Modulus And Conjugate Identity

Question

For any complex number z, the expression z multiplied by its conjugate z-bar is always equal to which real quantity by the fundamental modulus identity?

RankGuru · Accepted Answer

The product of a complex number with its conjugate is a foundational identity: if z = x + iy then z-bar = x - iy, a relationship used everywhere in JEE Advanced. Multiplying, z·z-bar = (x + iy)(x - iy) = x^2 - (iy)^2 = x^2 + y^2, since i^2 = -1. This quantity x^2 + y^2 is precisely the square of the modulus, |z|^2, and is always a non-negative real number. The identity converts conjugate products into real magnitudes, simplifying rationalization and modulus computations. Option 2 Re(z) equals z + z-bar, not the product. Option |z| omits the square. Option z^2 is generally complex and unequal to the real product. Hence z·z-bar = |z|^2. Plausibility check: for z = 3 + 4i, z·z-bar = 9 + 16 = 25 = 5^2 = |z|^2, numerically confirming that the conjugate product returns the squared modulus.

Modulus And Conjugate Identity

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