Number Sets - Math Steps, Examples & Questions

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Number sets

Here you will learn about number sets and how they are related and classified.

Students begin to learn about number sets in kindergarten and develop that knowledge from elementary school through high school.

What are number sets?

Number sets are groups of numbers that share the same definition. In elementary school, students work with the set of rational numbers. There are also subsets within rational numbers.

Let’s look at the number sets and how they are classified.

Number Sets	Description	Samples
Rational Numbers	Any number that can be expressed as a fraction where the numerator and denominator are integers (a ratio of integers). This includes positive and negative whole numbers, terminating decimals, and repeating decimals.	$-5, 0, 1, \cfrac{4}{5}\, , 1\cfrac{3}{4}\, , -1.2, 9.5, 0.\overline{3}$
Integers	Whole numbers and their opposites.	$\dots-3, -2, -2, 0, 1, 2, 3...$
Whole Numbers	Natural numbers and $0.$	$0, 1, 2, 3, 4, 5, 6 ...$
Natural Numbers	Positive numbers without fractions or decimals, also known as counting numbers.	$1, 2, 3, 4, 5, 6, ...$

In later grades, students work with the set of irrational numbers, real numbers, and others. A Venn diagram can also help you see the number sets.

What are number sets?

Common Core State Standards

How does this relate to $6$ th grade math?

Grade 6 – Number System (6.NS.C.5)
Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (for example, temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of $0$ in each situation.

Grade 6 – Number System (6.NS.C.6)
Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.

How to identify number sets

In order to identify number sets:

Recall the definition of the number set.
Show that the number fits or does not fit the definition.

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Number sets examples

Example 1: whole numbers

Is $-6$ a whole number?

Recall the definition of the number set.

Whole numbers are $0$ and positive numbers without fractions or decimals, also known as counting numbers.

2Show that the number fits or does not fit the definition.

$-6$ is negative, so it is not a whole number.

Example 2: integers

Is $905$ an integer?

Recall the definition of the number set.

Integers are whole numbers and their opposites.

Show that the number fits or does not fit the definition.

$905$ is positive and has no fraction or decimal, so it is a whole number, also making it an integer.

Example 3: rational numbers

Is $- \, \cfrac{3}{4}$ a rational number?

Recall the definition of the number set.

By definition, a rational number is any number that can be expressed as a fraction where the numerator and denominator are integers.

Show that the number fits or does not fit the definition.

$- \, \cfrac{3}{4}$ can be written as $\cfrac{-3}{4}$ or $\cfrac{3}{-4}.$

Either way, both the numerator and the denominator are integers, which make the fraction a rational number.

Example 4: identifying number sets

What number sets does $3 \, \cfrac{5}{8}$ belong to?

Recall the definition of the number set.

Natural numbers are counting numbers starting at $1.$

Whole numbers are natural numbers and $0.$

Integers are whole numbers and their opposites.

Rational numbers are any number that can be written as a fraction.

Show that the number fits or does not fit the definition.

$3 \, \cfrac{5}{8}$ has a fractional part, so it is NOT a natural number, whole number or integer.

$3 \, \cfrac{5}{8}=\cfrac{29}{8}$ so it is a rational number.

Example 5: identifying number sets

What number sets does $-27$ belong to?

Recall the definition of the number set.

Natural numbers are counting numbers starting at $1.$

Whole numbers are natural numbers and $0.$

Integers are whole numbers and their opposites.

Rational numbers are any number that can be written as a fraction.

Show that the number fits or does not fit the definition.

$-27$ is negative, so it is NOT a natural number or a whole number.

$-27$ does not have a fractional or decimal part, so it is an integer.

$-27=\cfrac{-27}{1}$ or $\cfrac{27}{-1}$ so it is also a rational number.

Example 6: identifying number sets

What number sets does $34, 582, 312$ belong to?

Recall the definition of the number set.

Natural numbers are counting numbers starting at $1.$

Whole numbers are natural numbers and $0.$

Integers are whole numbers and their opposites.

Rational numbers are any number that can be written as a fraction.

Show that the number fits or does not fit the definition.

$34, 582, 312$ is positive and it does not have a fractional or decimal part, so it is a natural number, whole number and integer.

$34,582,312=\cfrac{34,582,312}{1}$ so it is also a rational number.

Teaching tips for number sets

As students are learning to name the different types of number sets, have them create their own Venn Diagram to make connections between the different types of number sets.

Introduce the names of the number sets in a way that they build upon themselves. For example, starting with natural numbers, then whole numbers, then integers. This will help students understand that these number sets are all included in the rational number set as well.

Reinforce to students any other appropriate groupings, such as knowing cardinal numbers and positive integers are the same as the set of natural numbers or that referring to negative integers, is a subset with integers.

Easy mistakes to make

Mixing up the definitions
It is common to mix up definitions, but with time and reinforcement, students will learn the difference.

Including zero in the set of natural numbers
Zero is not included in the set of natural numbers. This set is also known as ‘the counting set’ and starts with the number $1.$

Thinking whole numbers cannot be rational numbers
Rational numbers are numbers that can be written as a fraction where the numerator and the denominator are integers. The number $5$ is rational because $5$ can be written as the fraction $\cfrac{5}{1}.$

Practice number sets questions

Whole numbers

Natural numbers

Rational numbers

Integers

Natural numbers are counting numbers starting at $1.$

Whole numbers are natural numbers and $0.$

Integers are whole numbers and their opposites.

Rational numbers are any number that can be written as a fraction.

$-9 \cfrac{2}{3}$ is negative and has a fractional part, so it is NOT a natural number, whole number or integer.

$-9 \cfrac{2}{3}=\cfrac{-29}{3}$ or $\cfrac{29}{-3},$ so it is a rational number.

$-9 \cfrac{2}{3}$ can also be written in decimal form as a repeating decimal:

$-9.6666666$

Whole numbers

Natural numbers

Integers

Rational numbers

Natural numbers are counting numbers that start at $1,$ so they do NOT include $0.$

$0$ is NOT in the set of natural numbers.

-\cfrac{3}{4}

78

-5.6

5 \cfrac{1}{3}

Integers are whole numbers and their opposites.

$78$ is a whole number, so it is also an integer.

-\cfrac{1}{2}

-5

45.6

45,678

Whole numbers are natural numbers (counting numbers) and $0.$

$45,678$ is positive and has no fractional or decimal part, so it is a whole number.

Natural numbers, whole numbers, integers, rational numbers

Only natural numbers

Natural numbers and whole numbers

Only integers

Natural numbers are counting numbers starting at $1.$

Whole numbers are natural numbers and $0.$

Integers are whole numbers and their opposites.

Rational numbers are any number that can be written as a fraction.

$56$ is a positive number with no fractional or decimal part, so it is a natural number, whole number and an integer.

$56=\cfrac{56}{1},$ so it is also a rational number.

Whole numbers, integers and rational numbers

Integers and rational numbers

Only rational numbers

Only integers

Natural numbers are counting numbers starting at $1.$

Whole numbers are natural numbers and $0.$

Integers are whole numbers and their opposites.

Rational numbers are any number that can be written as a fraction.

$-346$ is a negative number, so it is NOT a natural number or a whole number.

$-346$ does not have a fractional or decimal part, so it is an integer.

$-346=\cfrac{-346}{1}$ or $\cfrac{346}{-1},$ so it is also a rational number.

Number sets FAQs

What is the definition of an irrational number?

An irrational number cannot be written as a ratio of two integers. Instead, it is a non-repeating, non-terminating decimal. It has an infinite number of decimal positions where the digits follow no pattern. For example, the square root of $2 = 1.41421356237\dots$ There is no pattern, and the decimal digits go on forever.

Are there other sets besides the set of real numbers?

Yes, for example in upper mathematics, there is a set called the set of complex numbers. It includes real numbers and imaginary numbers. Complex numbers are made up of a real part and an imaginary part. Other sets of numbers include algebraic numbers, which are solutions to polynomials, and transcendental numbers, which are not.

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