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The Imaginary Unit i

No real number squares to a negative, so we invent one: i, with i squared equal to minus one. See how this single rule lets us take square roots of negatives and tame powers of i.

A number whose square is negative

Every real number you square comes out zero or positive: 3^2 = 9, and (-3)^2 = 9 too. So the equation x^2 = -1 has no real solution — there is simply no point on the number line that works. For centuries that was the end of the story. Then mathematicians tried a bold move: instead of giving up, invent a new number that fills the gap.

Define the imaginary unit i by the single rule i^2 = -1. That is the whole idea. From it, the square root of a negative number suddenly has meaning: sqrt(-1) = i, sqrt(-9) = sqrt(9)·sqrt(-1) = 3i, and so on. We call numbers like 3i, -2i, or 7i imaginary numbers.

sqrt(-25) = sqrt(25 · -1)
         = sqrt(25) · sqrt(-1)
         = 5 · i
         = 5i

check:  (5i)^2 = 25 · i^2 = 25 · (-1) = -25   ✓
Pull out the positive factor, then replace sqrt(-1) with i.

Powers of i cycle

Once you know i^2 = -1, the higher powers fall out by multiplying step by step, and they repeat in a cycle of four. This is the most useful computational fact about i.

i^1 = i
i^2 = -1
i^3 = i^2 · i = -i
i^4 = i^2 · i^2 = (-1)(-1) = 1
i^5 = i^4 · i = 1 · i = i      ← cycle restarts

rule: i^n depends only on n mod 4
  n ≡ 0 → 1     n ≡ 1 → i
  n ≡ 2 → -1    n ≡ 3 → -i
The pattern i, -1, -i, 1 repeats forever.
  1. To find i^n, divide n by 4 and keep the remainder r.
  2. Read off the answer: r = 0 gives 1, r = 1 gives i, r = 2 gives -1, r = 3 gives -i.
  3. Example: i^23. Since 23 = 4·5 + 3, the remainder is 3, so i^23 = -i.