Definition: The imaginary unit, written i, is the number whose square is negative 1, so i equals the square root of negative 1 and powers of i repeat every four steps.
Where does i come from?
No real number squared gives a negative, so we define i by the rule i2 = -1. This single rule lets us take square roots of negative numbers: the square root of -49 is 7i, because (7i)2 = 49 times i2 = -49.
What is the pattern of its powers?
- i1 = i
- i2 = -1
- i3 = -i
- i4 = 1
After i4 the cycle starts over. To simplify a big power, divide the exponent by 4 and keep only the remainder.
Worked example: for i57, 57 = 4 times 14 + 1, so the remainder is 1 and i57 = i.
What goes wrong?
A frequent error is writing the square root of -4 times the square root of -9 as the square root of 36, which gives 6. Convert to i first: 2i times 3i = 6i2 = -6. Mathbot always rewrites radicals of negatives before multiplying.