Imaginary Unit (i)
In short: The imaginary unit, written i, is the new number built so that squaring it gives −1, opening the door to square roots of negative numbers.
Algebra 2Numbers and OperationsN-CN.A.1
What is Imaginary Unit?
No ordinary number squares to a negative, so mathematicians invented one: the imaginary unit \(i\), fixed by \(i^2 = -1\), or \(i = \sqrt{-1}\). With it, each negative number gains a square root, for example \(\sqrt{-49} = 7i\).
Its powers loop through four values: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\), and the loop restarts. To tame a big exponent, divide it by 4 and look only at what is left over.
How to spot it
- A quadratic has a negative discriminant and the answer would need the square root of a negative number.
- A radical such as \(\sqrt{-18}\) must be simplified.
- An expression has a large exponent on \(i\), such as \(i^{27}\) or \(i^{50}\).
- A problem introduces numbers of the form \(a + bi\).
The method
- Pull the minus sign out of a radical: \(\sqrt{-n} = \sqrt{n} \cdot i\), then simplify \(\sqrt{n}\).
- For a power of the imaginary unit, divide the exponent by 4 and record the leftover: 0, 1, 2 or 3.
- Read the loop: leftover 0 means \(1\), leftover 1 means \(i\), leftover 2 means \(-1\), leftover 3 means \(-i\).
- Whenever \(i^2\) appears while multiplying, swap it for \(-1\).
Imaginary Unit: solved example
Problem
Simplify \(i^{27}\) and \(\sqrt{-72}\).
Solution
For \(i^{27}\): \(27 = 4 \times 6 + 3\), so the remainder is 3 and \(i^{27} = i^3 = -i\).
For \(\sqrt{-72}\): \(\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}\), so \(\sqrt{-72} = 6\sqrt{2}\,i\). The results are \(-i\) and \(6i\sqrt{2}\).
Try it
Question 1
Simplify \(i^{50}\).
Show the answer
\(50 = 4 \times 12 + 2\), so \(i^{50} = i^2 = -1\).
Question 2
Simplify \(\sqrt{-18}\).
Show the answer
\(\sqrt{18} = 3\sqrt{2}\), so \(\sqrt{-18} = 3i\sqrt{2}\).
Imaginary Unit vs. Complex Number
The imaginary unit is a single number, \(i\), with \(i^2 = -1\). A complex number is a combination built from it, \(a + bi\), with a real part \(a\) and an imaginary part \(b\). Think of \(i\) as the building block and \(3 + 5i\) as the finished structure.
Keep going
Lessons
Worksheets and quizzes
- Complex Numbers: Adding, Subtracting and Multiplying – Worksheet
- Complex Numbers: Adding, Subtracting and Multiplying – Quiz
Related words
Whole dictionary: A–Z · Algebra 2 · Numbers and Operations