Complex Numbers

i^2 = -1 extends ℝ to ℂ. Conjugates make division work.

The idea

Imaginary numbers enlarge the real number system so that negative numbers have square roots. Enlargement is the standard remedy for an unsolvable equation: negative numbers were introduced so that $x + 3 = 1$ could be solved, and fractions so that $2x = 1$ could. The equation $x^{2} = -1$ has no real solution, since every real number has a square at least $0$.

Define the imaginary unit $i$ by $i^{2} = -1.$ In every other respect $i$ follows the ordinary rules of arithmetic. A number $a + bi$ with $a$ and $b$ real is a complex number; $a$ is its real part and $b$ its imaginary part. Plotted in a plane whose horizontal axis is real and whose vertical axis is imaginary, $a + bi$ sits at the point $(a, b)$.

To add or multiply complex numbers, compute as with polynomials in $i$ — collect like terms, expand products — and replace $i^{2}$ by $-1$ wherever it appears.

Ways to work on it

• Walkthrough. i^2 = -1, complex arithmetic, conjugates.
• Practice. Powers of i, addition, or multiplication of complex numbers.
• Hardest. Divide two complex numbers and write the quotient in a + bi form.

Not sure where to start? Take the ten-question placement test.