The Imaginary Unit

Complex Analysis · Axiom Academy

Meet i, the number whose square is −1. One definition unlocks a four-step cycle that simplifies any power of i. Over the real numbers, x^2 = -1 has no solution — every real square is zero or positive, so the parabola y = x^2 never dips below the axis. We respond not by giving up but by defining a brand-new number that does the job. 2. Powers of i Cycle, Every 4 Steps Multiplying by i is the same move every time — a quarter-turn around the unit circle. Start at 1 and keep multiplying by i : you land on i , then -1 , then -i , then back to 1 . After four steps you are exactly where you began, so the powers repeat with period 4 . Because the cycle has length 4, only the remainder of n after dividing by 4 matters. Whole loops of four bring you back to 1 and can be thrown away, so i^n equals i^r , where r is the remainder 0,1,2, or 3 . Divide the exponent by 4 : , so the remainder is 3 . Five full loops of four wash out, leaving i^ 23 = i^ 3 = -i . , remainder 1 , so i^ 17 = i^ 1 = i . , remainder 2 , so i^ 50 = i^ 2 = -1 . , remainder 3 , so i^ 23 = i^ 3 = -i . , remainder 0 , so i^ 100 = i^ 0 = 1 . 4. Negative Powers Run the Cycle Backward If multiplying by i is a quarter-turn counter-clockwise, then dividing by i — multiplying by i^ -1 — is a quarter-turn clockwise . Starting from 1 and stepping clockwise gives -i , then -1 , then i : the same four values, traversed in reverse. multiply top and bottom by i and use i^2 = -1

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