Complex Numbers and the Complex Plane

The big idea

A complex number has the form \(a+bi\), where \(a\) and \(b\) are real numbers and \(i^2=-1\). The letter \(i\) is not a variable waiting to be solved; it names the imaginary unit. A complex number can also be treated as a point. Plot \(a+bi\) at \((a,b)\), with the real part \(a\) horizontally and the coefficient of \(i\) vertically. This two-way view lets algebra create a motion or orbit on a plane.

Worked example

Square \(1+i\):

\[ (1+i)^2=1+2i+i^2=1+2i-1=2i. \]

The starting number \(1+i\) is the point \((1,1)\). Its square \(2i\) is the point \((0,2)\). The magnitude symbol measures distance from zero:

\[ |1+i|=\sqrt{1^2+1^2}=\sqrt2, \qquad |2i|=\sqrt{0^2+2^2}=2. \]

This distance formula is the ordinary Pythagorean theorem applied to the point’s horizontal and vertical coordinates.

How children may show it

Students may expand products line by line, use the identity \((a+bi)^2=(a^2-b^2)+2ab\,i\), plot labeled points, or draw arrows between successive points. Ask them to keep the algebraic form and coordinate form side by side: \(a+bi\leftrightarrow(a,b)\). For magnitude, a small right triangle from the origin can make the square root formula visible.

Common mix-up

The most common arithmetic error is treating \(i^2\) as \(1\) or leaving it unreduced. Circle each \(i^2\) before simplifying and replace it with \(-1\). Students may also plot \(a+bi\) at \((b,a)\); return to the words “real part first, \(i\)-part second.” Finally, magnitude is a nonnegative real distance, not the sum \(a+b\) and not another complex number.

Try it together

Choose \(z=-1+i\). Plot it, square it, simplify using \(i^2=-1\), and plot the result. Then find the magnitude of both numbers. Repeat with \(z=-i\). If a next-step rule ever returns exactly to a point already visited, trace what the same rule must do afterward. This prepares the idea that a short repeated cycle can describe an infinite future.