2 + 3

*i*

The imaginary number

*i*is defined as the square root of minus one, so

*i*

^{2}= -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 + 3

*i*) × ( 4 + 5

*i*)

= 2 × 4 + 2 × 5

*i*+ 3

*i*× 4 + 3

*i*× 5

*i*

= 8 + 10

*i*+ 12

*i*- 15

= -7 + 22

*i*

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:

3

^{2}= 9

3

^{-2}= 1/9 = 0.11111111...

3

^{(1/2)}= √3 = 1.73205...

(-3)

^{(1/2)}= √(-3) = 1.73205

*i*

3

^{2i}= e

^{2ln(3)i}=

*cos*(2

*ln*(3)) +

*i*sin(2

*ln*(3)) = -0.5863 + 0.8101

*i*

3

^{2+2i}= 3

^{2}× 3

^{2i}= 5.2763 + 7.2911

*i*

The complex numbers are given the symbol: ℂ

-----

* the complex numbers include all real numbers. So 2 is the complex number 2 + 0

*i*. Even though some complex multiplications give answers that are part of the real numbers, those answers are still complex numbers. For example (2 +3

*i*) × (2 - 3

*i*) = 13 (a real number) = 13 + 0

*i*(a complex number).

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