i
High School
Definition
An italicized i is used to represent an imaginary number. Imaginary numbers give you a negative result when squared. Usually, any number you square should come out positive. Therefore i is used to help us imagine that we would be able to get a negative outcome.
Worked examples
\(i^2 = -1\)
This is the defining property of i — when you square it, you get negative one.
\(\sqrt{-16} = \sqrt{16 \cdot (-1)} = \sqrt{16} \cdot \sqrt{-1} = 4i\)
Use i to simplify square roots of negative numbers by factoring out \(\sqrt{-1} = i\).
\((3 + 2i)^2 = 9 + 12i + 4i^2 = 9 + 12i - 4 = 5 + 12i\)
When expanding with i, replace \(i^2\) with \(-1\) to simplify.
Common mistakes
- \(i^2 = 1\) → \(i^2 = -1\) The imaginary unit i is defined so its square is negative one, not positive one.
- \(\sqrt{-9} = -3\) → \(\sqrt{-9} = 3i\) The square root of a negative number is imaginary; factor out i, not a negative sign.
- \(i = \sqrt{-1}\) so \(i\) is undefined → \(i\) is defined as the imaginary unit where \(i^2 = -1\) i extends the real numbers; it's a valid mathematical object, not an error.
Where you'll use it next
You'll use i to solve quadratic equations with no real solutions, work with complex numbers in algebra and precalculus, analyze AC circuits in physics, and study signal processing in engineering.
Found in 1 StudyPug lesson
Introduction to imaginary numbers
12th Grade12thGrade 12 Math
Have you ever run into a situation where you want to find the square root of a negative number, but the calculator gives you nothing but a big error? Well, we will explore the meaning of this error from a different perspective. In this section, we will introduce the notion of imaginary numbers, and how to simplify them.
See also
Reviewed by Pat Cheng, M.Ed. — StudyPug Curriculum Lead · Last updated June 6, 2026