Difference between revisions of "Complex number"
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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.) | |
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+ | 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]]. | ||
As you can see, complex numbers enable us to remove the restriction of <math>x\ge 0</math> for the domain of <math>f(x)=\sqrt{x}</math>. | As you can see, complex numbers enable us to remove the restriction of <math>x\ge 0</math> for the domain of <math>f(x)=\sqrt{x}</math>. | ||
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== See also == | == See also == | ||
− | + | * [[Fundamental Theorem of Algebra]] | |
* [[Trigonometry]] | * [[Trigonometry]] | ||
* [[Real numbers]] | * [[Real numbers]] |
Revision as of 21:14, 7 July 2006
We come about the idea of complex numbers when we trying to solve equations such as . 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, , such that . Then we will have solutions to . It turns out that not only are we able to find the solutions of but we can now find all solutions to any polynomial using . (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 where and . The set of complex numbers is denoted by . The set of complex numbers contains the set of the real numbers, but is much larger. Every complex number has a real part, denoted by , or simply , and an imaginary part, denoted by , or simply . So, if , we can write , where is the imaginary unit.
As you can see, complex numbers enable us to remove the restriction of for the domain of .
The letters and are usually used to denote complex numbers.
Contents
[hide]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.
Simple Example
If and w = c+di,
- ,
- ,
Topics
Problems
- AIME