Imaginary numbers

Advanced algebra ii: activities Page 1 / 1

This module provides sample problems related to i and imaginary numbers.

Explain, using words and equations, why the equation $x^{2} = –1$ has no answer, but $x^{3} = –1$ does.

OK, so now we are going to use our imaginations. (Didn’t think we were allowed to do that in math class, did you?) Suppose there were an answer to $x^{2} = –1$ ? Obviously it wouldn’t be a number that we are familiar with (such as 5, $- \frac{3}{4}$ , or $π$ ). So, let’s just give it a new name: $i$ , because it’s imaginary . What would it be like?

Definition of i

The definition of the imaginary number

i

is that it is the square root of

–1

$i = \sqrt{- 1}$ or, equivalently, $i^{2} = –1$

Based on that definition, answer the following questions. In each case, don’t just guess—give a good mathematical reason why the answer should be what you say it is!

What is $i (–i)$ ? (*Remember that $–i$ means $–1 \times i$ .)

$i^{1}$
$i^{2}$
$i^{3}$
$i^{4}$
$i^{5}$
$i^{6}$
$i^{7}$
$i^{8}$
$i^{9}$
$i^{10}$
$i^{11}$
$i^{12}$

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$i^{100}$
$i^{101}$
$i^{102}$
$i^{103}$
$i^{104}$

Imaginary numbers

Definition of i

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