Imaginary Numbers

There is no such value such that when squared it results in a negative value; we therefore classify roots of negative numbers as "imaginary."

Learning Objective

Explain why some numbers need to be classified as imaginary

Key Points

There is no such value such that when squared it results in a negative value.More specifically, solving $x^2=-1$ for $x$ results in a "number" that would not be a real number, referred to as an imaginary number.
The imaginary number, $i$, is defined as the square root of -1: $i=\sqrt{-1}$.

Terms

imaginary number
The square root of -1.
radicand
The value under the radical sign.

Full Text

A radical expression represents the root of a given quantity. What does it mean, then, if the value under the radical is negative, such as in $\displaystyle \sqrt{-1}$? There is no real value such that when multiplied by itself it results in a negative value. This means that there is no real value of $x$ that would make $x^2 =-1$ a true statement.

That is where imaginary numbers come in. When the radicand (the value under the radical sign) is negative, the root of that value is said to be an imaginary number. Specifically, the imaginary number, $i$, is defined as the square root of -1: thus, $i=\sqrt{-1}$.

We can write the square root of any negative number in terms of $i$. Here are some examples:

$\sqrt{-25}=\sqrt{25\cdot-1}=\sqrt{25}\cdot\sqrt{-1}=5i$
$\sqrt{-18} = \sqrt{2\cdot9\cdot-1} = \sqrt{2} \cdot \sqrt{9} \cdot \sqrt{-1} = 3i\sqrt{2}$

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