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# Number Systems

Do you know when we were young we only knew there are odd numbers and even numbers? In this article we will learn about some other classification of numbers!

A number system is made up of different types of numbers that all have a certain property or properties. There are many different types of number systems:

• Natural (or counting) numbers

• Whole numbers

• Integers

• Rational numbers

• Irrational numbers

• Real numbers

• Complex numbers

## The number systems

### Natural (or counting) numbers

A natural number can be described as a positive whole number, they start from 1 and go onwards; 1, 2, 3, 4...

1, 44, 170, 5000

Natural numbers can be represented on a number line;

Natural Numbers, Thomas-Gay - StudySmarter Original

### Whole numbers

Whole numbers are all of the natural numbers plus zero. They start at 0 and go onwards; 0, 1, 2, 3, 4...

Whole numbers can also be represented on a number line;

Whole Numbers, Thomas-Gay - StudySmarter Originals

### Integers

An integer includes both positive and negative whole numbers, as well as zero. They include

-36, -2, 0, 5, 79

Integers can be shown on a number line;

Integers, Thomas-Gay - StudySmarter

### Rational numbers

A rational number can be written as a fraction, , where and are both integers and is not equal to 0. A decimal that terminates or has an indefinite repeating pattern can also be considered a rational number.

### Irrational numbers

Irrational numbers are numbers that cannot be written in the form , where and are both integers and does not equal 0. An irrational number is a number that doesn't have a terminating or reoccurring decimal.

The most well known example of an irrational number is pi (), this is a number that has a decimal that goes on forever without repeating any pattern;

### Complex numbers

A complex number can be written in the form , where and are both real numbers and is an imaginary unit.

## Real number system

The real numbers include all rational and irrational numbers, and thus all natural numbers, whole numbers, and integers. This can be shown in the diagram below, this is a Venn diagram where which indicates how they all link to one another;

Number Systems, Thomas-Gay - StudySmarter Originals

This diagram shows that each subset fits within the larger set, for example, it shows that all rational numbers are real numbers, and so on. All integers are rational numbers and real numbers. Since the natural numbers set sits within all of the other subsets, a natural number can be considered a whole number, integer, rational number and a real number.

It is useful to be able to identify which type of number it is that you have been given.

Which number system or systems does belong to?

Solution:

To identify this you are able to use your calculator to solve the square root to see what type of number it is;

Since the decimal does not terminate or repeat, this is an irrational number.

To which number system or systems does belong to?

Solution:

Once again you can solve the square root to identify the type of number;

Since is equal to 8, belongs to the number systems of, natural numbers, whole numbers, integers and rational numbers.

### Graphing real numbers

To graph real numbers simply means to place them in order on a number line. There are different indicators that can be added to the number line to represent the set of numbers that are being graphed. Let's work through some examples to explore each of these indicators;

Graph the set of numbers

Solution:

To do this you need to first start by drawing a number line, the first number that is included is 1. To show that x could equal one you draw a filled dot above the number. Since there is no end to the graph, rather than drawing all of the numbers onto the graph you can simply draw an arrow going past the last number to indicate it is continuing;

Graphing real numbers example, Thomas-Gay - StudySmarter Originals

Graph the set of numbers

Solution:

For this example you need to first start by drawing your number line, then you can begin to input the information. Since this time x is smaller than 10 but does not include 10, the circle above 10 will not be colored in.

Graphing real numbers example, Thomas-Gay - StudySmarter Originals

Graph the set of numbers

Solution:

For this example you need to first start by drawing your number line, then you can begin to input the information. Since the x is smaller than and includes , the circle above will be colored in.

Graphing real numbers example, Thomas-Gay - StudySmarter Originals

Graph the set of numbers

Solution:

To begin with, draw out your number line, then you can start to input the information. Since x is bigger then or equal to -6, the circle above the -6 should be colored in.

Graphing real numbers example, Thomas-Gay - StudySmarter Originals

## Number Systems - Key takeaways

• Numbers can belong to the set of whole numbers, integers, rational numbers, irrational numbers, real numbers and complex numbers.
• Rational numbers are numbers that are represented as a fraction, in the form , where a and b are integers and , a terminating or a reoccurring decimal.
• Irrational numbers have a decimal that do not terminate with no repeating pattern.
• Complex numbers are written in the form, , where a and b are real numbers and i is the imaginary unit.

The number system is made up of different types of numbers.

Some examples of different types of numbers that make up the number system include;

• Natural numbers
• Whole numbers
• Integers
• Rational numbers
• Irrational numbers
• Real numbers
• Complex numbers

It is a method of writing numbers which represents the different sets of numbers.

## Final Number Systems Quiz

Question

What are the 6 different types of numbers?

• Natural numbers
• Whole numbers
• Integers
• Rational numbers
• Irrational numbers
• Real numbers

Show question

Question

What is the difference between a natural number and a whole number?

A natural number are positive whole numbers that does not include 0, whereas whole numbers are positive numbers including 0.

Show question

Question

What is an integer?

An integer is a type of number than includes both positive and negative whole numbers.

Show question

Question

What is the difference between a rational number and an irrational number?

A rational number is a number that can be written as a fraction or a decimal that terminates or reoccurs, where as a irrational number can not be put into a fraction since the decimal is non terminating.

Show question

Question

What is a real number?

A real number is a number system made up of; natural numbers, whole numbers, integers, rational numbers and irrational numbers.

Show question

Question

What is an operation?

An operation is the term used for any additions, subtractions, multiplication, division and squaring of a number.

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Question

Why is the order of operations important?

The order of operations is important to make sure that you get the correct answer to your sum.

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Question

What are the 4 steps of order of operations?

1. Calculate anything inside grouping symbols.
2. Calculate any powers.
3. Calculate any multiplication or division.
4. Calculate any addition or subtraction.

Show question

Question

What is the acronym to help you remember the 4 steps for the order of operations?

P - Parentheses

E - Exponent

M/D - Multiplication or Division

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Question

Values that can be expressed as an infinite decimal expansion are termed what?

Real numbers

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Question

A type of real number, written as the ratio of two integers is?

Rational number

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Question

Which of the following is not a real number?

None of the options

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Question

Which of these is a rational number with infinite decimals?

All the options

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Question

What is the symbol notation for real numbers?

R

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Question

Apply the associative property of real numbers to the following example. If (a+b)+c and a+(b+c), what is the result if a = 0.91, b = 12 and c = 0?

(0.91 +12) +0 = 0.91 + (12 + 0)

12.91 = 12.91

Show question

Question

Applying the distributive property of real numbers as a × (b + c) = (a × b) + (a × c), what will be the value of each equation if a = 66, b = -3 and c = 14

66 ×  (-3 + 14) = (66 ×  (-3)) + (66 ×  14)

726 = 726

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Question

What are the types of real numbers?

Rational and irrational numbers

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Question

What is not a real number?

Imaginary numbers

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Question

What is an irrational number?

Irrational numbers are a type of real numbers that cannot be written as the ratio of two integers.

Show question

Question

What is the symbol notation for irrational numbers?

Q'

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Question

“The product and sum of two real numbers remain the same even after interchanging the order of the numbers”. What property of real numbers does this appropriately describe?

Commutative property

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Question

How are rational numbers expressed?

p/q, where p and q are integers and not equal to 0.

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Question

Are negative numbers real numbers?

Negative numbers are real numbers

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Question

How do you identify integers?

They are positive and negative non-fractional natural numbers, including zero.

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Question

Which of these is an integer multiplication rule?

The product of two positive integers will always be a positive integer.

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Question

What are odd consecutive integers?

These are a set of integers such that each integer in the set differs from the previous integer by a difference of 2 and each integer is an odd number

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Question

What is the product of -4 and 7?

- 28

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Question

A set of integers is denoted by ...

Z

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Question

Integer numbers that follow each other in a sequence or in order without gaps are called…

Consecutive

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Question

What are integers?

Integers are numbers without fractional components.

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Question

What is integer division?

Integer division is the division where the fractional part is discarded.

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Question

Which of these are examples of an integer?

0 and 2/2

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Question

What is the difference between whole numbers and integers?

Whole numbers are counting numbers including 0, while integers are positive and negative whole numbers.

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Question

Are these two examples of consecutive integers?

• {-3, -2, -1, 0, 1, 2..}

• {10, 11, 12, 13, 14..}

Yes

Show question

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