Imaginary (Complex) numbers

Imaginary numbers refer to irrational numbers that cannot exist in the real word, but which do exist in math. The most common imaginary number to see on the ACT is the imaginary number i, or the square root of -1.

When in Doubt, Try to Square It

The best piece of advice for problems involving i, especially those that involve calculating a real number or simplifying a fraction such that it only contains real numbers is:

• See if there is a way to square i. Since it is equal to -1, squaring i can take an imaginary number and make it a real number
• When i is part of a binomial in the denominator, it may require using a conjugate to simplify out i. See #55 on this list for a longer explanation of conjugates.

Powers of i

On the ACT, it’s beneficial to understand not only that ( i = \sqrt{-1} ), but also that there’s a simple way to determine what ( i ) raised to any power equals. ( i ) follows a cycle of four distinct values, based on the power it is raised to. These values consistently repeat in the same sequence, and as long as a student knows the starting point of the pattern, they can always calculate the value of ( i ) for any exponent.

Any number taken to the 0 power is equal to 1, and so i⁰ = 1, and so i¹ = i

Any square root squared is simply equal to the value underneath the square root, so i² = (i)² =  − 1

Since i² =  − 1 and i¹ = i, then i³ = i¹ • i²=  − 1 • i = − i

And finally, i² =  − 1 and so i⁴ = i² • i²=  − 1 •  − 1 = 1

And so the pattern of ilooks like: 1, i,  − 1,  − i.

To figure out the value of i at any given exponent, simply take the exponent and find it’s nearest factor of 4. The conversion chart then looks like:

Remainder After Factor	Value
0	1
1	i
2	− 1
3	− i

The Complex Plane

Occasionally ACT problems involving i will include the concept of the “complex plane”.

This means that the xy plane is replaced with a plane that includes:

• real numbers on the x-axis
• imaginary numbers such as i on the y-axis.

What is the sum of the complex numbers 3 – 4i and 5 + 3i? [ACT April 2018 Form A09 ]
F. 7	G. 27	H. -1 + 8i	J. 8 – i	K. 15 – 12i
J
(3 – 4i) + (5 + 3i) = (3 + 5) + (-4 + 3)i = 8 – i