Number Types

The ISEE will directly and indirectly test your knowledge of number types, so let’s take a look at the various number types. All the number types we will explore here are known as “real numbers.” There are also “imaginary numbers,” but you will learn about them later. For now, let’s take a look at the following number types:

Natural or Counting Numbers

First, let’s look at Natural numbers, also known as Counting numbers. Natural numbers only include positive whole numbers. They do not include negative numbers, decimals or fractions. They also do not include zero. You can think of these as the numbers we use to talk about things in the real world, like talking about how many people there are in the room. 

Examples

Whole Numbers

Whole numbers are much like natural numbers. They include positive numbers as well as zero, which is the major difference between the two. They do not include negative numbers, decimals or fractions. 

Examples

You can think of whole numbers and natural numbers as the very first numbers you learned when you were first learning about number lines.

Integers

Much like whole numbers, integers include positive numbers and zero. However, they also include negative numbers. They do not include decimals or fractions.

Examples

Negative numbers exist on the left side of 0 on number lines.

Rational Numbers

Rational numbers include any number that can be represented as a fraction or a ratio (that’s actually why they’re called rational numbers!) of two integers (except if both are 0). Rational numbers include integers, as well as decimals, that can be represented as fractions. 

Examples

Irrational Numbers

Irrational numbers are any numbers that cannot be represented as a fraction.

Examples

Notice that while all natural numbers are integers, not all integers are natural numbers, as by definition negative numbers are not natural numbers. The same can be said for numbers in different categories, i.e. 2 is a rational, integer, and natural number, -2 is a rational number and an integer, whereas -0.25 is only a rational number. In that way, some numbers will fall under the umbrella of multiple categories, while others may fit only one. 

So why is π, or pi, not considered a rational number?

As you may have noticed, 3.14 and 22/7 are listed as rational numbers, and it is very likely that you have used these numbers to represent pi in the past. However, they are not the true value of pi, which is derived from the ratio of a circle’s circumference to its diameter. These numbers are simply useful approximations. In reality, there is no fraction that would properly equal pi. In most cases, you use approximations of pi in your math problems. While you can approximate where irrational numbers might fall on a number line, you will never find the exact location like you can for the rest of the real numbers. 

Practice Problems

1. List five new examples of rational numbers:         
2. List five new examples of natural numbers:         
3. Why are some rational numbers not integers?
4. What is the main difference between whole numbers and natural numbers?
5. Give an example of an integer that is also a natural number. What is an integer that is not a natural number?

Answers to Practice Problems

1. Many possible answers (example: .25, 14, 1.8, -2.4, 18)
2. Many possible answers (example: 2, 3, 5, 6, 9)
3. Any fraction is not an integer, but is a rational number. Thus, any fraction would be rational but not an integer. 
4. Whole numbers are just like natural numbers, except they also include the number 0 (while natural numbers do not). 
5. Many possible answers (example: 6 is both an integer and natural number. -3 is an integer but is not a natural number)