Complex numbers are “complex” because they include a real number (a) and an imaginary number (bi). Thus, complex numbers are represented as the binomial a + bi, and as you will see, they act like binomials in calculations.
To reiterate, complex numbers are a real number added to an imaginary number, written in the form a + bi, where a is the real number and bi is the imaginary part.
It is important to note that a and/or b can equal 0. Therefore, the classificaiton of complex numbers encompasses all real numbers, imaginary numbers, and combinations thereof.
It may help you to conceptualize complex numbers by graphing them on a complex plot, known as an Argand plane. This plane looks similar to the cartesian plane, but in this case, the x-axis represents real numbers, and the y-axis represents imaginary numbers. Instead of x-coordinates and y-coordinates, we use the real component and the imaginary component of each complex number to graph the coordinates. The imaginary axis acts as an extension of the number line to two dimensions, since imaginary numbers do not exist on the normal number line.
The graph above will make it easier to understand why adding and subtracting complex numbers is actually quite straightforward. If you want to compute (4 + 3i) + (3 + 2i) all you have to do is combine like terms. That is:
Returning to the cartesian plot, adding or subtracting a complex number is simply about shifting coordinates along the real axis (x) and the imaginary axis (y). Let’s try another one:
Multiplying complex numbers is just like multiplying binomials. For example:
Dividing complex numbers is also like binomials. How would you reduce the following fraction?
While this fraction may look like it is already reduced, a complex number in its simplest form will have no imaginary numbers in its denominator. We can eliminate the imaginary number by multiplying by the conjugate.
The conjugate of an expression is the expression given when you switch the sign between the two terms of the expression. You may have utilized conjugates when dealing with binomials.
When we multiply a binomial expression by its conjugate, by using FOIL, we get the difference of squares, which is helpful when dealing with radicals or complex numbers.
The conjugate for complex numbers looks very similar:
In fractions that contain radicals, they are not truly in their simplest form until there are no radicals in the denominator. Similarly, in fractions that contain complex numbers, we prefer to eliminate the complex numbers from the denominator. We can do this by multiplying by the conjugate of the denominator- a “fancy form of 1.”
Fancy Form of 1
Any number over itself equals 1, and the same holds true for complex numbers. By multiplying a fraction containing complex numbers in the denominator by the conjugate of the denominator over itself, we are essentially multiplying by 1, so the value of the expression doesn’t actually change.
By multiplying by the conjugate, we have eliminated any imaginary numbers from the denominator, and the fraction is in its simplest form.