Complex Number Arithmetic

This lesson covers the four basic arithmetic operations on complex numbers: addition, subtraction, multiplication, and division. Mastering these operations is essential, as they underpin every other topic in the complex numbers unit.

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Addition and Subtraction

To add or subtract complex numbers, combine the real parts and imaginary parts separately:

$(a + bi) + (c + di) = (a + c) + (b + d)i$(a+bi)+(c+di)=(a+c)+(b+d)i

$(a + bi) - (c + di) = (a - c) + (b - d)i$(a+bi)−(c+di)=(a−c)+(b−d)i

Worked Example 1: Compute $(3 + 2i) + (1 - 5i)$(3+2i)+(1−5i).

$(3 + 2i) + (1 - 5i) = (3 + 1) + (2 + (-5))i = 4 - 3i$(3+2i)+(1−5i)=(3+1)+(2+(−5))i=4−3i

Worked Example 2: Compute $(7 - i) - (4 + 3i)$(7−i)−(4+3i).

$(7 - i) - (4 + 3i) = (7 - 4) + (-1 - 3)i = 3 - 4i$(7−i)−(4+3i)=(7−4)+(−1−3)i=3−4i

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Multiplication

To multiply complex numbers, expand using the distributive law (like multiplying brackets) and use the fact that $i^2 = -1$i2=−1:

$(a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i$(a+bi)(c+di)=ac+adi+bci+bdi2=(ac−bd)+(ad+bc)i