# Express and Plot Complex Numbers

### Learning Objectives

- Express square roots of negative numbers as multiples of
*i* - Plot complex numbers on the complex plane

**imaginary number**. The imaginary number [latex]i[/latex] is defined as the square root of negative 1.

[latex]\sqrt{-1}=i[/latex]

So, using properties of radicals,[latex]{i}^{2}={\left(\sqrt{-1}\right)}^{2}=-1[/latex]

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

[latex]\begin{array}{l} \sqrt{-25}=\sqrt{25\cdot \left(-1\right)}\hfill \\ \text{ }=\sqrt{25}\sqrt{-1}\hfill \\ \text{ }=5i\hfill \end{array}[/latex]

We use 5*i*and not [latex]-\text{5}i[/latex] 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*+

*bi*where

*a*is the real part and

*bi*is the imaginary part. For example, [latex]5+2i[/latex] is a complex number. So, too, is [latex]3+4\sqrt{3}i[/latex]. 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.

### A General Note: Imaginary and Complex Numbers

A**complex number**is a number of the form [latex]a+bi[/latex] where

*a*is the real part of the complex number.*bi*is the imaginary part of the complex number.

*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.

### How To: Given an imaginary number, express it in standard form.

- Write [latex]\sqrt{-a}[/latex] as [latex]\sqrt{a}\sqrt{-1}[/latex].
- Express [latex]\sqrt{-1}[/latex] as
*i*. - Write [latex]\sqrt{a}\cdot i[/latex] in simplest form.

### Example: Expressing an Imaginary Number in Standard Form

Express [latex]\sqrt{-9}[/latex] in standard form.Answer: [latex-display]\sqrt{-9}=\sqrt{9}\sqrt{-1}=3i[/latex-display] In standard form, this is [latex]0+3i[/latex].

### Try It

Express [latex]\sqrt{-24}[/latex] in standard form.Answer: [latex]\sqrt{-24}=0+2i\sqrt{6}\\[/latex]

## Licenses & Attributions

### CC licensed content, Original

- Revision and Adaptation.
**Provided by:**Lumen Learning**License:**CC BY: Attribution.

### CC licensed content, Shared previously

- Introduction to Complex Numbers.
**Authored by:**Sousa, James.**License:**CC BY: Attribution. - Question ID 61706.
**Authored by:**Day, Alyson.**License:**CC BY: Attribution.**License terms:**IMathAS Community License CC-BY + GPL. - Question ID 65709.
**Authored by:**Kaslik,Pete, mb Lippman,David.**License:**CC BY: Attribution.**License terms:**IMathAS Community License CC-BY + GPL. - College Algebra.
**Provided by:**OpenStax**Authored by:**Abramson, Jay et al..**Located at:**https://openstax.org/books/college-algebra/pages/1-introduction-to-prerequisites.**License:**CC BY: Attribution.**License terms:**Download for free at http://cnx.org/contents/[email protected].