Complex Numbers

A complex number can be written in the form a + bi where a and b are real numbers and i² = –1.

Examples: The following are examples of complex numbers.

2 + 3i

3/2 + (–2i) = 3/2 – 2i

 –π + 2.6i

0 + 5i = 5i                    This is called a pure imaginary number.

17 + 0i = 17                 This is called a real number.

0 + 0i = 0                     This is zero.

Powers of i

By definition, when we raise any number to the 0 power, we get 1. So in particular, we have i⁰ = 1.

Similarly, when we raise any number to the power of 1, we just get that number. So i¹ = i.

By the definition of i, we have i² = –1.

i³ = (i²)(i) = (–1)i = –i.

i⁴ = (i²)(i²) = (–1)(–1) = 1.

i⁵ = (i⁴)(i) = (1)(i) = i.

⋯

Notice that the pattern begins to repeat.

Starting with i⁰ = 1, we have

i⁰ = 1               i¹ = i              i² = –1             i³ = –i

i⁴ = 1               i⁵ = i              i⁶ = –1             i⁷ = –i

i⁸ = 1               i⁹ = i              i¹⁰ = –1             i¹¹ = –i

⋯

In other words, when we raise i to a nonnegative integer, there are only four possible answers:

1, i, –1, or –i

To decide which of these values is correct, we can find the remainder upon dividing the exponent by 4.

Example: Compute i⁵³.

I will provide a solution to this question tomorrow. Meanwhile, please post your attempted solutions in the comments below.

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