Imaginary Numbers

Eventually, you are going to encounter a situation where you will have to deal with square roots of negative numbers. How can this be done ? After all, a positive number squared or a negative number squared will always equal a positive number.

Mathematicians have designated a special number ‘i’ which is equal to the square root of minus 1. Then, it follows that i² = -1. To determine the square root of a negative number (-16 for example), take the square root of the absolute value of the number (square root of 16 = 4) and then multiply it by ‘i’. So, the square root of -16 is 4i.
As a double check, we can square 4i (4*4 = 16 and i*i =-1), producing -16.
All negative square roots are called “imaginary numbers” (now you know where that letter ‘i’ comes from).

Complex Numbers

When a number has the form a + bi (a real number plus an imaginary number) it is called a “complex number”. How do complex numbers “crop up” in mathematics? A good example would be the roots of the quadratic equation x² -6x + 25 = 0 where the 2 roots are 3 + 4i and 3 – 4i. Can we be sure these are the roots of the equation?
As a double-check, using those roots, we can “rebuild” the original equation by

(X – 3 -4i) * (X – 3 + 4i) = X² -3x +4Xi -3X +9 -12i -4Xi +12i -(4i)²This reduces to: X² -3x -3X +9 -(16)*i²Since i²= -1 then -(16)*i² becomes -(-16) = 16 and so:X² -6X +25 =0

Complex Number Multiplication

Addition and subtraction of complex numbers pretty much follow the rules of basic arithmetic and so we won’t discuss these. Multiplication starts getting a little tricky. Consider:

(5 + 6i) * (7 + 8i)This equals 35 + 40i + 42i + 48i²As we saw above, i² = -1 so 48i² = -48So answer= -13 + 82i

Complex Number Division

Were you wondering – is division more difficult than multiplication? Sure is. First we must define a new term – conjugate, whereby the conjugate of a + bi = a-bi.
Example – the conjugate of (3 + 4i) is (3 – 4i).
The main principle to remember in complex number division is that we multiply the “top” and “bottom” of the fraction by the conjugate of the denominator.
Time for an example: let’s divide (9 +3i) by (7 +5i)

(9 + 3i)
—————
(7 + 5i)The denominator is (7 + 5i) and its conjugate is (7- 5i)Multiplying top and bottom by the conjugate:((9 + 3i)* (7 – 5i))
—————————
((7 + 5i) * (7 – 5i))Which equals
(78 – 24 i)
—————
74Which equals
78                 24i
——   minus ——
74                 74Which equalsAnswer = 1.054054054054054 & -0.32432432432432434 i

Square Root of a Complex Number

Now we move on to even greater difficulty.
Time to define another term – modulus, whereby the modulus of a complex number a + bi equals the square root of (a² + b²).
The modulus of a complex number is generally represented by the letter ‘r’ and so:

r = Square Root (a² + b²)Next we’ll define these 2 quantities:

y = Square Root ((r-a)/2)x = b/2yFinally, the 2 square roots of a complex number are:

root ₁ = x + yiroot ₂ = -x – yiAn example should make this procedure much clearer.
Find the square root of 12 + 16i

r = Square Root (12² + 16²)r = Square Root (144 + 256) = 20y = Square Root ((20-12)/2) = 2x = 16/(2*2) = 4root ₁ = 4 + 2iroot ₂ = -4 – 2iEven though you have a calculator that can do these calculations for you, now you know the procedures for complex number arithmetic.

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