We're going to allow ourselves to take square roots of negative integers. Let's write $i$ (for imaginary) for $\sqrt{-1}$, and go on from there.

A complex number is one like $4+ 2i$, or $-\frac{7}{5} + 5i$, or $\sqrt{3} - \pi i$ -- that is, of the form $a + bi$ where $a$ and $b$ are real numbers, the familiar numbers from the number line. We call $a$ the real part and $b$ the imaginary part of $a + bi$.

We can illustrate real numbers on the number line. We can extend this to show complex numbers on a plane, called the Argand diagram.

We can add and subtract complex numbers. For example,

$(4 + 2i) + (-\frac{7}{5} + 5i) = \frac{13}{5} + 7i$

and

$(4 + 2i) - (-\frac{7}{5} + 5i) = \frac{27}{5} - 3i$.

We have created an online GeoGebra worksheet for you to explore.

Use the Geogebra interactivity to find some pairs of complex numbers whose sum is a real number. What do you notice?

Can you explain it algebraically?

Use the Geogebra interactivity to find some pairs of complex numbers whose sum is an imaginary number. What do you notice?

Can you explain it algebraically?

In general, what would you need to add to $a + bi$ to get a real number? Or an imaginary number?