Rational and Real Numbers

Learning Objectives

• Identify the subset(s) of the real numbers that a given number belongs to.
• Locate points on a number line.
• Compare rational numbers.
• Identify rational and irrational numbers.

You’ve worked with fractions and decimals, like `3.8` and `21 2/3`. These numbers can be found between the integer numbers on a number line. There are other numbers that can be found on a number line, too. When you include all the numbers that can be put on a number line, you have the real number line. Let's dig deeper into the number line to see what those numbers look like and where they fall on the number line.

The fraction `16/3`, mixed number `5 1/3`, and decimal `5.33`… (or `5.bar(3)`) all represent the same number. This number belongs to a set of numbers that mathematicians call rational numbersNumbers that can be written as the ratio of two integers, where the denominator is not zero. . Rational numbers are numbers that can be written as a ratio of two integers. Regardless of the form used, `5.bar(3)` is rational because this number can be written as the ratio of `16` over `3`, or `16/3`.

Examples of rational numbers include the following.

`0.5`, as it can be written as `1/2`

`2 3/4`, as it can be written as `11/4`

`-1.6`, as it can be written as `-1 6/10=(-16)/10`

`4`, as it can be written as `4/1`

`-10`, as it can be written as `(-10)/1`

All of these numbers can be written as the ratio of two integers.

You can locate these points on the number line.

In the following illustration, points are shown for `0.5` or `1/2`, and for `2.75` or `2 3/4=11/4`.

As you have seen, rational numbers can be negative. Each positive rational number has an opposite. The opposite of `5.bar(3)` is `- 5.bar(3)`, for example.

Be careful when placing negative numbersNumbers less than `0`. on a number line. The negative sign means the number is to the left of `0`, and the absolute value of the number is the distance from `0`. So to place `-1.6` on a number line, you would find a point that is ` |-1.6| ` or `1.6` units to the left of `0`. This is more than `1` unit away, but less than `2`.

When two whole numbersThe numbers `0`, `1`, `2`, `3`, …., or all natural numbers plus `0`. are graphed on a number line, the number to the right on the number line is always greater than the number on the left.

The same is true when comparing two integersThe numbers …, `-3,` `-2,` `-1,` `0,` `1,` `2,` `3`… or rational numbers. The number to the right on the number line is always greater than the one on the left.

Here are some examples.

There are also numbers that are not rational. Irrational numbersNumbers that cannot be written as the ratio of two integers—the decimal representation of an irrational number is nonrepeating and nonterminating. cannot be written as the ratio of two integers.

Any square root of a number that is not a perfect square, for example `sqrt2`, is irrational. Irrational numbers are most commonly written in one of three ways: as a root (such as a square root), using a special symbol (such as `pi`), or as a nonrepeating, nonterminating decimal.

Numbers with a decimal part can either be terminating decimalsNumbers whose decimal parts do not continue indefinitely but end eventually—these are all rational numbers. or nonterminating decimalsNumbers whose decimal parts continue forever (without ending in an infinite sequence of zeros)—these decimals can be rational (if they repeat) or irrational (if they are nonrepeating).. Terminating means the digits stop eventually (although you can always write zeros at the end). For example, `1.3` is terminating, because there’s a last digit. The decimal form of `1/4` is `0.25`. Terminating decimals are always rational.

Nonterminating decimals have digits (other than `0`) that continue forever. For example, consider the decimal form of `1/3`, which is `0.3333`…. The `3`s continue indefinitely. Or the decimal form of `1/11`, which is `0.090909`…: the sequence “`09`” continues forever.

In addition to being nonterminating, these two numbers are also repeating decimalsNumbers whose decimal parts repeat a pattern of one or more digits—these are all rational numbers.. Their decimal parts are made of a number or sequence of numbers that repeats again and again. A nonrepeating decimalNumbers whose decimal parts continue without repeating—these are irrational numbers. has digits that never form a repeating pattern. The value of `sqrt(2),` for example, is `1.414213562`…. No matter how far you carry out the numbers, the digits will never repeat a previous sequence.

If a number is terminating or repeating, it must be rational; if it is both nonterminating and nonrepeating, the number is irrational.

The setA collection or group of things such as numbers. of real numbersAll rational or irrational numbers. is made by combining the set of rational numbers and the set of irrational numbers. The real numbers include natural numbersAlso called counting numbers, the numbers `1`, `2`, `3`, `4`, … or counting numbersAlso called natural numbers, the numbers `1`, `2`, `3`, `4`, ..., whole numbers, integers, rational numbers (fractions and repeating or terminating decimals), and irrational numbers. The set of real numbers is all the numbers that have a location on the number line.

The set of real numbers is all numbers that can be shown on a number line. This includes natural or counting numbers, whole numbers, and integers. It also includes rational numbers, which are numbers that can be written as a ratio of two integers, and irrational numbers, which cannot be written as a the ratio of two integers. When comparing two numbers, the one with the greater value would appear on the number line to the right of the one with the lesser value.