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 [latex]i[/latex] is defined as the square root of negative 1.

So, using properties of radicals,

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

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.

If [latex]b=0[/latex], then [latex]a+bi[/latex] is a real number. If [latex]a=0[/latex] 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.

**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 1: Expressing an Imaginary Number in Standard Form

Express [latex]\sqrt{-9}[/latex] in standard form.

### Solution

[latex]\sqrt{-9}=\sqrt{9}\sqrt{-1}=3i[/latex]

In standard form, this is [latex]0+3i[/latex].