The real & complex number systems
The idea
The number system you use grew in layers: whole numbers, negatives, fractions, then irrationals like √2 and π, which together fill the real number line completely. Complex numbers add one more layer by introducing i, a number defined by i² = −1. Every complex number has the form a + bi with real numbers a and b, and the rule i² = −1 is the only new fact you need — everything else is ordinary algebra. The payoff is that every quadratic equation now has solutions, even ones like x² + 9 = 0 with no real roots.
Treat a + bi like a binomial: add and subtract by combining like parts, multiply by distributing and replacing i² with −1. Division uses the conjugate: multiplying top and bottom by a − bi turns the denominator real, because (a + bi)(a − bi) = a² + b². The misconception to drop is that imaginary numbers are fake or optional decorations. They are coordinates: a + bi is the point (a, b) in the complex plane, and engineers lean on them daily to describe rotations and oscillations.
Worked example
Write the quotient (5 − 2i)/(3 + i) in the standard form a + bi, where a and b are real numbers.
- Division by a complex number is undone by its conjugate. Multiply the top and bottom by 3 − i, the conjugate of the denominator, which changes the form but not the value of the fraction.
- Expand the numerator: (5 − 2i)(3 − i) = 15 − 5i − 6i + 2i². Since i² = −1, the last term is −2, giving 15 − 11i − 2 = 13 − 11i.
- Expand the denominator: (3 + i)(3 − i) = 9 − i² = 9 + 1 = 10, a real number — exactly the point of using the conjugate.
- Divide each part by 10: the quotient is 13/10 − (11/10)i, or 1.3 − 1.1i.
- Check by multiplying back: (1.3 − 1.1i)(3 + i) = 3.9 + 1.3i − 3.3i − 1.1i² = 3.9 − 2i + 1.1 = 5 − 2i, the original numerator, so the answer is right.
Answer. The quotient is 1.3 − 1.1i, that is, a = 13/10 and b = −11/10.
Check your understanding
- Why does multiplying a complex number by its conjugate always produce a real number, and how is that like rationalizing a denominator with radicals?
- How would you explain to a friend what i actually is, without using the word imaginary?
- Where do the real numbers sit inside the complex plane, and what does that picture say about how the two systems relate?
- What pattern do the powers i, i², i³, i⁴ follow, and how could that repeating cycle help you simplify i^37 quickly?
Build the foundations first
The real & complex number systems builds on these concepts. If any feel shaky, start there.