what is the significance of (a+bi)(a-bi)? It's Algibra 2

i is the “imaginary” square root of minus one (√-1 = ±i); and any “real number” becomes “imaginary” when it is multiplied by i.

The sum of a “real number” and an “imaginary number” is said to be a “complex number”;

Thus:

(a + bi) is a “complex number”;

 i.e. it is the sum of a real part (a) and an imaginary part (b)

(a - bi) fairly obviously, is also a “complex number”;

 but note in particular that it has the same magnitude real part,
 and equal but opposite imaginary part when compared to the previous (a + bi).

Any two complex numbers of this specific form (equal real, but opposite imaginary parts), are said to be: “COMPLEX CONJUGATES” of one another.

The important thing to note about complex conjugates is that when multiplied together, the product is entirely real.

(a + bi) (a - bi) = a² + abi - abi + b²

and it is evident that the two imaginary parts cancel one another out thus leaving an entirely real product.

Thus in general, the product of any two complex conjugates is always real.

This result has important applications, and should be memorized as a standard form:

(a + bi) (a - bi) = (a² + b²)

or, in words: “the product of two complex conjugates is equal to the sum of the squares of their real and imaginary parts”.