The subset comprising fractions is called the Rational Numbers. Included within this subset are the Integers, Whole Numbers, and Natural Numbers.

All Natural Numbers, Whole Numbers, and Integers can be written in fraction form, even though they are not typically considered fractional values. For example, 3 can be written as «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»3«/mn»«mn»1«/mn»«/mfrac»«/math».

«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨normal¨»§#8474;«/mi»«mo»=«/mo»«mfenced open=¨{¨ close=¨}¨»«mrow»«mi»q«/mi»«mo»|«/mo»«mi»q«/mi»«mo»=«/mo»«mfrac»«mi»a«/mi»«mi»b«/mi»«/mfrac»«mo»,«/mo»«mo»§#160;«/mo»«mi»a«/mi»«mo»,«/mo»«mo»§#160;«/mo»«mi»b«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»§#8484;«/mi»«mo»,«/mo»«mo»§#160;«/mo»«mi»b«/mi»«mo»§#8800;«/mo»«mn»0«/mn»«/mrow»«/mfenced»«/math»  
 The b-value in the denominator cannot equal zero because the value of the fraction would be undefined.

The set notation above is read as:

The Rational Numbers are defined as numbers that can be written as fractions in the form a over b, where a and b are Integers, and where b cannot equal zero.

  

Some examples of fractions are included on the number line above.

Fractions can also be written as repeating or terminating decimals, and vice versa. As such, the following are all examples of Rational Numbers:

 A line over numbers in a decimal represents a repeating digit or set of digits.
«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»0«/mn»«mo».«/mo»«mover»«mn»49«/mn»«mo»§#175;«/mo»«/mover»«mo»=«/mo»«mn»0«/mn»«mo».«/mo»«mn»49494949«/mn»«mo».«/mo»«mo».«/mo»«mo».«/mo»«/math»

«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»1«/mn»«mn»3«/mn»«/mfrac»«mo»,«/mo»«mo»§#160;«/mo»«mfrac»«mn»5«/mn»«mn»4«/mn»«/mfrac»«mo»,«/mo»«mo»-«/mo»«mfrac»«mn»9«/mn»«mn»7«/mn»«/mfrac»«mo»,«/mo»«mo»§#160;«/mo»«mn»0«/mn»«mo».«/mo»«mover»«mn»49«/mn»«mo»§#175;«/mo»«/mover»«mo»,«/mo»«mo»§#160;«/mo»«mn»18«/mn»«mo».«/mo»«mn»953«/mn»«/math»




Rational Numbers
are numbers that can be expressed as fractions of the form «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mi»a«/mi»«mi»b«/mi»«/mfrac»«/math», where a and b are Integers, and b must not equal zero (Expressed as decimals, they either terminate or repeat.)  «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨normal¨»§#8474;«/mi»«mo»=«/mo»«mfenced open=¨{¨ close=¨}¨»«mrow»«mi»q«/mi»«mo»|«/mo»«mi»q«/mi»«mo»=«/mo»«mfrac»«mi»a«/mi»«mi»b«/mi»«/mfrac»«mo»,«/mo»«mo»§#160;«/mo»«mi»a«/mi»«mo»,«/mo»«mo»§#160;«/mo»«mi»b«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»§#8484;«/mi»«mo»,«/mo»«mo»§#160;«/mo»«mi»b«/mi»«mo»§#8800;«/mo»«mn»0«/mn»«/mrow»«/mfenced»«/math»

 

There are many other Real Numbers that do not belong to any of the Real Number subsets discussed earlier in this lesson. Consider Pi, «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»§#960;«/mi»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mn»3«/mn»«mo».«/mo»«mn»141592654«/mn»«mo».«/mo»«mo».«/mo»«mo».«/mo»«/math» .

Pi represents the ratio of the circumference of a circle to its diameter. Pi does not terminate and does not repeat. As such, it cannot be written as a fraction.

Pi is an Irrational Number. Irrational Numbers do not terminate and do not have a repeating decimal pattern.

«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover»«mi mathvariant=¨normal¨»§#8474;«/mi»«mo»§#175;«/mo»«/mover»«mo»=«/mo»«mfenced open=¨{¨ close=¨}¨»«mrow»«mi»non«/mi»«mo»-«/mo»«mi»terminating«/mi»«mo»,«/mo»«mo»§#160;«/mo»«mi»non«/mi»«mo»-«/mo»«mi»repeating«/mi»«mo»§#160;«/mo»«mi»decimals«/mi»«/mrow»«/mfenced»«/math»

Irrational Numbers still lie on a number line and belong to the Real Number system, but they are a subset apart from the Natural Numbers, Whole Numbers, Integers, and Rational Numbers.

Irrational Numbers
numbers that cannot be expressed as fractions (Expressed as decimals, they neither terminate, nor repeat.)

«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover»«mi mathvariant=¨normal¨»§#8474;«/mi»«mo»§#175;«/mo»«/mover»«mo»=«/mo»«mfenced open=¨{¨ close=¨}¨»«mrow»«mi»non«/mi»«mo»-«/mo»«mi»terminating«/mi»«mo»,«/mo»«mo»§#160;«/mo»«mi»non«/mi»«mo»-«/mo»«mi»repeating«/mi»«mo»§#160;«/mo»«mi»decimals«/mi»«/mrow»«/mfenced»«/math»


The Real Numbers system includes all Rational Numbers and Irrational Numbers. «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨normal¨»§#8477;«/mi»«mo»=«/mo»«mi mathvariant=¨normal¨»§#8474;«/mi»«mo»§#160;«/mo»«mo»§#8746;«/mo»«mo»§#160;«/mo»«menclose notation=¨top¨»«mi mathvariant=¨normal¨»§#8474;«/mi»«/menclose»«/math»

Refer to the visual organization below. This is called a Venn diagram.

  

Real Numbers
the set of Rational Numbers and Irrational Numbers