Complex conjugate

The complex conjugate of a number a + bi is a − bi, obtained by reversing the sign of its imaginary part. Conjugation reflects complex numbers across the real axis.

Complex number: A number expressible as a + bi, where a and b are real and i² = −1. Conjugation acts on every complex number by preserving its real part and reversing its imaginary part.
Rationalization: An algebraic technique that removes radicals or complex terms from a denominator by multiplying by a suitable expression. Multiplying by a complex conjugate rationalizes denominators containing imaginary terms.
Imaginary unit: The number i satisfying i² = −1. The sign reversal in conjugation changes the coefficient of i.
Additive inverse: The number that sums with a given number to produce zero. Negation reverses both coordinates, whereas conjugation reverses only the imaginary coordinate.
Complex plane: A plane in which real and imaginary parts serve as horizontal and vertical coordinates. Conjugation is reflection across this plane’s real axis.
Real part: The real coordinate of a complex number a + bi, equal to a. Adding a number to its conjugate isolates twice its real part.
Real number: A number belonging to the continuous number system represented on the number line. Real numbers are exactly the complex numbers unchanged by conjugation.
Multiplicative inverse: For a nonzero number, the number whose product with it is one. The inverse of a nonzero complex number uses its conjugate but also divides by its squared modulus.
Modulus of a complex number: The nonnegative distance from a complex number to zero in the complex plane. Multiplying a number by its conjugate gives the square of its modulus.
Imaginary part: The coefficient b of i in a complex number a + bi. Subtracting a number’s conjugate isolates twice i times its imaginary part.

Linked from 11 pages