Complex Numbers

In mathematics, a complex number is an expression of the form a + bi, where a and b are real numbers, and i represents the imaginary number defined as i2 = -1. (In other words, i is the square root of -1.) The real number a is called the real part of the complex number, and the real number b is the imaginary part. For example, 3 + 2i is a complex number, with real part 3 and imaginary part 2. Complex numbers are often used in applied mathematics, control theory, signal analysis, fluid dynamics and other fields.

On this page you can read about complex operations and the complex plane. There are functions availabe for calculations with complex numbers, these are described on the page Complex functions.

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Operations

You can add up, subtract, multiply and divide complex numbers.
Operator
Description
Example
+
Plus, addition (2+6i) + (4+3i)
       Ans = 6 + 9i
-
Minus, subtraction (2+6i) - (4+3i)
       Ans = -2 + 3i
*
Multiplication (2+6i) * (4+3i)
       Ans = -10 + 30i
/
Division (-10+30i) / (4+3i)
       Ans = 2 + 6i
^
Power i^2
       Ans = -1
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Complex plane

Complex Plane A complex number can be viewed as a point or a position vector on a two-dimensional coordinate system called the complex plane.

If you project the complex number
z = a+bi as a vector in the complex plane, the angle of this vector is given by it's argument and the length is given by it's modulus (the absolute value). By mirroring z in the real axis you will get the complex conjugate of z.

You can read more about the functions for complex values on the page Complex Functions.

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Definition

Complex operations are calculated as follows.
Operation Calculation
Addition

Subtraction

Multiplication

Division

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Examples

'Examples of using complex numbers

'Definition of the complex number i
i^2
       Ans = -1
Sqrt(-1)
       Ans = i

'Sqrt(-4) = Sqrt(4)*Sqrt(-1) = Sqrt(4)*i = 2i
Sqrt(-4)
       Ans = 2i
Sqrt(-25) + 3.5
       Ans = 3.5 + 5i

'Define a few variables with complex values and use them
z1 = 2+6i
       z1 = 2 + 6i
z2 = 4+3i
       z2 = 4 + 3i
z1+z2
       Ans = 6 + 9i
z1 / (z2-3)
       Ans = 2
3 * z1
       Ans = 6 + 18i
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See Also

Complex Functions