Complex conjugate

In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. For example, the complex conjugate of a + b i {displaystyle a+bi} is a − b i . {displaystyle a-bi.} In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. For example, the complex conjugate of a + b i {displaystyle a+bi} is a − b i . {displaystyle a-bi.} In polar form, the conjugate of r e i φ {displaystyle re^{ivarphi }} is r e − i φ {displaystyle re^{-ivarphi }} . This can be shown using Euler's formula. The product of a complex number and its conjugate is a real number: a 2 + b 2 {displaystyle a^{2}+b^{2}}  or  r 2 {displaystyle r^{2}} in polar coordinates. Complex conjugates are important for finding roots of polynomials. According to the complex conjugate root theorem, if a complex number is a root to a polynomial in one variable with real coefficients (such as the quadratic equation or the cubic equation), so is its conjugate. The complex conjugate of a complex number z {displaystyle z} is written as z ¯ {displaystyle {overline {z}}} or z ∗ {displaystyle z^{*}!} . The first notation, a vinculum, avoids confusion with the notation for the conjugate transpose of a matrix, which can be thought of as a generalization of the complex conjugate. The second is preferred in physics, where dagger is used for the conjugate transpose, while the bar-notation is more common in pure mathematics. If a complex number is represented as a 2×2 matrix, the notations are identical. In some texts, the complex conjugate of a previous known number is abbreviated as 'c.c.'. For example, writing e i φ + c.c. {displaystyle e^{ivarphi }+{ ext{c.c.}}} means e i φ + e − i φ {displaystyle e^{ivarphi }+e^{-ivarphi }} . The following properties apply for all complex numbers z and w, unless stated otherwise, and can be proved by writing z and w in the form a + ib.