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)

Here's an image of the Unit Circle: (Note: You can also see this image at Wikipedia.org)

Each point shows you the rectangular coordinates, which is how long the vertical & horizontal sides of a right triangle would be if its hypotenuse was 1 unit long. Both rectangular coordinates are positive in Quadrant I, which is the one at the top right; although at least 1 of them is negative in the other quadrants, it doesn't affect the absolute values of the sides of the imaginary right triangle. A minus sign(-) simply means that the length goes in the opposite direction! Also, the angles (also known as arguments) of the complex numbers are shown in both radians & degrees. (Angles shown in radians often include the irrational number known as pi(π) in their expressions! That's because they're based on the formula for the circumference of a circle in terms of the radius.)

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