Complex numbers · Grade 11
In Grade 11 you learn: complex numbers.
Complex numbers extend the real numbers with i, where i² = −1, so that every quadratic equation has solutions. Written as a + bi, they are points on an Argand diagram. In modulus–argument form they turn multiplication into rotation and scaling. That is why engineers and physicists rely on them.
Key facts
- i^2 = −1. Add and multiply like algebra, replacing i^2 by −1.
- The conjugate of z = a + bi is z* = a − bi, and z × z* = a² + b². To divide, multiply top and bottom by the conjugate of the bottom.
- De Moivre’s theorem: (r(cos θ + i sin θ))^n = r^n(cos nθ + i sin nθ).
- A polynomial with real coefficients has its non-real roots in conjugate pairs.