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In this section, you will:
  • Express square roots of negative numbers as multiples of  i.
  • Plot complex numbers on the complex plane.
  • Add and subtract complex numbers.
  • Multiply and divide complex numbers.

The study of mathematics continuously builds upon itself. Negative integers, for example, fill a void left by the set of positive integers. The set of rational numbers, in turn, fills a void left by the set of integers. The set of real numbers fills a void left by the set of rational numbers. Not surprisingly, the set of real numbers has voids as well. For example, we still have no solution to equations such as

x 2 + 4 = 0

Our best guesses might be +2 or –2. But if we test +2 in this equation, it does not work. If we test –2, it does not work. If we want to have a solution for this equation, we will have to go farther than we have so far. After all, to this point we have described the square root of a negative number as undefined. Fortunately, there is another system of numbers that provides solutions to problems such as these. In this section, we will explore this number system and how to work within it.

Expressing square roots of negative numbers as multiples of i

We know how to find the square root of any positive real number. In a similar way, we can find the square root of a negative number. The difference is that the root is not real. If the value in the radicand is negative, the root is said to be an imaginary number     . The imaginary number i is defined as the square root of negative 1.

1 = i

So, using properties of radicals,

i 2 = ( 1 ) 2 = 1

We can write the square root of any negative number as a multiple of i . Consider the square root of –25.

25 = 25 ( 1 )           = 25 1           = 5 i

We use 5 i and not 5 i because the principal root of 25 is the positive root.

A complex number is the sum of a real number and an imaginary number. A complex number is expressed in standard form when written a + b i where a is the real part and b i is the imaginary part. For example, 5 + 2 i is a complex number. So, too, is 3 + 4 3 i .

Showing the real and imaginary parts of 5 + 2i. In this complex number, 5 is the real part and 2i is the complex part.

Imaginary numbers are distinguished from real numbers because a squared imaginary number produces a negative real number. Recall, when a positive real number is squared, the result is a positive real number and when a negative real number is squared, again, the result is a positive real number. Complex numbers are a combination of real and imaginary numbers.

Imaginary and complex numbers

A complex number    is a number of the form a + b i where

  • a is the real part of the complex number.
  • b i is the imaginary part of the complex number.

If b = 0 , then a + b i is a real number. If a = 0 and b is not equal to 0, the complex number is called an imaginary number . An imaginary number is an even root of a negative number.

Given an imaginary number, express it in standard form.

  1. Write a as a 1 .
  2. Express 1 as i .
  3. Write a i in simplest form.

Expressing an imaginary number in standard form

Express 9 in standard form.

9 = 9 1 = 3 i

In standard form, this is 0 + 3 i .

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Express 24 in standard form.

24 = 0 + 2 i 6

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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