Difference between revisions of "Complex number"
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− | The '''complex numbers''' arise when we try to solve [[equation]]s such as <math> x^2 = -1 </math>. | + | The '''complex numbers''' arise when we try to solve [[equation]]s such as <math> x^2 = -1 </math>. |
− | 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> is the [[imaginary unit]]. 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, since <math>a = a + 0i</math>, but it is much larger. Every complex number <math> z </math> has a ''[[real part]]'' denoted <math>\Re(z)</math> or <math>\mathrm{Re}(z)</math> and an ''[[imaginary part]]'' denoted <math> \Im(z)</math> or <math> \mathrm{Im}(z)</math>. Note that the imaginary part of a complex number is real: for example, <math>\Im(3 + 4i) = 4</math>. So, if <math>z\in \mathbb C</math>, we can write <math>z=\mathrm{Re}(z)+i\mathrm{Im}(z)</math>. (<math>z</math> and <math>w</math> are traditionally used in place of <math>x</math> and <math>y</math> as [[variable]]s when dealing with complex numbers, while <math>x</math> and <math>y</math> (and frequently also <math>a</math> and <math>b</math>) are used to represent real values such as the real and imaginary parts of complex numbers. This [[mathematical convention]] is often broken when it is inconvenient, so be sure that you know what set variables are taken from when dealing with the complex numbers.) | + | ==Derivation== |
+ | 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.) | ||
+ | |||
+ | ==Formal Definition== | ||
+ | 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> is the [[imaginary unit]]. 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, since <math>a = a + 0i</math>, but it is much larger. | ||
+ | |||
+ | ==Parts== | ||
+ | Every complex number <math> z </math> has a ''[[real part]]'' denoted <math>\Re(z)</math> or <math>\mathrm{Re}(z)</math> and an ''[[imaginary part]]'' denoted <math> \Im(z)</math> or <math> \mathrm{Im}(z)</math>. Note that the imaginary part of a complex number is real: for example, <math>\Im(3 + 4i) = 4</math>. So, if <math>z\in \mathbb C</math>, we can write <math>z=\mathrm{Re}(z)+i\mathrm{Im}(z)</math>. (<math>z</math> and <math>w</math> are traditionally used in place of <math>x</math> and <math>y</math> as [[variable]]s when dealing with complex numbers, while <math>x</math> and <math>y</math> (and frequently also <math>a</math> and <math>b</math>) are used to represent real values such as the real and imaginary parts of complex numbers. This [[mathematical convention]] is often broken when it is inconvenient, so be sure that you know what set variables are taken from when dealing with the complex numbers.) | ||
As you can see, complex numbers enable us to remove the restriction of <math>x\ge 0</math> from the [[domain]] of the [[function]] <math>f(x)=\sqrt{x}</math> (although some additional considerations are necessary). | As you can see, complex numbers enable us to remove the restriction of <math>x\ge 0</math> from the [[domain]] of the [[function]] <math>f(x)=\sqrt{x}</math> (although some additional considerations are necessary). | ||
− | |||
== Operations == | == Operations == | ||
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== Simple Example == | == Simple Example == | ||
− | If <math>z=a+bi</math> and <math> | + | If <math>z=a+bi</math> and <math>w = c + di</math>, |
* <math>\mathrm{Re}(z)=a</math>,<math>\mathrm{Im}(z)=b</math> | * <math>\mathrm{Re}(z)=a</math>,<math>\mathrm{Im}(z)=b</math> |
Revision as of 18:06, 25 November 2007
The complex numbers arise when we try to solve equations such as .
Contents
[hide]Derivation
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, , such that . If we add this new number to the reals, we will have solutions to . It turns out that in the system that results from this addition, we are not only able to find the solutions of but we can now find all solutions to every polynomial. (See the Fundamental Theorem of Algebra for more details.)
Formal Definition
We are now ready for a more formal definition. A complex number is a number of the form where and is the imaginary unit. The set of complex numbers is denoted by . The set of complex numbers contains the set of the real numbers, since , but it is much larger.
Parts
Every complex number has a real part denoted or and an imaginary part denoted or . Note that the imaginary part of a complex number is real: for example, . So, if , we can write . ( and are traditionally used in place of and as variables when dealing with complex numbers, while and (and frequently also and ) are used to represent real values such as the real and imaginary parts of complex numbers. This mathematical convention is often broken when it is inconvenient, so be sure that you know what set variables are taken from when dealing with the complex numbers.)
As you can see, complex numbers enable us to remove the restriction of from the domain of the function (although some additional considerations are necessary).
Operations
- Addition
- Subtraction
- Multiplication
- Division
- Absolute value/Modulus/Magnitude (denoted by ). This is the distance from the origin to the complex number in the complex plane.
- Conjugation
- The argument function
Simple Example
If and ,
- ,
- ,
Topics
Problems
- AIME