AI Tutor
What happens when an equation like x2=−1x^2 = -1 forces you to take the square root of a negative number?
In the 1500s, Italian mathematician Gerolamo Cardano encountered these "impossible" roots while solving cubic equations and treated them as temporary computational stepping stones.

The Imaginary Unit and Standard Form

We define the imaginary unit ii as the number whose square equals −1-1, meaning i=−1i = \sqrt{-1} and i2=−1i^2 = -1.
A complex number combines a real number and an imaginary number in the standard form a+bia + bi, where aa is the real part and bb is the imaginary coefficient.
How do we combine two complex numbers if they have both real and imaginary components?

Adding and Subtracting

To add or subtract complex numbers, you treat ii just like an algebraic variable: combine the real parts together and the imaginary parts together.
For addition, (a+bi)+(c+di)=(a+c)+(b+d)i(a + bi) + (c + di) = (a + c) + (b + d)i, and for subtraction, (a+bi)−(c+di)=(a−c)+(b−d)i(a + bi) - (c + di) = (a - c) + (b - d)i.
Let's test this with addition: (4−7i)+(−2+3i)=(4−2)+(−7+3)i=2−4i(4 - 7i) + (-2 + 3i) = (4 - 2) + (-7 + 3)i = 2 - 4i.
What happens when complex numbers multiply each other and create higher powers of ii?