Complex number
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