Imaginary and Complex Numbers

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)² This reduces to: X² -3x -3X +9 -(16)*i² Since 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 + 48i² As we saw above, i² = -1 so 48i² = -48 So 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 don't you think ?

(9 + 3i) ÷ (7 + 5i) Multiplying top and bottom by the conjugate: ((9 + 3i)* (7 - 5i)) ÷ ((7 + 5i) * (7 - 5i)) Which equals (78 - 24 i) ÷ 74 Equals (78 ÷ 74) - (24i ÷ 74) Answer = 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 the quantities
y = Square Root ((r-a)/2) x = b/2y.
Finally, the 2 square roots of a complex number are:
root ₁ = x + yi root ₂ = -x - yi An example should make this procedure much clearer.
Find the square root of 12 + 16i.
r = Square Root (12² + 16²) r = Square Root (144 + 256) = 20 y = Square Root ((20-12)/2) = 2 x = 16/(2*2) = 4 root ₁ = 4 + 2i root ₂ = -4 - 2i

Even 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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