Start Learning Complex Numbers 3 | Absolute Value, Conjugate, Argument [dark version]

Complex conjugation and polar coordinates form the backbone of practical complex-number calculations—especially for finding absolute values and arguments and for rewriting numbers in a form that makes multiplication easier. A complex number is written as Z = X1 + iX2, where X1 is the real part and X2 is the imaginary part. On the complex plane, the distance from the origin to the point (X1, X2) is the absolute value (also called the modulus), denoted |Z|. Using the Pythagorean theorem, |Z| = √(X1^2 + X2^2), a nonnegative real number.

The conjugate is the key tool that links this geometric length to algebra. Flipping the point across the real axis keeps the real part the same but changes the sign of the imaginary part: if Z = X1 + iX2, then the conjugate is \bar{Z} = X1 − iX2. Multiplying Z by \bar{Z} causes the imaginary cross-terms to cancel: (X1 + iX2)(X1 − iX2) = X1^2 + X2^2. Since |Z|^2 equals X1^2 + X2^2, this gives the powerful identity Z\bar{Z} = |Z|^2. That relationship becomes a recurring calculation shortcut when working with many complex numbers.

Polar coordinates then translate the same number into “length and direction.” The absolute value gives the radius, while an angle—called the argument of Z, often written as φ—fixes the direction. For the first quadrant (where X1 > 0 and X2 > 0), φ can be computed as φ = arctan(X2/X1). The angle is not universally given by that simple formula in every region of the plane, but the method is consistent once the correct quadrant is handled. With |Z| and φ, the complex number can be reconstructed as Z = |Z|(cos φ + i sin φ). This polar form is especially useful later because it streamlines multiplication and other operations.

An example makes the workflow concrete. For Z = 3 + 3i, the conjugate is \bar{Z} = 3 − 3i. Multiplying gives Z\bar{Z} = 18, so |Z| = √18 = 3√2. The argument comes from arctan(3/3) = arctan(1) = π/4 (45°). Therefore, in polar form, Z = 3√2(cos(π/4) + i sin(π/4)). The notes also foreshadow a further rewrite—combining cos and i sin into an exponential expression—hinting at why these foundations matter for complex analysis and more advanced calculations.