The Retired Professor Simplifies Complex and Imaginary Numbers

If a square root of a negative number is taken, we quickly run into a problem.  No real number when multiplied by itself gives a negative number.  So, a new number called i is created and defined to be the square root of -1.  If we square this number, i², we get -1.  When we cube i we get -i, and when we raise i to the fourth power we get 1.  As powers of i increase we go through the cycle of these four values.

If we multiply i by a coefficient we get what is called an imaginary number. 

The name is misleading, these are actual numbers, and are of great importance.

A number consisting of a real term and an imaginary term of the form a + bi is called a complex number.

Complex numbers can be graphed using two axes, the horizontal axis labeled x and the vertical axis labeled iy.  So, think of complex numbers as two-dimensional numbers.  It is as simple as that.

When we add, or subtract, complex numbers we add, or subtract, the real part and add, or subtract, the imaginary part.

3 + 4i is to be added to 5 + 2i.  Adding 3 and 5 we get 8, and 4 plus 2 gives 6.

The sum above is 8 + 6i.

If we multiply the numbers (3 + 4i)(5 + 2i) we must treat the numbers as though they are binomials, and multiply as we would multiply binomials.

(3)(5) + (3)(2i) + (4i)(5) + (4i)(2i) = 15 + 6i + 20i + 8i²

Since i² is -1, we get 15 + 6i + 20i – 8 = 7 + 26i

Definition:  The conjugate of a complex number has the opposite sign for the imaginary term.

The numbers 4 – 2i and 4 + 2i are conjugates of each other.

It is desirable to not have i in a denominator.  To fix the problem, multiply any complex number in the denominator by 1 in the form of its conjugate divided by itself.

As an example, start with the number below, and remove any imaginary number from its denominator.

(2 + 3i) ÷ (4 – 2i)

(2 + 3i) (4 + 2i) = 8 + 4i + 12i + 6i² = 2 + 16i = 1 + 8i

(4 – 2i) (4 + 2i)    16 + 8i – 8i – 4i²        20            10

Notice the imaginary parts all disappear from the denominator.  This will always happen.

Think about the factoring rule  

Absolute Value of a Complex Number

Finally, what is the absolute value of a complex number?  Well, the absolute value of a real number the distance from the point to the origin.  The same thing is true for absolute values of complex numbers.  Recall that complex numbers are two-dimensional, so we have an x distance and a y distance.  Since the axes are at right angles, the absolute value of a complex number requires the Pythagorean Theorem. 

Find │3 – 4i│. 

(3² + (-4)²)^1/2 = (9 + 16)^1/2 = 25^1/2 = 5

Using a graphing calculator for complex numbers is easy.  The following is for the TI-83 and TI-84 calculators. 

First, one must select a + bi under MODE.  Once that is done, look for the i key, it is a shift key.  Using the key allows you to enter a complex number.  When entering a complex number be certain to use parentheses, or an operation will only impact part of a complex number.  Use the operations +, -, *, and /, along with the absolute value command, abs(, as you would with real numbers.  Be certain to close the parentheses with ) once you finish entering a complex number in the absolute value command.  Your answer will be in complex number form.  This is a great tool for checking how accurate your work is, but do not substitute the calculator for actual needed practice.

There is also a conjugate command, which is hidden.  Most people will find the conjugate easy enough not to need it, but if you want to see it look under MATH and select the complex numbers.