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Imaginary Unit — Definition, Formula & Examples

The imaginary unit, written as ii, is a number defined so that i2=−1i^2 = -1. It lets you take square roots of negative numbers, which aren't possible using only real numbers.

The imaginary unit ii is the element of the complex number system C\mathbb{C} satisfying the equation i2=−1i^2 = -1. Every complex number can be expressed in the form a+bia + bi, where aa and bb are real numbers and ii is the imaginary unit.

Key Formula

i=−1,i2=−1i = \sqrt{-1}, \quad i^2 = -1
Where:
  • ii = The imaginary unit

How It Works

The powers of ii cycle in a repeating pattern of four: i1=ii^1 = i, i2=−1i^2 = -1, i3=−ii^3 = -i, i4=1i^4 = 1, and then the cycle restarts. To simplify any power ini^n, divide nn by 4 and use the remainder to find the result. You can also use ii to rewrite square roots of negative numbers: −k=ik\sqrt{-k} = i\sqrt{k} for any positive real number kk. This conversion is the first step in most problems involving complex arithmetic.

Worked Example

Problem: Simplify i27i^{27}.
Divide the exponent by 4: Find the remainder when 27 is divided by 4.
27÷4=6 remainder 327 \div 4 = 6 \text{ remainder } 3
Use the remainder: Since the remainder is 3, i27=i3i^{27} = i^3.
i3=i2⋅i=(−1)(i)=−ii^3 = i^2 \cdot i = (-1)(i) = -i
Answer: i27=−ii^{27} = -i

Why It Matters

The imaginary unit is essential in Algebra 2 and Precalculus whenever you solve quadratic equations with negative discriminants. In electrical engineering, ii (often written jj) is used to model alternating current circuits. It also appears throughout physics and signal processing to represent oscillations and waves.

Common Mistakes

Mistake: Writing −4=2\sqrt{-4} = 2 instead of −4=2i\sqrt{-4} = 2i.
Correction: A negative number under a square root cannot produce a real result. Factor out ii first: −4=4⋅−1=2i\sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2i.