- Recognize rational numbers in a list of numbers
- Simplify fractions and fractional expressions
Rational 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] |
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 number
A rational number is a number that can be written in the form [latex]{\Large\frac{p}{q}}[/latex], where [latex]p[/latex] and [latex]q[/latex] are integers and [latex]q\ne o[/latex].
All fractions, both positive and negative, are rational numbers.
A few examples of rational numbers are:
[latex]\Large\frac{4}{5}\normalsize ,-\Large\frac{7}{8}\normalsize ,\Large\frac{13}{4}\normalsize ,\text{and}-\Large\frac{20}{3}[/latex]
Are Integers Rational Numbers?
To decide if an integer is a rational number, we try to write it as a ratio of two integers. An easy way to do this is to write it as a fraction with denominator one.
[latex]3=\Large\frac{3}{1}\normalsize ,\space-8=\Large\frac{-8}{1}\normalsize ,\space0=\Large\frac{0}{1}[/latex]
Since any integer can be written as the ratio of two integers, all integers are rational numbers. Remember that all the counting numbers and all the whole numbers are also integers, and so they, too, are rational.
What about decimals? Are they rational? Let’s look at a few to see if we can write each of them as the ratio of two integers. We’ve already seen that integers are rational numbers. The integer [latex]-8[/latex] could be written as the decimal [latex]-8.0[/latex]. So, clearly, some decimals are rational.
Think about the decimal [latex]7.3[/latex]. Can we write it as a ratio of two integers? Because [latex]7.3[/latex] means [latex]7\Large\frac{3}{10}[/latex], we can write it as an improper fraction, [latex]\Large\frac{73}{10}[/latex]. So [latex]7.3[/latex] is the ratio of the integers [latex]73[/latex] and [latex]10[/latex]. It is a rational number. In general, any decimal that ends after a number of digits such as [latex]7.3[/latex] or [latex]-1.2684[/latex] is a rational number. We can use the place value of the last digit as the denominator when writing the decimal as a fraction.
Write each as the ratio of two integers:
- [latex]-15[/latex]
- [latex]6.81[/latex]
- [latex]-3\Large\frac{6}{7}[/latex]
Every rational number can be written both as a ratio of integers and as a decimal that either stops or repeats. Let’s dive into exactly what that means.
Let’s look at the decimal form of the numbers we know are rational. We have seen that every integer is a rational number, since [latex]a=\Large\frac{a}{1}[/latex] for any integer, [latex]a[/latex]. We can also change any integer to a decimal by adding a decimal point and a zero.
Integer [latex]-2,-1,0,1,2,3[/latex]
Decimal [latex]-2.0,-1.0,0.0,1.0,2.0,3.0[/latex]
These decimal numbers stop.
We have also seen that every fraction is a rational number. We can change any fraction to a decimal by dividing the numerator by the denominator.
Ratio of Integers [latex]\Large\frac{4}{5}\normalsize ,\Large\frac{-7}{8}\normalsize ,\Large\frac{13}{4}\normalsize ,\Large\frac{20}{3}[/latex]
Decimal Forms [latex]0.8,-0.875,3.25,6.666\ldots \text{ or } 6.\overline{66}[/latex]
These decimals either stop or repeat.
A line over a decimal number indicates that the number is a repeating decimal, meaning the digit or digits underneath the line will continue indefinitely.
For example, [latex]0.\overline{3}[/latex] represents the number [latex]0.3333\ldots[/latex], where the [latex]3[/latex] repeats forever. This notation provides a compact way to write repeating decimals without having to show the repeating part to an excessive number of places.