# Cuckoo For Conjugates!

## When you divide a complex number by its conjugate, the absolute value of the quotient is always equal to 1!

Remember that this is the formula for the absolute value of a complex number:

## Examples:

(1 + *i*) ÷ (1 - *i*) = *i*

(1 - *i*) ÷ (1 + *i*) = -*i*

|*i*| = 1; |-*i*| = 1

## Obviously, the absolute value of *i* is 1; multiplying *i* by any real number just multiplies the absolute value, so |*xi*| = |*x*|. (Don't forget that absolute value is always positive!)

(2 + 3*i*) ÷ (2 - 3*i*) = -5/13 + 12/13*i*

(2 - 3*i*) ÷ (2 + 3*i*) = -5/13 - 12/13*i*

|-5/13 + 12/13*i*| = 1; |-5/13 - 12/13*i*| = 1

It doesn't matter if the real or imaginary parts of the complex numbers are rational or not, the quotient's absolute value will always be **1** as long as the divisor & dividend are conjugates! Remember that the conjugate of **a + b***i* is **a - b***i*.

(3/2 + *i*/3) ÷ (3/2 - *i*/3) = 77/85 + 36/85*i*

(3/2 - *i*/3) ÷ (3/2 + *i*/3) = 77/85 - 36/85*i*

|77/85 + 36/85*i*| = 1; |77/85 - 36/85*i*| = 1

### Note: If the imaginary part is a fraction, you can put *i* in the numerator, especially if it's a *unit* fraction.

Since their absolute value equals **1**, the quotients are reciprocals of each other! The only difference is the sign between the real & imaginary parts! In other words, if a complex number has an absolute value of **1**, then its reciprocal is its conjugate!

I thought about using irrational numbers in this final example, but it made things too complicated! So I used fractions instead!

(-3/2 + 5/6*i*) ÷ (-3/2 - 5/6*i*) = 28/53 - 45/53*i*

(-3/2 - 5/6*i*) ÷ (-3/2 + 5/6*i*) = 28/53 + 45/53*i*

|28/53 - 45/53*i*| = 1; |28/53 + 45/53*i*| = 1

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© Derek Cumberbatch