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Numbers that lie between integers on the number line can be written as decimal numbers. These consist of an integer part, a decimal point as a separator, and a non-zero decimal part written to the right of the decimal point. Consider the number $12.346$ as an example.

The integer part of this number is $12.$ Since there is a decimal part, $0.346,$ the number is greater than $12$ but less than $13.$ Therefore, when plotting $12.346$ on a number line, the point will lie between $12$ and $13.$
It is important to note that these decimals can have very different values, depending on their place value.

A repeating decimal number, or **recurring decimal number**, is a number in decimal form in which some digits after the decimal point repeat infinitely. The digits repeat their values at regular intervals and the infinitely repeated part is not zero. When writing the decimal, a line is drawn over the repeating portion to express such a number.

Repeating Decimal Numbers | ||
---|---|---|

Number | Notation | Fraction |

$0.666666…$ | $0.6$ | $32 $ |

$1.533333…$ | $1.53$ | $1523 $ |

$5.373737…$ | $5.37$ | $99532 $ |

A terminating decimal number is a number in decimal form with a finite number of digits. Terminating numbers can be written as fractions, which means that they are rational numbers.

Terminating Decimal Numbers | ||
---|---|---|

Number | Fraction | |

$0.5$ | $21 $ | |

$1.53$ | $100153 $ | |

$52.372$ | $25013093 $ |

The set of irrational numbers is formed by all numbers that *cannot* be expressed as the ratio between two integers.
*not* repeating and non-terminating.

$2 ,3 ,5 ,e,π $

Irrational numbers are real numbers, but they cannot be expressed as fractions. Also, the decimal expansion of irrational numbers is $2 π =1.41421356237…=3.14159265359… $

In other words, a number is irrational if it is not rational. Although this number set does not have its own symbol, it is sometimes represented with a combination of other symbols. $R−QorR∖Q $