Complex Conjugate Complex Conjugate In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs.For example, 3 + 4i and 3 − 4i are complex conjugates. An alternative notation for the complex conjugate is . However, the notation avoids confusion with the notation for the conjugate transpose of a matrix, which can be thought of as a generalization of complex conjugation. The star-notation is preferred in physics 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. Complex numbers are considered points in the complex plane, a variation of the Cartesian coordinate system where both axes are real number lines that cross at the origin, however, the -axis is a product of real numbers multiplied by +/- . On the illustration, the -axis is called the real axis, labeled Re, while the -axis is called the imaginary axis, labeled Im.
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The plane defined by the Re and Im axes represents the space of all possible complex numbers. In this view, complex conjugation corresponds to reflection of a complex number at the x-axis, equivalent to a degree rotation of the complex plane about the Re axis. In polar form, the conjugate of is . This can be shown using Euler's formula. Pairs of complex conjugates are significant because the imaginary unit is qualitatively indistinct from its additive and multiplicative inverse , as they both satisfy the definition for the imaginary unit: . Thus in most "natural" settings, if a complex number provides a solution to a problem, so does its conjugate, such as is the case for complex solutions of the quadratic formula with real coefficients. We say that a Complex Number is formed by the Combination of the real and the imaginary part. Before learning about complex numbers, let us first look at the imaginary numbers. If we have any number, whose Square is a negative number, then we say that the number is Imaginary Number, which is represented by I, called iota. So we write root ( -9) = 3i, where root (-1) = i. So we say that the complex number is of the form of Z = ( a + ib), where a and b are the Real Numbers and I = root (-1 ). The Set of the complex numbers is represented by the alphabet C. Now we will learn about complex conjugate, where we say that if the complex number z is represented as a + bi then its mathematical complex conjugate is represented as Ż = a – ib.
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To find the complex conjugates of the complex number z= 2 + 3i, we will simply write it as conj (Z) = 2-3i Another important thing to be remembered is that if the complex number z and its conjugate are added, then the imaginary part of the two numbers cancels out, as one is a positive number and the other is a negative. Let us try to add the above given z and its conjugate, then we have: Z + conjugate ( Z) = 2 + 3i + 2 – 3i = 4 + 3i – 3i = 4 Ans Let us look at a special situation of finding the conjugate: If we have z = i^3, the to find its conjugate we precede as follows: We know that z = i^3 = -I [ as we know that i^2 = 1] So z = 0 – i Thus we write conjugate ( z) = 0 + i = i Ans
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