The ACT® Math section seldom asks questions that deal solely with number definitions. Occasionally, they might ask something like, "this sort of special case of number belongs to which of the following classifications of number: whole, integer, rational, irrational."
You're more likely to see a question that asks "which of the following whole numbers…" or "rational numbers…" or "irrational numbers…" For these questions, you need to know the definitions so you can understand what's being asked.
- Whole Number – non-negative, non-decimal (e.g., 0, 1, 2, 3)
- Integer – non-decimal, non-fraction (e.g., -2, -1, 0, 1, 2)
- Rational – can be written as a decimal or fraction (e.g., 1/2, 0.75, -3)
- Irrational – can't be written as a decimal or fraction (e.g., π, √2)
Imaginary Numbers
Imaginary numbers are a real concept you need to know for the ACT® Math. The name might sound confusing – because how can a number be imaginary? It's actually simpler to conceptualize than you might realize.
Basically, a bunch of mathematicians said there's this number that doesn't actually exist. But if we all pretend that it does, it makes a lot of things possible. So let's all agree that i equals the square root of negative one. This concept, while abstract, has practical applications in various fields.
From that agreement sprang much of modern life: circuitry, electronics, flight, smartphones, etc. The concept is important for math. It comes up on the ACT® Math section quite a bit. It's pretty easy to work with. Imaginary numbers work off of our exponent rules.
Consider this example
So, if i = √-1, then what happens when we raise the square root of negative one to the second power? Let's work it out step by step.
More Practice
- What would happen to i if we raised it to the 3rd power?
- The 4th power?
- The 5th power?
- What about i to the 0th power?
Let's work through each of these:
- i² Since i = √-1: i² = (√-1)² When you square the square root of a number, you get the original number: i² = -1
- i³ We can find i³ by multiplying i² by i: i³ = i² · i i³ = -i
- i⁴ We can find i⁴ by multiplying i³ by i: i⁴ = i³ · i i⁴ = -i · i i⁴ = -i² Since i² = -1: i⁴ = -(-1) i⁴ = 1
- i⁵ We can find i⁵ by multiplying i⁴ by i: i⁵ = i⁴ · i i⁵ = 1 · i i⁵ = i
- i⁰ Any number (except zero) raised to the power of zero is 1: i⁰ = 1
| Power | Value |
|---|---|
| i⁰ | 1 |
| i¹ | i |
| i² | -1 |
| i³ | -i |
| i⁴ | 1 |
| i⁵ | i |
| i⁶ | -1 |
| i⁷ | -i |
How can I apply this on the test?
Imaginary number questions tend to be simple applications of the exponent rules. Basically, these questions are usually like the example above.
The ACT® Math section can also give you questions that involve algebra. That's merely an extension of the exponent rules you'll be able to figure out based on the logic on this page.
Complex Numbers
Complex numbers also sound, well… complex. For the context of the ACT®, you can consider the name complex numbers a misnomer. Though, that may not be true of complex numbers in real life.
Complex numbers are numbers that consist of a real part and an imaginary part. They are typically written in the form a + bi, where a is the real part and b is the imaginary part. For example, 3 + 4i is a complex number.
Complex numbers involve a real number and an imaginary number put together. Hence the name "complex numbers." The ACT® Math section seldom does anything more with complex numbers than asking you to add them.
This is simple algebra. Rather than worry about what "i" equals, mathematically, you can just add like terms.
Consider this example
Add the following complex numbers: 3 + 4i and 5 + 3i
Click for the Answer
8 + 7i
How do I apply this to the ACT® Math Section?
You're likely to get at least one question like the above per ACT® Math section. You're unlikely to get anything more complicated. That's not to say it's impossible. If you're not looking for a score of greater than 32, it's really not worth studying random advanced topics in math unless you've mastered everything else first, since the last few points can be the hardest to get.
Otherwise, know how to add imaginary numbers as like terms. you'll be able to get this question right when you see it on the ACT® Math section.