Imaginary and complex numbers
Imaginary numbers are built on i, defined so that i² = −1, and a complex number combines a real and an imaginary part in the form a + bi. Together they let every polynomial equation have a solution.
Imaginary numbers arise from a question ordinary arithmetic cannot answer: what is the square root of a negative number? Mathematicians define the imaginary unit i by i² = −1, so √(−9) = 3i. A complex number has the form a + bi, where a is the real part and b is the imaginary part — for example, 4 − 7i. Real numbers are just complex numbers with b = 0, so the complex numbers contain every number you've already worked with.
Arithmetic with complex numbers follows familiar algebra plus one substitution rule. Add and subtract by combining like parts: (3 + 2i) + (1 − 5i) = 4 − 3i. Multiply using the distributive property and replace i² with −1: (2 + i)(3 − i) = 6 − 2i + 3i − i² = 7 + i. Powers of i cycle with period four — i, −1, −i, 1 — so i⁴⁷ reduces to i³ = −i by dividing the exponent by 4 and keeping the remainder.
Complex numbers matter because they complete algebra: with them, every quadratic has two roots. When the quadratic formula produces a negative discriminant, the equation has no real solutions but two complex ones that come as a conjugate pair, a + bi and a − bi. Multiplying conjugates gives the real number a² + b², the trick used to simplify division by a complex number.
Imaginary and complex numbers are tested on the ACT and ASVAB math sections — usually simplifying powers of i, doing complex arithmetic, or solving quadratics with negative discriminants — and appear in the advanced topics of the AMC 12, where problems draw on conjugates, moduli, and roots of polynomials.
Key takeaways
- The imaginary unit i is defined by i² = −1, so square roots of negative numbers become imaginary numbers.
- A complex number has the form a + bi, with a real part and an imaginary part.
- Powers of i cycle every four: i, −1, −i, 1.
- Quadratics with a negative discriminant have two complex roots that form a conjugate pair.
- The ACT, ASVAB, and AMC 12 all test complex-number arithmetic and quadratic applications.