Complex Arithmetic

Background - Algebraic

Addition and Subtraction

Rule: (a + bi) + (c + di) = (a + c) + (b + d)i

Example: (1 + 3i) + (5 - 7i) = (1 + 5) + (3 - 7)i = 6 - 4i

Example: (2 + (1/2)i) - (-1 + 2i) = (2 - (-1)) + (1/2 - 2)i = 3 - (3/2)i

Multiplication

Rule: (a + bi)*(c + di) = ac + a(di) + (bi)c + (bi)(di) = (ac - bd) + (ad + bc)i

Example: (1 + 3i)*(5 - 7i) = (1*5 - 3*(-7)) + (1*(-7) + 3*5)i = 26 + 8i

Example: (2 + (1/2)i)*(-1 + 2i) = (2*(-1) - (1/2)*2) + (2*2 + (1/2)*(-1))i = -3 + (7/2)i

Division is a bit trickier, and uses the fact that (a + bi)*(a - bi) = a2 + b2.

The number a - bi is called the conjugate of a + bi.

Division

Rule: (a + bi)/(c + di) = ((a + bi)/(c + di))*((c - di)/(c - di))

= ((a + bi)*(c - di))/((c + di)*(c - di)) = (ac + bd)/(c2 + d2) + ((bc - ad)/(c2 + d2))i

Example: (1 + 3i)/(5 - 7i) = (1*5 + 3*(-7))/(52 + 72) + ((3*5 - 1*(-7))/(52 + 72))i = -(8/37) + (11/37)i

Example: (2 + (1/2)i)/(-1 + 2i) = (2*(-1) + (1/2)*2)/(12 + 22) + (((1/2)*(-1) - 2*2)/(12 + 22))i = -(1/5) - (9/10)i

If you think you need some practice, refer to the algebraic problems in the sample.

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