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Number Systems

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 Number Systems StudySmarterNatural 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 Number Systems StudySmarterWhole 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 Number Systems StudySmarterIntegers, 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 Diagram StudySmarter

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;

Worked example Number Systems StudySmarterGraphing 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.

Worked example Number Systems StudySmarter

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.

Number Systems Graphing real numbers StudySmarter

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.

Number Systems Graphing real numbers StudySmarter

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.

Frequently Asked Questions about Number Systems

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?

Show answer

Answer

  • 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?

Show answer

Answer

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?


Show answer

Answer

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?


Show answer

Answer

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?


Show answer

Answer

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? 

Show answer

Answer

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

Show question

Question

Why is the order of operations important?

Show answer

Answer

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

Show question

Question

What are the 4 steps of order of operations?

Show answer

Answer

  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?


Show answer

Answer

P - Parentheses 

E - Exponent 

M/D - Multiplication or Division 

A/S - Addition or Subtraction

Show question

Question

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


Show answer

Answer

Real numbers

Show question

Question

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


Show answer

Answer

Rational number

Show question

Question

Which of the following is not a real number? 

Show answer

Answer

None of the options

Show question

Question

Which of these is a rational number with infinite decimals?


Show answer

Answer

All the options

Show question

Question

What is the symbol notation for real numbers?


Show answer

Answer

R

Show question

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?


Show answer

Answer

(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 


Show answer

Answer

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

    726 = 726

Show question

Question

What are the types of real numbers?


Show answer

Answer

Rational and irrational numbers 

Show question

Question

What is not a real number? 


Show answer

Answer

Imaginary numbers

Show question

Question

What is an irrational number? 

Show answer

Answer

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?


Show answer

Answer

Q'

Show question

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?


Show answer

Answer

Commutative property

Show question

Question

How are rational numbers expressed?


Show answer

Answer

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

Show question

Question

Are negative numbers real numbers?


Show answer

Answer

Negative numbers are real numbers

Show question

Question

How do you identify integers? 

Show answer

Answer

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


Show question

Question

Which of these is an integer multiplication rule? 

Show answer

Answer

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

Show question

Question

What are odd consecutive integers? 

Show answer

Answer

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 

Show question

Question

What is the product of -4 and 7?


Show answer

Answer

- 28

Show question

Question

A set of integers is denoted by ... 


Show answer

Answer

Z

Show question

Question

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


Show answer

Answer

Consecutive

Show question

Question

What are integers?

Show answer

Answer

Integers are numbers without fractional components.


Show question

Question

What is integer division?

Show answer

Answer

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

Show question

Question

Which of these are examples of an integer? 


Show answer

Answer

0 and 2/2

Show question

Question

What is the difference between whole numbers and integers?


Show answer

Answer

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

Show question

Question

Are these two examples of consecutive integers?

  • {-3, -2, -1, 0, 1, 2..}

  • {10, 11, 12, 13, 14..}

Show answer

Answer

Yes

Show question

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