Definition: A complex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit satisfying i squared equals negative 1.
What is a complex number?
Real numbers cannot solve x2 = -9, so mathematicians added a new number, i, and built every number of the form a + bi. The value a is the real part and b is the imaginary part. Every real number is complex with b = 0.
How do you add and multiply them?
Add real parts together and imaginary parts together. To multiply, distribute as you would with binomials, then replace i2 with -1.
Worked example: (4 + 3i)(2 – 5i) = 8 – 20i + 6i – 15i2 = 8 – 14i + 15 = 23 – 14i.
Mathbot tip: collect the pieces in two piles, one with i and one without, before you simplify.
What mistake is common?
Many students forget that i2 is -1, not +1, and leave an i2 term in the final answer. Another slip is calling 5i “not a real answer” when it is a perfectly valid complex number with real part 0.