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# Defining and Classifying Number Types

Shakiyla Huggins, Luke Winspur
• Author
Shakiyla Huggins

Shakiyla Huggins earned an MS in Applied Mathematics from NC A&T State University and an MA in Teaching from Lenoir Rhyne University. She has teacher licensure for high school math as well as a graduate certificate in Online Teaching and Instructional Design. She has taught k-12 and college.

• Instructor
Luke Winspur

Luke has taught high school algebra and geometry, college calculus, and has a master's degree in education.

Learn about classification of numbers. Identify the many types of numbers, including whole, natural, rational, irrational, even, odd, prime, composite, and integers. Updated: 10/30/2021

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#### How many types of number are there?

There are many different types of numbers. The most commonly referenced are the following nine types: whole, natural, rational, irrational, even, odd, prime, composite, and integers.

#### What is classified as a rational number?

A rational number is any number that can be written as a fraction or a ratio of two integers. However, the denominator of the fraction, cannot be equal to zero.

There are a finite number of numbers that we use in both mathematics and real-world situations. Several numbers can be used interchangeably (real numbers), while some have stringent rules and regulations (complex numbers).

• Real numbers - numbers that can be identified on a number line and can be used to measure distance.
• Complex numbers - numbers with both a real and imaginary component, therefore, cannot be identified on the number line.

It is important to be sure that each real number is correctly used. To properly use real numbers, a system identifies the properties of each type of number. This classification system is called the real number system.

The real number system is a system that separates all numbers into five categories.

1. rational numbers
2. integers
3. whole numbers
4. natural or counting numbers
5. irrational numbers

The image below shows the five groups within the real number system and how they relate to each other.

Once each number is appropriately classified within the real number system, other categories of numbers are applied based on the rules of the main category. The five types of real numbers can also contain even and odd numbers or prime and composite numbers.

Let's take a moment to explore each of the different types of numbers below.

### Real Numbers

Real numbers, as mentioned above, are numbers that are identified on a number line. A real number consists of every single number that is used to measure distance. Examples of real numbers include: {eq}10,000 {/eq}, {eq}\frac{3}{4} {/eq}, {eq}0 {/eq}, {eq}-365,980,125 {/eq},{eq}\sqrt{51} {/eq}, {eq}\pi {/eq} and {eq}0.\bar{3} {/eq}.

Real numbers consist of rational numbers, integers, whole numbers, natural or counting numbers, and irrational numbers.

### Natural Numbers

Natural numbers are all of the counting numbers. Think about a young child and their process of learning how to count. Children are first taught the counting numbers, one through ten. Once they have mastered one through ten, they are then taught numbers eleven through twenty, and so on.

However, they are NOT taught zero, correct? That is because zero is NOT a natural or counting number. Natural numbers are the numbers that are used to count.

Examples of natural numbers include {eq}1 {/eq}, {eq}32 {/eq}, {eq}541 {/eq}, {eq}6000 {/eq}, {eq}70,156 {/eq}, {eq}802,963 {/eq} or {eq}7,853,742 {/eq}

### Whole Numbers

Whole numbers are all counting numbers, as well as zero. It is imperative to understand that zero is a whole number; it is not considered a counting number. Therefore we can conclude that all natural numbers are whole numbers. Zero is the only difference between natural numbers and whole numbers.

Examples of whole numbers include {eq}0 {/eq}, {eq}7 {/eq}, {eq}1025 {/eq}, {eq}1,259,874 {/eq}, or {eq}5,589,876,425, {/eq}.

### Even and Odd

Even and odd numbers are numbers that fit into the category of integers. We will discuss integers in a few moments. But, for now, an even or odd number is a whole number that is either positive or negative. Additionally, since all whole numbers are also real numbers, even and odd numbers are also real numbers.

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## Natural Numbers

This is a lesson on the different classifications of numbers, which is basically a big word for just saying that numbers are part of families and numbers have different homes depending on what kind of number they are, just like people do.

I'll start by talking about myself a little bit. A lot of the time when I meet someone new, they ask me where I live. You don't give them your specific address. I don't tell them 202 Calvert Drive; that doesn't really mean much. But I do tell them that I live in Cupertino. That's probably the most specific name I use to describe where I live.

On the numbers side of things, the most basic number is the number 1, and the most specific descriptor of where it lives is called the natural numbers. The natural numbers are all the numbers that you learn when you're a baby, like 1, 2, 3, 4, 5, 6 and on and on. The natural numbers are also sometimes called the counting numbers because they're the first numbers you learn how to count.

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• 0:06 Natural Numbers
• 1:10 Whole Numbers
• 2:16 Integers
• 2:59 Rational Numbers
• 4:17 Irrational Numbers
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There are trillions of different types of numbers that are used every single day. Every number falls into the category of a real number or a complex number. The Real Number System is a classification system that separates the different types of real numbers. There are five subsets of the real number system: natural numbers, whole numbers, integers, rational numbers, and irrational numbers. Within these five subsets, other types of numbers can also be classified, including but not limited to even, odd, prime, and composite numbers. As a result of the real number system having subsets, many numbers can be used interchangeably. The smallest subset is natural numbers. Natural numbers are also considered to be whole numbers, integers, and rational numbers. The next subset is whole numbers. Whole numbers are also considered to be integers and rational numbers. The next subset is integers. Integers are also considered to be rational numbers. The last subset is irrational numbers, which fall into a category of its own.

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## Whole Numbers

Back to me real fast, although I do live in Cupertino, Cupertino is part of Santa Clara County. This means that since I'm a resident of Cupertino, I'm also a resident of Santa Clara County. But the reverse isn't necessarily true, because not everyone that lives in Santa Clara County lives in Cupertino, so you have to be careful going back and forth.

Numbers are the exact same way. If we get a tiny bit less specific, what we come to are called the whole numbers. All of the natural numbers are part of the whole numbers, just like all the people who live in Cupertino also live in Santa Clara County. But there is one whole number that is not a natural number. That is the number 0. So when we're talking about the whole numbers, we're talking about the numbers that start with 0 and starting going up 1, 2, 3 and on and on.

## Integers

Taking a step further back in my situation, Cupertino and Santa Clara County are both parts of California. Again, not everyone that lives in California lives in Santa Clara or, even more specifically, Cupertino. But if you live in Cupertino or Santa Clara, you're definitely living in California.

Taking a step out in terms of the numbers brings us to what are called the integers. Again, not all of the integers are whole numbers and natural numbers. But all of the whole numbers and natural numbers are integers. The integers now also add in the negatives: -1, -2, -3 and on and on in the negative direction as well.

## Rational Numbers

Taking another step back with me, I think most people are aware that California is a state that is part of the U.S. So I am also a resident of the United States. Again, everyone that is a resident of California is also a resident of the U.S. But not everyone that lives in the U.S. lives in California.

That trend continues with the numbers. As we take another step back, we come to what are called the rational numbers. The new additions to the club are fractions. This means we could have things like -1/2 or 1/3 or 3/4 or maybe 11/7.

The list becomes a little bit harder to write but, again, you could imagine that there are a lot of different kinds of numbers in here. It's still true that all of the previous numbers we mentioned are rationals, but not all of the rationals, specifically these fractions, are part of the numbers in the integers, whole numbers or natural numbers.

Video Transcript

## Natural Numbers

This is a lesson on the different classifications of numbers, which is basically a big word for just saying that numbers are part of families and numbers have different homes depending on what kind of number they are, just like people do.

I'll start by talking about myself a little bit. A lot of the time when I meet someone new, they ask me where I live. You don't give them your specific address. I don't tell them 202 Calvert Drive; that doesn't really mean much. But I do tell them that I live in Cupertino. That's probably the most specific name I use to describe where I live.

On the numbers side of things, the most basic number is the number 1, and the most specific descriptor of where it lives is called the natural numbers. The natural numbers are all the numbers that you learn when you're a baby, like 1, 2, 3, 4, 5, 6 and on and on. The natural numbers are also sometimes called the counting numbers because they're the first numbers you learn how to count.

## Whole Numbers

Back to me real fast, although I do live in Cupertino, Cupertino is part of Santa Clara County. This means that since I'm a resident of Cupertino, I'm also a resident of Santa Clara County. But the reverse isn't necessarily true, because not everyone that lives in Santa Clara County lives in Cupertino, so you have to be careful going back and forth.

Numbers are the exact same way. If we get a tiny bit less specific, what we come to are called the whole numbers. All of the natural numbers are part of the whole numbers, just like all the people who live in Cupertino also live in Santa Clara County. But there is one whole number that is not a natural number. That is the number 0. So when we're talking about the whole numbers, we're talking about the numbers that start with 0 and starting going up 1, 2, 3 and on and on.

## Integers

Taking a step further back in my situation, Cupertino and Santa Clara County are both parts of California. Again, not everyone that lives in California lives in Santa Clara or, even more specifically, Cupertino. But if you live in Cupertino or Santa Clara, you're definitely living in California.

Taking a step out in terms of the numbers brings us to what are called the integers. Again, not all of the integers are whole numbers and natural numbers. But all of the whole numbers and natural numbers are integers. The integers now also add in the negatives: -1, -2, -3 and on and on in the negative direction as well.

## Rational Numbers

Taking another step back with me, I think most people are aware that California is a state that is part of the U.S. So I am also a resident of the United States. Again, everyone that is a resident of California is also a resident of the U.S. But not everyone that lives in the U.S. lives in California.

That trend continues with the numbers. As we take another step back, we come to what are called the rational numbers. The new additions to the club are fractions. This means we could have things like -1/2 or 1/3 or 3/4 or maybe 11/7.

The list becomes a little bit harder to write but, again, you could imagine that there are a lot of different kinds of numbers in here. It's still true that all of the previous numbers we mentioned are rationals, but not all of the rationals, specifically these fractions, are part of the numbers in the integers, whole numbers or natural numbers.

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