### Learning Outcomes

- Identify rational numbers from a list of numbers
- Identify irrational numbers from a list of numbers

counting numbers | [latex]1,2,3,4\dots [/latex] |

whole numbers | [latex]0,1,2,3,4\dots[/latex] |

integers | [latex]\dots -3,-2,-1,0,1,2,3,4\dots [/latex] |

### Rational Numbers

What type of numbers would you get if you started with all the integers and then included all the fractions? The numbers you would have form the set of rational numbers. A rational number is a number that can be written as a ratio of two integers.### Rational Numbers

A rational number is a number that can be written in the form [latex]\frac{p}{q}[/latex], where [latex]p[/latex] and [latex]q[/latex] are integers and [latex]q\ne o[/latex].

### example

Write each as the ratio of two integers:1. [latex]-15[/latex]2. [latex]6.81[/latex]3. [latex]-3\frac{6}{7}[/latex]Solution:

1. | |

[latex]-15[/latex] | |

Write the integer as a fraction with denominator 1. | [latex]\frac{-15}{1}[/latex] |

2. | |

[latex]6.81[/latex] | |

Write the decimal as a mixed number. | [latex]6\frac{81}{100}[/latex] |

Then convert it to an improper fraction. | [latex]\frac{681}{100}[/latex] |

3. | |

[latex]-3\frac{6}{7}[/latex] | |

Convert the mixed number to an improper fraction. | [latex]-\frac{27}{7}[/latex] |

### try it

[ohm_question]145911[/ohm_question]

Rational Numbers | ||
---|---|---|

Fractions | Integers | |

Number | [latex]\frac{4}{5},-\frac{7}{8},\frac{13}{4},\frac{-20}{3}[/latex] | [latex]-2,-1,0,1,2,3[/latex] |

Ratio of Integer | [latex]\frac{4}{5},\frac{-7}{8},\frac{13}{4},\frac{-20}{3}[/latex] | [latex]\frac{-2}{1},\frac{-1}{1},\frac{0}{1},\frac{1}{1},\frac{2}{1},\frac{3}{1}[/latex] |

Decimal number | [latex]0.8,-0.875,3.25,-6.\overline{6}[/latex] | [latex]-2.0,-1.0,0.0,1.0,2.0,3.0[/latex] |

### Irrational Numbers

Are there any decimals that do not stop or repeat? Yes. The number [latex]\pi [/latex] (the Greek letter pi, pronounced ‘pie’), which is very important in describing circles, has a decimal form that does not stop or repeat.[latex-display]\pi =\text{3.141592654.......}[/latex-display]Similarly, the decimal representations of square roots of numbers that are not perfect squares never stop and never repeat. For example,[latex-display]\sqrt{5}=\text{2.236067978.....}[/latex-display]A decimal that does not stop and does not repeat cannot be written as the ratio of integers. We call this kind of number an irrational number.### Irrational Number

An irrational number is a number that cannot be written as the ratio of two integers. Its decimal form does not stop and does not repeat.

- stops or repeats, the number is rational.
- does not stop and does not repeat, the number is irrational.

### example

Identify each of the following as rational or irrational:1. [latex]0.58\overline{3}[/latex]2. [latex]0.475[/latex]3. [latex]3.605551275\dots [/latex]

Answer: Solution:1. [latex]0.58\overline{3}[/latex]The bar above the [latex]3[/latex] indicates that it repeats. Therefore, [latex]0.58\overline{3}[/latex] is a repeating decimal, and is therefore a rational number.2. [latex]0.475[/latex]This decimal stops after the [latex]5[/latex], so it is a rational number.3. [latex]3.605551275\dots[/latex]The ellipsis [latex](\dots)[/latex] means that this number does not stop. There is no repeating pattern of digits. Since the number doesn't stop and doesn't repeat, it is irrational.

### try it

[ohm_question]145910[/ohm_question]

### example

Identify each of the following as rational or irrational:1. [latex]\sqrt{36}[/latex]2. [latex]\sqrt{44}[/latex]

Answer: Solution:1. The number [latex]36[/latex] is a perfect square, since [latex]{6}^{2}=36[/latex]. So [latex]\sqrt{36}=6[/latex]. Therefore [latex]\sqrt{36}[/latex] is rational.2. Remember that [latex]{6}^{2}=36[/latex] and [latex]{7}^{2}=49[/latex], so [latex]44[/latex] is not a perfect square.This means [latex]\sqrt{44}[/latex] is irrational.

### try it

[ohm_question]145915[/ohm_question]

## Licenses & Attributions

### CC licensed content, Original

- Determine Rational or Irrational Numbers (Square Roots and Decimals Only).
**Authored by:**James Sousa (Mathispower4u.com) for Lumen Learning.**License:**CC BY: Attribution. - Question ID 145910, 145915, 145911.
**Authored by:**Lumen Learning.**License:**CC BY: Attribution.

### CC licensed content, Shared previously

- Real Numbers.
**Authored by:**James Sousa (Mathispower4u.com).**License:**CC BY: Attribution.

### CC licensed content, Specific attribution

- Prealgebra.
**Provided by:**OpenStax**License:**CC BY: Attribution.**License terms:**Download for free at http://cnx.org/contents/[emailprotected].