electors for President and Vice President of the United States, Representatives in Congress, the executive and judicial officers of a state, or the members of the legislature thereof, is denied to any of the male inhabitants of such state, being twenty-one years of age, and citizens of the United States, or in any way abridged, except for participation in rebellion, or other crime, the basis of representation therein shall be reduced in the proportion which the number of such male citizens shall bear to the whole number of male citizens twenty-one years of age in such state.

Section 3. No person shall be a Senator or Representative in Congress, or elector of President and Vice President, or hold any office, civil or military, under the United States, or under any state, who, having previously taken an oath, as a member of Congress, or as an officer of the United States, or as a member of any state legislature, or as an executive or judicial officer of any state, to support the Constitution of the United States, shall have engaged in insurrection or rebellion against the same, or given aid or comfort to the enemies thereof. But Congress may by a vote of two-thirds of each House, remove such disability.

Section 4.
 * 1) The validity of the public debt of the United States, authorized by law, including debts incurred for payment of pensions and bounties for services in suppressing insurrection or rebellion, shall not be questioned.
 * 2) But neither the United States nor any state shall assume or pay any debt or obligation incurred in aid of insurrection or rebellion against the United States, or any claim for the loss or emancipation of any slave; but all such debts, obligations and claims shall be held illegal and void.

Section 5. The Congress shall have power to enforce, by appropriate legislation, the provisions of this article.

Amendment XV - Rights Not to Be Denied on Account of Race (1870)
Section 1. The right of citizens of the United States to vote shall not be denied or abridged by the United States or by any state on account of race, color, or previous condition of servitude.

Section 2. The Congress shall have power to enforce this article by appropriate legislation.

Amendment XVI - Income Tax (1913)
The Congress shall have power to lay and collect taxes on incomes, from whatever source derived, without apportionment among the several states, and without regard to any census or enumeration.

Amendment XVII - Election of Senators (1913)
The Senate of the United States shall be composed of two Senators from each state, elected by the people thereof, for six years; and each Senator shall have one vote. The electors in each state shall have the qualifications requisite for electors of the most numerous branch of the state legislatures. When vacancies happen in the representation of any state in the Senate, the executive authority of such state shall issue writs of election to fill such vacancies: Provided, that the legislature of any state may empower the executive thereof to make temporary appointments until the people fill the vacancies by election as the legislature may direct. This amendment shall not be so construed as to affect the election or term of any Senator chosen before it becomes valid as part of the Constitution.

Amendment XVIII - Prohibition (1919)
Section 1. After one year from the ratification of this article the manufacture, sale, or transportation of intoxicating liquors within, the importation thereof into, or the exportation thereof from the United States and all territory subject to the jurisdiction thereof for beverage purposes is hereby prohibited.

Section 2. The Congress and the several states shall have concurrent power to enforce this article by appropriate legislation.

Section 3. This article shall be inoperative unless it shall have been ratified as an amendment to the Constitution by the legislatures of the several states, as provided in the Constitution, within seven years from the date of the submission hereof to the states by the Congress.

Amendment XIX - Women's Right to Vote (1920)
The right of citizens of the United States to vote shall not be denied or abridged by the United States or by any state on account of sex. Congress shall have power to enforce this article by appropriate legislation.

Amendment XX - Presidential Term and Succession (1933)
Section 1. The terms of the President and Vice President shall end at noon on the 20th day of January, and the terms of Senators and Representatives at noon on the 3d day of January, of the years in which such terms would have ended if this article had not been ratified; and the terms of their successors shall then begin.

Section 2. The Congress shall assemble at least once in every year, and such meeting shall begin at noon on the 3d day of January, unless they shall by law appoint a different day.

Section 3. If, at the time fixed for the beginning of the term of the President, the President elect shall have died, the Vice President elect shall become President. If a President shall not have been chosen before the time fixed for the beginning of his term, or if the President elect shall have failed to qualify, then the Vice President elect shall act as President until a President shall have qualified; and the Congress may by law provide for the case wherein neither a President elect nor a Vice President elect shall have qualified, declaring who shall then act as President, or the manner in which one who is to act shall be selected, and such person shall act accordingly until a President or Vice President shall have qualified.

Section 4. The Congress may by law provide for the case of the death of any of the persons from whom the House of Representatives may choose a President whenever the right of choice shall have devolved upon them, and for the case of the death of any of the persons from whom the Senate may choose a Vice President whenever the right of choice shall have devolved upon them.

Section 5. Sections 1 and 2 shall take effect on the 15th day of October following the ratification of this article.

Section 6. This article shall be inoperative unless it shall have been ratified as an amendment to the Constitution by the legislatures of three-fourths of the several states within seven years from the date of its submission.

Amendment XXI - Repeal of Prohibition (1933)
Section 1. The eighteenth article of amendment to the Constitution of the United States is hereby repealed.

Section 2. The transportation or importation into any state, territory, or possession of the United States for delivery or use therein of intoxicating liquors, in violation of the laws thereof, is hereby prohibited.

Section 3. This article shall be inoperative unless it shall have been ratified as an amendment to the Constitution by conventions in the several states, as provided in the Constitution, within seven years from the date of the submission hereof to the states by the Congress.

Amendment XXII - Two Term Limit on President (1951)
Section 1. No person shall be elected to the office of the President more than twice, and no person who has held the office of President, or acted as President, for more than two years of a term to which some other person was elected President shall be elected to the office of the President more than once. But this article shall not apply to any person holding the office of President when this article was proposed by the Congress, and shall not prevent any person who may be holding the office of President, or acting as President, during the term within which this article becomes operative from holding the office of President or acting as President during the remainder of such term.

Section 2. This article shall be inoperative unless it shall have been ratified as an amendment to the Constitution by the legislatures of three-fourths of the several states within seven years from the date of its submission to the states by the Congress.

Amendment XXIII - Presidential Vote in District of Columbia (1961)
Section 1. The District constituting the seat of government of the United States shall appoint in such manner as the Congress may direct:

A number of electors of President and Vice President equal to the whole number of Senators and Representatives in Congress to which the District would be entitled if it were a state, but in no event more than the least populous state; they shall be in addition to those appointed by the states, but they shall be considered, for the purposes of the election of President and Vice President, to be electors appointed by a state; and they shall meet in the District and perform such duties as provided by the twelfth article of amendment.

Section 2. The Congress shall have power to enforce this article by appropriate legislation.

Amendment XXIV - Poll Tax (1964)
Section 1. The right of citizens of the United States to vote in any primary or other election for President or Vice President, for electors for President or Vice President, or for Senator or Representative in Congress, shall not be denied or abridged by the United States or any state by reason of failure to pay any poll tax or other tax.

Section 2. The Congress shall have power to enforce this article by appropriate legislation.

Amendment XXV - Presidential Succession (1967)
Section 1. In case of the removal of the President from office or of his death or resignation, the Vice President shall become President.

Section 2. Whenever there is a vacancy in the office of the Vice President, the President shall nominate a Vice President who shall take office upon confirmation by a majority vote of both Houses of Congress.

Section 3. Whenever the President transmits to the President pro tempore of the Senate and the Speaker of the House of Representatives his written declaration that he is unable to discharge the powers and duties of his office, and until he transmits to them a written declaration to the contrary, such powers and duties shall be discharged by the Vice President as Acting President.

Section 4. Whenever the Vice President and a majority of either the principal officers of the executive departments or of such other body as Congress may by law provide, transmit to the President pro tempore of the Senate and the Speaker of the House of Representatives their written declaration that the President is unable to discharge the powers and duties of his office, the Vice President shall immediately assume the powers and duties of the office as Acting President.

Thereafter, when the President transmits to the President pro tempore of the Senate and the Speaker of the House of Representatives his written declaration that no inability exists, he shall resume the powers and duties of his office unless the Vice President and a majority of either the principal officers of the executive department or of such other body as Congress may by law provide, transmit within four days to the President pro tempore of the Senate and the Speaker of the House of Representatives their written declaration that the President is unable to discharge the powers and duties of his office. Thereupon Congress shall decide the issue, assembling within forty-eight hours for that purpose if not in session. If the Congress, within twenty-one days after receipt of the latter written declaration, or, if Congress is not in session, within twenty-one days after Congress is required to assemble, determines by two-thirds vote of both Houses that the President is unable to discharge the powers and duties of his office, the Vice President shall continue to discharge the same as Acting President; otherwise, the President shall resume the powers and duties of his office.

Amendment XXVI - Right to Vote at Age 18 (1971)
Section 1. The right of citizens of the United States, who are 18 years of age or older, to vote, shall not be denied or abridged by the United States or any state on account of age.

Section 2. The Congress shall have the power to enforce this article by appropriate legislation.

Amendment XXVII - Compensation of Members of Congress (1992)
No law, varying the compensation for the services of the Senators and Representatives, shall take effect, until an election of Representatives shall have intervened.