Imaginary Numbers on the ACT
Imaginary numbers were discovered or invented as a way to take the square root of a negative number. You can’t do that by real numbers. If you take the square root of -1, you get imaginary number i.
√-1 = i
How to Simplify Imaginary Numbers
For example, suppose you need to find the square root of -16.
Now, how do you get the number -16? One way is to multiply 16 and -1.
√-16 = √-1 × √16
You already know that √-1 is i, and √16 is 4. So,
√-16 = 4i
Powers of Imaginary Numbers
Suppose you need to figure out what i2 is. Since you already know that i is √-1, when you square a square root, both of them cancel out, leaving you with the answer of -1.
i2 = -1
For greater powers like i3, think of it as building on the power you already know.
i3 = i2 × i
i3 = -i
Similarly,
i4 = 1
The powers of i work in a cycle.
Complex Numbers
Complex numbers are like binomials. They have two parts: a real number and an imaginary number. Some examples of complex numbers are: (3 + 4i), (-2, -i), or (7 – 3i). You can think of a complex number as a + bi. In this case, the a and b can be substituted by any real number.
Addition and Subtraction
If you want to add or subtract, you get this:
Combine like terms: a and c and then, b and d, taking the i at the end.
Multiplication
You multiply the terms of a binomial or complex number in this order: First, Outer, Inner, Last (FOIL). For example,
(5 + 3i) × (4 + 5i)
- First: 5 × 4 = 20
- Outer: 5 × 5i = 25i
- Inner: 3i × 4 = 12i
- Last: 3i × 5i = 15i2
So, answer = 20 + 25i + 12i + 15i2
Now, i2 = -1. So, we get
Answer = 20 + 25i + 12i - 15 = 5 + 37i
Division
According to the laws of Mathematics, you can’t have a radical in the denominator of a fraction. So, first you have to simplify in order to solve the problem. In order to divide complex numbers, multiply it by the complex conjugate of the denominator. For example,
