Difference between revisions of "Complex number"

Revision as of 09:57, 8 July 2006

The complex numbers arise when we try to solve equations such as $x^2 = -1$ . We know (from the trivial inequality) that the square of a real number cannot be negative, so this equation has no solutions in the real numbers. However, it is possible to define a number, $i$ , such that $i = \sqrt{-1}$ . If we add this new number to the reals, we will have solutions to $x^2 = -1$ . It turns out that in the system that results from this addition, we are not only able to find the solutions of $x^2 = -1$ but we can now find all solutions to every polynomial. (See the Fundamental Theorem of Algebra for more details.)

We are now ready for a more formal definition. A complex number is a number of the form $a + bi$ where $a,b\in \mathbb{R}$ and $i = \sqrt{-1}$ . The set of complex numbers is denoted by $\mathbb{C}$ . The set of complex numbers contains the set $\mathbb{R}$ of the real numbers, but is much larger. Every complex number has a real part, denoted by $\Re$ , or simply $\mathrm{Re}$ , and an imaginary part, denoted by $\Im$ , or simply $\mathrm{Im}$ . So, if $z\in \mathbb C$ , we can write $z=\mathrm{Re}(z)+i\mathrm{Im}(z)$ , where $i$ is the imaginary unit.

As you can see, complex numbers enable us to remove the restriction of $x\ge 0$ for the domain of $f(x)=\sqrt{x}$ .

The letters $z$ and $\omega$ are usually used to denote complex numbers.

Operations

Addition
Subtraction
Multiplication
Division
Absolute value/Modulus/Magnitude (denoted by $|z|$ ). This is the distance from the origin to the complex number in the complex plane.

Simple Example

If $z=a+bi$ and w = c+di,

$\mathrm{Re}(z)=a$ , $\mathrm{Im}(z)=b$
$|z|=\sqrt{a^2+b^2}$
$\mathrm{Re}(w)=c$ , $\mathrm{Im}(w)=d$
$|w|=\sqrt{c^2+d^2}$
$z+w=(a+c)+(b+d)i$
$z-w=(a-c)+(b-d)i$

Topics

Problems

AIME
- 1984 #8
- 1985 #3
- 1988 #11
- 1989 #14
- 1990 #10
- 1992 #10
- 1994 #8
- 1994 #13
- 1995 #5
- 1996 #11
- 1997 #11
- 1997 #14
- 1998 #13
- 1999 #9
- 2000 Alternate #9
- 2002 #12
- 2004 #13
- 2005 Alternate #9

Revision as of 21:14, 7 July 2006 (view source) Joml88 (talk \| contribs) ← Older edit		Revision as of 09:57, 8 July 2006 (view source) JBL (talk \| contribs) Newer edit →
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−	~~We come about the idea of~~ '''complex numbers''' when we ~~trying~~ to solve ~~equations~~ such as <math> x^2 = -1 </math>. We know that ~~it's absurd for~~ the square of a real number to be negative so this equation has no solutions in real numbers. However, ~~if we~~ define a number, <math> i </math>, such that <math> i = \sqrt{-1} </math>. ~~Then~~ we will have solutions to <math> x^2 = -1 </math>. It turns out that not only ~~are we~~ able to find the solutions of <math> x^2 = -1 </math> but we can now find ''all'' solutions to ''~~any~~'' polynomial ~~using <math> i </math>~~. (See the [[Fundamental Theorem of Algebra]] for more details.)	+	The '''complex numbers''' arise when we try to solve [[equation]]s such as <math> x^2 = -1 </math>. We know (from the [[trivial inequality]]) that the square of a [[real number]] cannot be [[negative]], so this equation has no solutions in the real numbers. However, it is possible to define a number, <math> i </math>, such that <math> i = \sqrt{-1} </math>. If we add this new number to the reals, we will have solutions to <math> x^2 = -1 </math>. It turns out that in the system that results from this addition, we are not only able to find the solutions of <math> x^2 = -1 </math> but we can now find ''all'' solutions to ''every'' polynomial. (See the [[Fundamental Theorem of Algebra]] for more details.)

	We are now ready for a more formal definition. A complex number is a number of the form <math> a + bi </math> where <math> a,b\in \mathbb{R} </math> and <math> i = \sqrt{-1} </math>. The set of complex numbers is denoted by <math>\mathbb{C}</math>. The set of complex numbers contains the set <math>\mathbb{R}</math> of the [[real number]]s, but is much larger. Every complex number has a '''real part''', denoted by <math>\Re</math>, or simply <math>\mathrm{Re}</math>, and an '''imaginary part''', denoted by <math>\Im</math>, or simply <math>\mathrm{Im}</math>. So, if <math>z\in \mathbb C</math>, we can write <math>z=\mathrm{Re}(z)+i\mathrm{Im}(z)</math>, where <math>i</math> is the [[imaginary unit]].		We are now ready for a more formal definition. A complex number is a number of the form <math> a + bi </math> where <math> a,b\in \mathbb{R} </math> and <math> i = \sqrt{-1} </math>. The set of complex numbers is denoted by <math>\mathbb{C}</math>. The set of complex numbers contains the set <math>\mathbb{R}</math> of the [[real number]]s, but is much larger. Every complex number has a '''real part''', denoted by <math>\Re</math>, or simply <math>\mathrm{Re}</math>, and an '''imaginary part''', denoted by <math>\Im</math>, or simply <math>\mathrm{Im}</math>. So, if <math>z\in \mathbb C</math>, we can write <math>z=\mathrm{Re}(z)+i\mathrm{Im}(z)</math>, where <math>i</math> is the [[imaginary unit]].

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