Complex Numbers

The complex numbers are the combination of the real numbers with the imaginary numbers. They are written as the real part plus the imaginary part. For example:

2 + 3i

The imaginary number i is defined as the square root of minus one, so i² = -1. Multiplication of complex numbers follows the same rules as the real numbers, you just have to keep track of each part of the multiplication:

(2 + 3i) × ( 4 + 5i)
= 2 × 4 + 2 × 5i + 3i × 4 + 3i × 5i
= 8 + 10i + 12i - 15
= -7 + 22i

We can add, subtract, multiply and divide any two complex numbers and get a complex number^* back (except dividing by zero). And, unlike the real numbers, we can raise any complex number to any other complex number without restriction.

For example:

3² = 9
3^-2 = 1/9 = 0.11111111...
3^(1/2) = √3 = 1.73205...
(-3)^(1/2) = √(-3) = 1.73205i
3²ⁱ = e^2ln(3)i = cos(2ln(3)) + i sin(2ln(3)) = -0.5863 + 0.8101i
3²⁺²ⁱ = 3² × 3²ⁱ = 5.2763 + 7.2911i

The complex numbers are given the symbol: ℂ

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* the complex numbers include all real numbers. So 2 is the complex number 2 + 0i. Even though some complex multiplications give answers that are part of the real numbers, those answers are still complex numbers. For example (2 +3i) × (2 - 3i) = 13 (a real number) = 13 + 0i (a complex number).