Complex Number Kookiness 11: Unit Circle Superiority!

When you divide a complex number by its absolute value (also called magnitude), you'll get a quotient in the unit circle that has the exact same argument as the 1st complex number!

Examples:

(1 + i) ÷ √(2) = √(2)/2 + √(2)/2i;

arg(1 + i) = 45° & arg(√(2)/2 + √(2)/2i) = 45°

(3 + 4i) ÷ 5 = 3/5 + 4/5i;

arg(3 + 4i) = 53.13010235...° & arg(3/5 + 4/5i) = 53.13010235...°

(They're the exact same irrational number of degrees, even though you can't see all the digits! There are far too many digits to count after the decimal point anyway; in fact, there's infinitely many!)

(-(4 + 1/2) - 9√(3)/2i) ÷ 9 = -1/2 - √(3)/2i;

arg(-9/2 - 9√(3)/2i) = -120° & arg(-1/2 - √(3)/2i) = -120°

Note: If you go counterclockwise instead of clockwise with this last example, the angle (or argument) is 240°. Graphing calculators refer to the clockwise angle when complex numbers are in Quadrants III or IV! Also, I typed the mixed number as an improper fraction for simplification! (-(4 + 1/2) = -9/2)

Since absolute value is ALWAYS positive, it doesn't matter which quadrant the 1st complex number is in! (So, there's never any need to make the divisor negative!) The magnitude will also ALWAYS be a real number, so you never have to divide by an imaginary/complex number in this math trick either. If you pick a complex number that's already in the unit circle as the dividend, then you would have to divide by 1, which is the number of multiplicative identity! That just makes the quotient the same as the dividend!

Also, to conlcude this Web page, don't forget the formula for the absolute value of a complex number:

Back to Index Page Back to Math Trick Menu

© Derek Cumberbatch