Number Sets

For the ACT^®, be familiar with these number sets and their basic properties.

1. Whole numbers are also known as “counting numbers“:

0, 1, 2, 3, 4, 5, 6, …

2. Integers are positive and negative whole numbers, including zero:

…, −3, −2, −1, 0, 1, 2, 3, …

3. Rational numbers are all numbers that can be written as fractions (like 1/4), terminating decimals (like 0.75), and repeating decimals (like 0.3333…). Note that every integer is also a rational number. For example, −3 can be written as a fraction: −3/1.

4. Irrational numbers are all numbers that cannot be expressed as a terminating or repeating decimal. Two familiar examples are π and √2. In fact, for any positive whole number that isn’t a perfect square (like 9 or 16), the square root is irrational.

5. Real numbers When we group together all rational and irrational numbers, we get the real numbers. An interesting property of the real numbers is that between any two real numbers, there are infinitely many irrational numbers! This may seem like a random fact (and a strange one at that), but some recent official ACT^® practice questions ask about ideas just like this.

Number Sets in ACT^® Questions

While one or two questions on the ACT^® may ask about number sets, many will focus on other math ideas and simply require you to understand the concepts behind the sets above. Let’s look at an example of how this might look in an ACT^® question.

Example

The sum of two consecutive integers is 119. What is the product of the integers?

1. 3,422
2. 3,540
3. 13,806
4. 14,042
5. 14,280

Integers are positive or negative whole numbers or zero. Since integers can’t be fractions, consecutive integers are exactly 1 apart. That is, if the smallest integer is x, then the next one is x + 1.

Given the fact that the sum is 119:

x + (x + 1) = 119
2x + 1 = 119
2x = 118
x = 59

The two consecutive integers are 59 and 60. Their product is 59 × 60 = 3,540, or answer choice B.

Practice

Which of the following numbers are irrational? Select all that apply.

A.√169 B.√16 C.√7 D.π/2 E.1/8 F.−5.17778

Ans, C, D

C. 7 is not a perfect square. Since 7 is a positive whole number that is not a perfect square, the square root of 7 is irrational.

D. π is irrational. An important property of real numbers is that the product of any irrational and rational number is itself irrational. π/2 is the product of π and 1/2.

2. Suppose a and b are integers. Which of the following must also be an integer?

A. (3a + b)/2 B.a(2b + 1) C.ab√2 D.a/b E.1/b(5a)

1. QuestionWhich of the following statements are true? Select all that apply.
   1. A.There are infinitely many irrational numbers between 0 and 1.
   2. B.Every square root is an irrational number.
   3. C.If a number can be written as a decimal, then it is a rational number.
   4. D.Zero is neither rational nor irrational.
   5. E.Every irrational number is a real number.
   6. F.Every integer is a rational number.