Complex Numbers and Quadratic Equations — Class 11 Mathematics

The real number system is extended so that equations like have solutions. Defining with , a complex number has the form where and . Two complex numbers are equal iff their real and imaginary parts match. Addition, subtraction and multiplication follow term-by-term rules, e.g. , and every non-zero has a multiplicative inverse . Powers of cycle: . The modulus and conjugate satisfy . Addition obeys the closure, commutative, associative laws, with additive identity and additive inverse ; multiplication has identity and is distributive over addition. The square root of a negative number (with ) is , but the rule fails when both are negative. Geometrically a complex number corresponds to the point in the Argand (complex) plane, where the real axis and imaginary axis replace the coordinate axes, is the distance from the origin and is the mirror image of in the real axis. Useful properties include and . These tools provide solutions to quadratic equations whose discriminant is negative.