Complex arithmetic

Japanese school year: Math II

What you learn

This section introduces the imaginary unit and covers the fundamental rules for adding, subtracting, multiplying, and dividing complex numbers. These arithmetic skills are widely used in advanced algebra, physics, and engineering. Familiarity with real number arithmetic, polynomial expansion, and rationalizing denominators is helpful beforehand.

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Key points

This defines the imaginary unit $i$ , which equals $-1$ when squared. It serves as the foundational rule for complex numbers, allowing you to replace any occurrence of $i^2$ with $-1$ .

i^2 = -1

This formula shows how to multiply two complex numbers, $a+bi$ and $c+di$ . You expand the terms as in standard algebra, replace $i^2$ with $-1$ , and group the real part $ac-bd$ and imaginary part $(ad+bc)i$ .

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

This formula calculates the division of complex numbers. Multiplying both numerator and denominator by the conjugate $c-di$ eliminates $i$ from the denominator, turning it into a real number.

\frac{a+bi}{c+di} = \frac{(ac+bd) + (bc-ad)i}{c^2+d^2} \quad (c^2+d^2 \neq 0)

This formula shows the product of a complex number $z=a+bi$ and its complex conjugate $\overline{z}=a-bi$ . The imaginary unit $i$ cancels out, yielding a purely real value equal to the sum of squares $a^2+b^2$ (or $|z|^2$ ).

z\overline{z} = a^2 + b^2 \quad (z = a+bi)

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Practice 1Coming soon