Rational Numbers
6.1 The Set of Rational Numbers
Rational Numbers
Rational numbers, $\mathbb{Q}$, are used to help us understand pieces of wholes, proportions, rates, and much more. They help solve problems that natural numbers and integers are not able to.
As we have developed our understanding of the different sets of numbers, we have slowly filled in the number line. First, we started with whole numbers, $\mathbb{W}$ which helped us fill out the right side of the number line. Then we added in integers, $\mathbb{Z}$, which filled out the left side of the number line. However, there are still gaps to be filled.
For example, say we start with $6$ cookies and we want to divide them into two. Thats no issue for us, $6\div 2 = 3$, we'll make two groups of $3$.
$$\HUGE \unicode{x25CF} \unicode{x25CF} \unicode{x25CF} \quad \unicode{x25CF} \unicode{x25CF} \unicode{x25CF}$$
But if we wanted to split the $3$ into two parts we have an issue. We can't really split up the group of $3$ into $2$ even parts.
$$\HUGE \unicode{x25CF} \unicode{x25CF} \unicode{x25CF}$$
But it is possible to split the cookies into pieces. Say, we cut each single cookie in half.
$$\HUGE \unicode{x25D6}\unicode{x25D7}\unicode{x25D6}\unicode{x25D7}\unicode{x25D6}\unicode{x25D7}$$
Now there is no problem creating two separate groups. This is what rational numbers are for us, a way to talk about pieces of things.
$$\HUGE \unicode{x25D6} \unicode{x25D6} \unicode{x25D6} \quad \unicode{x25D7} \unicode{x25D7} \unicode{x25D7}$$
Definition: Rational Number
A rational number is any number that can be written as
$$\frac{a}{b}$$
where $a,b\in\mathbb{Z}$ and $b\ne 0$.
Rational numbers are also called fractions. This term really only refers to fully reduce rational numbers that are not integers, but we'll talk more about that later. Lets focus on the simplest type of rational number first.
Definition: Unit Fraction
A fraction of the form $\frac{1}{a}$ with integer $a\ne 0$ is a unit fraction.
To understand how rational numbers are written, consider how we divided those $3$ cookies. We took each cookie and divided it into $2$ pieces.
$$\HUGE \overbrace{\unicode{x25CF}}^{1} \rightarrow \overbrace{\unicode{x25D6}\unicode{x25D7}}^{\frac{1}{2}+\frac{1}{2}}$$
The bottom of the rational number, the denominator, is telling us how many of these pices it takes to make the whole. While the top of the rational number, the numerator, is telling us how many of those pieces we have.
Putting it all together: $3\div 2$ means we cut every unit unto $2$ pieces, and we had $3$ of those units to cut. Each of these pieces are $\frac{1}{2}$ and we have $3$ of them, so
$$3\div 2 = 3\cdot \frac{1}{2} = \frac{3}{2}$$
$$\HUGE \unicode{x25D6} \unicode{x25D6} \unicode{x25D6}$$
On the Number Line
To find a rational number on the numberline, we first need to divide each interval into the number of pieces we're working with. If we're working with halves, we would mark every integer on the numberline. Then for each space between integers we would further divide it into halves.
Example: Rationals on the Number Line
Place each on the number line: $\frac{1}{5}$, $\frac{-17}{4}$, $\frac{-11}{3}$, $\frac{4}{7}$, $\frac{57}{2}$.
Show Solution
Use denominator as the partition size and numerator as the number of parts.
- $\frac{1}{5}$: split one unit into $5$ equal parts, move $1$ part right from $0$.
- $\frac{-17}{4}$: cut into fourths, count $17$ in the negative direction
- $\frac{-11}{3}$: split one unit into $3$ parts, move $11$ parts left from $0$.
- $\frac{4}{7}$: split into sevenths, move 4 parts right.
- $\frac{57}{2}$ You might notice that we can speed up the counting by dividing up our halves. $57$ halves is the same is $56+1$ halves. $$\frac{56}{2}+\frac{1}{2}$$
The reason why this helps us is because $\frac{56}{2}$ means $56\div 2=28$. So the number we're tyring to represent is $28+\frac{1}{2}$. We can move over to $28$ and then count one more $\frac{1}{2}$.
The key habit is always: partition by denominator first.
MyOpenMath: Rational points on a number line
Equivalent Fractions and Simplifying
Now for an idea that serves as the backbone of working with rational numbers: their representations are not unique. Having two halves means we have one whole. $\frac{2}{2}=1$. It is also true that three thirds will be one whole. $\frac{3}{3}=1$. That would seem to imply that $\frac{2}{2}=\frac{3}{3}$, because they're both one.
Definition: Equivalent Fractions
Two fractions are equivalent if they represent the same number on the number line.
Example: Equivalent Fractions
Plot both $\frac{7}{3}$ and $\frac{14}{6}$ on a number line.
Show Solution
Theorem: Fundamental Law of Fractions
If $n\ne 0$, then
$$\frac{a}{b}=\frac{na}{nb}.$$
Here we can get to the crux of the issue. If we write the denominator of a rational number as its prime factorization, that tells us the different ways we could divide the denominator into.
$$\frac{14}{6}=\frac{14}{2\cdot 3}$$
So we can think about that denominator as $2$ groups of $3$ or $3$ groups of $2$. But factoring the top reveals more information.
$$\frac{14}{6}=\frac{2\cdot 7}{2\cdot 3}$$
The numerator can also be thought of in groups of $2$. Because both the numerator and the denominator can be though of in groups of $2$, we can simplify the problem by only looking at one of the groups of two.
$$\frac{2\cdot 7}{2\cdot 3}=\cancel{\frac{2}{2}}\cdot\frac{7}{3}=\frac{7}{3}$$
So whenever both the numerator and the denominator share a factor, that factor can be discarded to simplify the number.
Definition: Simplest Form
A rational number $\frac{a}{b}$ is in simplest form when $\gcd(a,b)=1$.
Example: Find Equivalent Rational Numbers
Find an equivalent rational number for each: $\frac{3}{4}$, $\frac{10}{15}$, $\frac{7}{8}$, and $2$.
Show Solution
- This one is already in its simplest form, so we'll need to introduce common factors for the top and bottom to find an equivalent number. $$\frac{3}{4}=\frac{2}{2}\cdot\frac{3}{4}=\frac{6}{8}$$
- This number isn't reduced, so we can find its simplest form which would be an equivalent number. $$\frac{10}{15}=\frac{2\cdot 5}{3\cdot 5}=\frac{2}{3}$$
- $\frac{7}{8}=\frac{5\cdot 7}{5\cdot 8}=\frac{35}{40}$
- $2$ is interesting because its a whole number. But whole numbers can be written as rational numbers, specifically ones who only need $1$ piece to make the whole. $$2=\frac{2}{1}=\frac{2\cdot 9}{1\cdot 9}=\frac{18}{9}$$
MyOpenMath: Equivalent Rational Number
MyOpenMath: Another Equivalent Rational Number
Comparing Rational Numbers
Lets talk about comparing rational numbers, but using an area model approach. Say we have a young student who claims $\frac{1}{3}$ is larger than $\frac{1}{2}$ and draws this picture to show us. What would you say with the student?
The first noticable thing is that one circle is much larger than the other. When working with fractions, we need to be talking about pieces of the same thing. $\frac{1}{2}$ of a mile is not the same thing as $\frac{1}{2}$ of a foot. So the comparison in this picture is misleading.
Heres a Geogebra link that can help us visualize comapring fractions. Use it to compare $\frac{1}{2}$ and $\frac{1}{3}$. In short, to compare fractions we can use our fundamental law of fractions to get a common denominator. If we change our fractions so that we're talking about pieces of the same size, then we can compare.
Fact: Comparing via Common Denominator
Given two rational numbers $\frac{a}{b}$ and $\frac{c}{d}$, (both $b$ and $d$ not $0$) we can always use the funamental law of fractions to rewrite them as $\frac{ad}{bd}$ and $\frac{bc}{bd}$. At this point, the denominators are the same and we can compare them by just comparing the numerators.
If both numbers are positive,
- $ad>bc$ implies $\frac{a}{b}>\frac{c}{d}$
- $ad<bc$ implies $\frac{a}{b}<\frac{c}{d}$
- $ad=bc$ implies $\frac{a}{b}=\frac{c}{d}$
If both numbers are negative, we can reverse the inequality. If one is negative and the other positive, it is clear the negative number is smaller.
This is the origin of cross-multiplying to solve proportions, which we'll cover a little more closely later on. But to finish our example:
Example: Comparing Rational Numbers
Decide the if $\frac{1}{2}$ is greater than, less than, or equal to $\frac{1}{3}$.
Show Solution
and since $3>2$, we have $\frac{1}{2}>\frac{1}{3}$
MyOpenMath: Comparing rational numbers
Denseness of the Rationals
Being able to divide unit it to more and more pieces while keeping the value of a fraction the same brings is to an interesting property that is unique to rational numbers. Between any two rational numbers, there is always another rational number.For $\mathbb{N}, \mathbb{W}$ or $\mathbb{Z}$ it is not the case that you can find a number between them. For example given $2<3$, there are no integers between $2$ and $3$, but there is a rational number between them.
Theorem: Denseness of the Rationals
If $a,b\in\mathbb{Q}$ and $a<b$, then there exists $c\in\mathbb{Q}$ such that $a<c<b$.
Its very straight forward, lets take two fractions that seem close together: $\frac{3}{5}<\frac{4}{5}$. All we have to do is use the fundamental law of fractions to make that denominator a little finer.
$$\frac{3}{5}\cdot \frac{2}{2}<\frac{4}{5}\cdot \frac{2}{2}$$ $$\frac{6}{10}<\frac{8}{10}$$
Now we can see the number right in the middle.
$$\frac{6}{10}<\frac{7}{10}<\frac{8}{10}$$
And there is nothing to stop us from from doing it ad infinitum.
Example: Finding rational numbers
Find a rational number between each of the following pairs of numbers.
- $\frac{9}{11}$ and $\frac{10}{11}$
- $\frac{1}{2}$ and $\frac{7}{9}$
Show Solution
- Using our law of fractions, $$\frac{9}{11}<\frac{10}{11}$$ $$\frac{9}{11}\cdot \frac{2}{2}<\frac{10}{11}\cdot \frac{2}{2}$$ $$\frac{18}{22}<\frac{20}{22}$$ $\frac{19}{22}$ is a good choice here, but $\frac{20}{22}$ works as well. Both are between the given fractions.
- Well, to start, its hard to compare $\frac{1}{2}$ and $\frac{7}{9}$ because they have diffrent denominators. So lets get a common denominator. $$\frac{1}{2}\cdot \frac{9}{9} = \frac{9}{18}$$ $$\frac{7}{9}\cdot \frac{2}{2} = \frac{14}{18}$$ And that gives us a large number of choices. $\frac{10}{18}$ is as good a pick as any.
MyOpenMath: Denseness of the Rationals
One more fun thing before we end this section: consider two rational numbers, and we'll keep the denominators positive. So if we need a negative rational number we can keep the negative sign in the numerator.
$$\frac{a}{b}<\frac{c}{d}$$
We can always find a number between them using the mediant.
Definition: Mediant
For $\frac ab$ and $\frac cd$, the mediant is
$$\frac{a+c}{b+d}.$$
If $\frac ab<\frac cd$ with positive denominators, then
$$\frac ab<\frac{a+c}{b+d}<\frac cd.$$
Example: Find numbers in between
Find two rational numbers between $\frac{7}{18}$ and $\frac{1}{2}$.
Show Solution
Well, we know that $\frac{7}{18}<\frac{1}{2}$ (why?). And according to the definition above, $$\frac{7+1}{18+2}=\frac{8}{20}$$ should be between them. Lets check.
$$7\cdot 20 =140 < 144=8 \cdot 18$$ means $\frac{7}{18}<\frac{8}{20}$ and $$8\cdot 2 =16<20=1\cdot 20$$
means $\frac{8}{20}<\frac{1}{2}$.
This can be a quick way to find a rational number between two other numbers if getting a common denominator is too time consuming.