1-1 The need for the imaginary number and the powers of i

Why number systems are extended

• We studied the natural, then integer, then rational, then real numbers; each system was enlarged so that equations unsolvable in the previous one could be solved.
• x² + 1 = 0 has no real solution, because the square of any real number is non-negative. That forced the introduction of the imaginary unit.

The imaginary unit

• i² = −1, i = √(−1), and √(−a) = i√a for a > 0.
• i¹ = i, i² = −1, i³ = −i, i⁴ = 1 — after i⁴ the values repeat, so i⁵ = i, i⁶ = −1, and so on.
• Rule for any positive integer n: divide n by 4 and use the remainder: iⁿ = i^remainder (i⁰ = 1 when the remainder is 0).

Worked example

• i²⁷ = i²⁴ · i³ = (i⁴)⁶ · (−i) = 1 · (−i) = −i.
• i⁴ + i⁴⁰ + i⁴⁰⁰ = 1 + 1 + 1 = 3.