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 = 251/2 = 5