Step beyond the real line with the imaginary unit i.
Not every square root has a real value. When we encounter √(−1), we introduce the imaginary unit i, which allows us to work with square roots of negative numbers. This expands our number system to the complex numbers.
Using i, we can simplify square roots of any negative number. The powers of i cycle with period 4, which is a useful pattern to memorize.
Powers of i (cycle of 4)
Divide the exponent by 4 and use the remainder to find any power of i.
Simplify √(−72). Factor out −1 and simplify: √(−72) = √(36 · 2 · (−1)) = 6i√2.
Remember This!
To find i^n for large n, divide n by 4 and look at the remainder: remainder 0 → 1, remainder 1 → i, remainder 2 → −1, remainder 3 → −i.
What is i^23?