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 any 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 -1.

\sqrt{-1}=i

So, using properties of radicals,

i^{2}=(\sqrt{-1})^{2}=-1

We can write the square root of any negative number as a multiple of i. Consider the square root of -49.

\begin{aligned}\sqrt{-49} &=\sqrt{49 \cdot(-1)} \\&=\sqrt{49} \sqrt{-1} \\&=7 i\end{aligned}

We use 7 i and not -7 i because the principal root of 49 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 is the imaginary part. For example, 5+2 i is a complex number. So, too, is 3+4 i \sqrt{3}.



Imaginary numbers differ from real numbers in that a squared imaginary number produces a negative real number. Recall that when a positive real number is squared, the result is a positive real number and when a negative real number is squared, the result is also a positive real number. Complex numbers consist 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 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 a pure imaginary number. An imaginary number is an even root of a negative number.


HOW TO

Given an imaginary number, express it in the standard form of a complex number.

1. Write \sqrt{-a} as \sqrt{a} \sqrt{-1}.2. Express \sqrt{-1} as i.
3. Write \sqrt{a} \cdot i in simplest form.

 

EXAMPLE 1

Expressing an Imaginary Number in Standard Form
Express \sqrt{-9} in standard form.


Solution

\begin{aligned}\sqrt{-9} &=\sqrt{9} \sqrt{-1} \\&=3 i\end{aligned}

In standard form, this is 0+3 i.


TRY IT #1

Express \sqrt{-24} in standard form.