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Mathematics and Statistics · Ch 3 — Complex Numbers

Algebra of Complex Numbers — Addition, Subtraction, Multiplication and Division

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Algebra of Complex Numbers — Addition, Subtraction, Multiplication and Division

Complex numbers are added, subtracted and multiplied exactly like ordinary algebraic expressions in ii, using the single extra rule i2=−1i^{2}=-1. Division is handled with the conjugate (Section 4).

Addition and subtraction — combine like parts

(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.

Real parts combine with real parts, imaginary parts with imaginary parts.

Multiplication — expand, then replace i2i^{2}

Treat the two factors as binomials and expand:

(a+bi)(c+di)=ac+adi+bci+bd i2=ac+adi+bci−bd,(a+bi)(c+di)=ac+adi+bci+bd\,i^{2}=ac+adi+bci-bd,

so

(a+bi)(c+di)=(ac−bd)+(ad+bc)i.(a+bi)(c+di)=(ac-bd)+(ad+bc)i.

Note

The i2=−1i^{2}=-1 step is where the sign flips

The bd i2bd\,i^{2} term becomes −bd-bd, which is why the real part of the product is ac−bdac-bd (a difference), not ac+bdac+bd. Forgetting to replace i2i^{2} is the single most common error in multiplication.

Division — multiply by the conjugate …

Definition 1Sum/difference of complex numbers

(a+bi)±(c+di)=(a±c)+(b±d)i(a+bi)\pm(c+di)=(a\pm c)+(b\pm d)i — add or subtract real and imaginary par …

Definition 2Product of complex numbers

(a+bi)(c+di)=(ac−bd)+(ad+bc)i(a+bi)(c+di)=(ac-bd)+(ad+bc)i, obtained by expanding and using …

Definition 3Division by the conjugate

a+bic+di=(a+bi)(c−di)c2+d2\dfrac{a+bi}{c+di}=\dfrac{(a+bi)(c-di)}{c^{2}+d^{2}} — multiply numerator and denominator by the denominator's conjugate to mak …

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