Key takeaway: The imaginary unit, written i, is defined as the number whose square is negative 1, which makes square roots of negative numbers possible and underlies every complex number.
Grade 11 – Algebra 2N-CN.A.1Numbers and Operations
Definition of Imaginary Unit
The imaginary unit \(i\) is the new number invented to fill a gap: no real number squared is negative, so \(i\) is defined by \(i^2 = -1\), meaning \(i = \sqrt{-1}\).
With the imaginary unit, any square root of a negative number becomes a real number times \(i\); for example, \(\sqrt{-49} = 7i\). Its powers repeat in a loop of length four, so even a huge exponent shrinks to a small one once you look at what is left after dividing by 4.
Formula and notation
The defining rule and the four-step cycle:
\[ i^2 = -1, \qquad i^3 = -i, \qquad i^4 = 1 \]
For a positive real \(a\), \(\sqrt{-a} = i\sqrt{a}\). To reduce \(i^n\), divide \(n\) by 4; the leftover picks the value from the cycle in the figure below. Every use of the imaginary unit follows from the line \(i^2 = -1\).
Key facts
- The first four powers cycle through i, -1, -i and 1, then repeat forever.
- The square root of a negative number equals i times the root of its positive.
- Rewrite radicals of negatives with i before multiplying them.
- Imaginary numbers are not on the real number line; they get their own axis.
Imaginary Unit example
Simplify \(i^{47}\) and \(\sqrt{-72}\), then find \(\sqrt{-4}\cdot\sqrt{-9}\).
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Reduce the power: \(47 = 4(11) + 3\), so \(i^{47} = i^{3} = -i\).
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Factor the radicand: \(\sqrt{-72} = i\sqrt{36\cdot 2} = 6i\sqrt{2}\).
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Rewrite first, then multiply: \(\sqrt{-4}\cdot\sqrt{-9} = (2i)(3i) = 6i^2 = -6\).
\(i^{47} = \) \(-i\), \(\sqrt{-72} = \) \(6i\sqrt{2}\), and the product is \(-6\).
Common mistake
Watch out
Ben combines the radicals first: \(\sqrt{-4}\cdot\sqrt{-9} = \sqrt{36} = 6\).
Why it is wrong
The rule \(\sqrt{a}\sqrt{b} = \sqrt{ab}\) is safe only when both radicands are nonnegative. Convert to \(i\) first: \((2i)(3i) = 6i^2 = -6\). The sign changes because \(i^2 = -1\).
Related terms and lessons
Browse the glossary: A–Z list · Algebra 2 vocabulary · Numbers and Operations