A beginner-friendly walkthrough of the imaginary unit i, covering why the square root of a negative number needs a new kind of number, how to simplify expressions like the square root of negative 9, and the four-step repeating pattern of powers of i.
What Is the Imaginary Unit \(i\)?
Every number you squared before this lesson gave a positive result, or zero. \(3^2 = 9\), \((-3)^2 = 9\), and \(0^2 = 0\). There is no real number that, when squared, gives a negative number. So mathematicians invented one. It is called the imaginary unit, written \(i\), and it is defined by one simple rule:
\(i^2 = -1\)
Equivalently, \(i = \sqrt{-1}\). That single definition is the entire foundation of this topic. Once you accept that \(i\) exists and behaves like any other number in algebra (you can add it, multiply it, and combine it with real numbers), you can simplify expressions that used to have no answer at all.
Simplifying Square Roots of Negative Numbers
Because \(i^2 = -1\), any square root of a negative number can be rewritten using \(i\). The trick is to pull the negative sign out separately from the rest of the number, using the same idea you already know from working with mixed radicals.
For example, to simplify \(\sqrt{-9}\), split it into two factors:
Notice the pattern: pull out \(\sqrt{-1}\) and replace it with \(i\), then simplify whatever real radical is left over. Every square root of a negative number can be written in this form, a real number multiplied by \(i\).
Powers of \(i\): The Cyclic Pattern
Once you know \(i^2 = -1\), every higher power of \(i\) can be found using the same exponent rules you already use for real numbers, including the product law of exponents and the power of a power idea behind the power of a product rule. Working out the first few powers reveals something useful:
\(i^1 = i\)
\(i^2 = -1\)
\(i^3 = i^2 \times i = -1 \times i = -i\)
\(i^4 = i^2 \times i^2 = (-1) \times (-1) = 1\)
\(i^5 = i^4 \times i = 1 \times i = i\)
After the fourth power, the values start repeating. This means the powers of \(i\) cycle through only four possible results: \(i\), \(-1\), \(-i\), and \(1\), forever.
The four possible values of a power of i repeat every four steps.
To find any power of \(i\), divide the exponent by 4 and look only at the remainder. For example, to find \(i^{27}\), divide 27 by 4 to get a remainder of 3, so \(i^{27} = i^3 = -i\).
Worked Example
Simplify \(\sqrt{-16} + i^6\).
First simplify the radical: \(\sqrt{-16} = \sqrt{16}\times\sqrt{-1} = 4i\).
Next simplify the power: \(i^6 = i^4 \times i^2 = 1 \times (-1) = -1\).
Combine the results: \(4i + (-1) = -1 + 4i\).
Why Imaginary Numbers Matter
The imaginary unit \(i\) is not just a curiosity, it is the missing piece that lets every quadratic equation have a solution, even when its graph never touches the \(x\)-axis. When a real number is combined with an imaginary one, such as \(-1 + 4i\) from the example above, the result is called a complex number. From here, you can explore how complex numbers are plotted on the complex numbers and complex planes, which extends this idea into a full coordinate system.