Exemplos

Rational Numbers

Here are some examples! $$\dfrac{2}{3},\quad5,\quad4.5,\quad9\frac{1}{7},\quad6.\overline{4444}$$

For the record, $6.\overline{4444}$ means a $6$, then a decimal point, then infinitely many $4$s. It can also be written as the mixed number $6\frac{4}{9}$. Rational numbers are a subset of the real numbers. All real numbers that are not rational are called irrational.

Rational numbers have some really cool properties, like the reflexive, symmetric, and transitive properties. Check them out!

(Video) What are Rational Numbers?

by Don't Memorise

This video by Don't Memorise covers the basics you'll want to know.

$\large{\text{Summary}}$

When you hear the word "rational," what do you think of? Chances are, you thought of logic, of what it means to be a "rational" human being or make "rational" decisions.

But there's another word buried in there that might sound familiar to you: "ration." Forget about the "-al" at the end for a moment.

Now, when you hear the word "ration," what do you think of? Well, taken literally, a ration is essentially a fixed amount of a certain good (like food). Rations are enforced when there is not enough for everyone to have as much as they feel they need. In other words, there is a strict limit on how much food one may purchase, receive, or consume. In history, many countries rationed food for fighting soldiers during World War II.

But you came here to learn about math so let's focus on that. At the family dinner table.

Say. your family orders a ten-slice pizza for dinner. A whole pizza is broken up into ten slices of the same size. Your family might dictate that everyone can only have one out of those ten slices. The reason may be to save some leftovers for the next day. Everybody loves reheated pizza, after all.

Indeed, a ration of something, like pizza, is almost certainly a part of a whole. And when we talk about "parts of a whole" in the context of math (and especially when we use phrases like "one out of ten"), what are we talking about?

That's right, our favorite numbers with numerators and denominators, fractions. When we talk about rational numbers, we are talking about numbers that can be represented as fractions, or parts of a whole.

This includes $0$ (which can be represented as $\frac{0}{1}$, or $\frac{0}{2}$, or $\frac{0}{3}$, and so on), decimals like $0.25$ or $0.75$ (which are equivalent to the fractions $\frac{1}{4}$ and $\frac{3}{4}$, respectively), integers ($5=\frac{5}{1}$, $-1=\frac{-1}{1}$, et cetera), and numbers with infinitely repeating digits ($0.\overline{3}=\frac{1}{3}$, $0.\overline{6}=\frac{2}{3}$, and so on). All other numbers, that don't have this pattern, are called irrational. This includes square root expressions such as $\sqrt{2}$.

Finally, note that when you take the square root of a non-square number, the result is never a rational number. That's because you cannot express the result as a fraction with integer numerator and denominator.

Abundant Examples!

All Integers

The rational numbers include the integers. Think about it—you can express any integer as a fraction. All you have to do is let the integer be the numerator and $1$ (a nonzero integer) be the denominator. Here are a few examples: $$\begin{align} 5 &= \dfrac{5}{1} \\ -12 &=\dfrac{-12}{1}\\ 0 &= \dfrac{0}{1}\end{align}$$ In the last case, the denominator could be any nonzero number.

Fractions, mixed numbers, and decimals are all a bit more difficult to handle.

Ratios vs. Rational Numbers

When learning about a new topic in math, it helps to relate back to things you've already seen. The word ``rational" actually contains another very common word in math: the first five letters, ratio. Ratios compare two numbers or quantities using division. Sometimes we use a $:$ symbol in place of the division $\div$ symbol. For instance,

$$1 \ \text{ chicken} : 10 \ \text{ eggs} \Leftrightarrow \frac{1 \text{ chicken}}{10 \text{ eggs}}$$

While ratios are very important in solving certain word problems, for right now there's something else to consider. Ratios often have units, but if we take those away we're left with a number that's rational. That's assuming the bottom number is not a zero. In fact, we're left with a fraction, which we'll talk about next.

Certain Fractions and Mixed Numbers

As you saw in the graphic, all fractions of the form $\frac{p}{q},$ where $p$ and $q$ are integers and $q\neq 0$, are rational numbers. In this case, the numerator and denominator are $p$ and $q$, respectively.

For example, $\frac{-7}{22}$ is suitable because $-7$ and $22$ are integers and $22$ is nonzero.

The number $3\frac{1}{8}$ is also fine even though it is a mixed number. If we convert this mixed number to a fraction, we get $3\frac{1}{8} = \frac{25}{8}.$ This is what we want because $25$ and $8$ are integers and $8$ is nonzero. We could replace the denominator with any nonzero number and the same thing would happen every time!

Some fractions and mixed numbers are not rational, even though most of them are. For example, fractions such as $\frac{155}{0}$ causes problems. Sure, there's a numerator and a denominator but that alone isn't enough. Both $155$ and $0$ are integers, but remember, the denominator cannot be $0.$ Dividing by $0$ yields an undefined answer. By the same logic, $4\frac70$ also presents a problem. This number is undefined too. If you type either of these numbers into a calculator, you'll get an error message.

In addition, $\frac{\pi}{4}$ is not rational. Indeed, $4$ is a nonzero integer, but the numerator of the fraction, $\pi,$ is called irrational. Put another way, the digits of $\pi$ go on forever and have no pattern. Do you see a pattern here?

$$\pi = 3.14159265359\cdots$$

Certain Decimals

Decimals that terminate (do not go on forever) or repeat, are rational. We can convert these decimals to fractions that don't have a $0$ in the denominator. Other decimals are examples of irrational numbers, such as $\pi$ mentioned before.

Consider the number $1.8.$ This decimal terminates, because it only has one place value past the decimal. Thus we can convert it to a fraction like so: $$\begin{align} 1.8 &=\frac{1.8}{1}\\ &=\frac{1.8}{1} \cdot \frac{10}{10}\\ &=\frac{1.8 \cdot 10}{1 \cdot 10}\\ &=\frac{18}{10} \end{align}$$ We know that $18$ and $10$ are integers and $10$ is nonzero, so $1.8 = \frac{18}{10}$ is rational. While we can reduce the fraction $\frac{18}{10}$ to simplest form, there's no need to. That's because we were able to conclude $\frac{18}{10}$ is rational even without reducing it.

Next, consider $3.\bar{3}= 3.333\cdots$ We can convert this repeating decimal to the mixed number $3\frac{1}{3}.$ Then we can convert this mixed number to $\frac{10}{3},$ which is rational. The numerator and denominator are both integers, and the denominator is not zero.

So $3.\bar{3} = \frac{10}{3}$ is rational.

If a decimal does not terminate or repeat, it is called irrational. A famous example is $$\pi = 3.14159265359\cdots$$ which does not repeat. Another famous example is the square root expression

$$\sqrt{2} = 1.41421356237\cdots$$

This number, like most other square roots, is irrational.

Some Fun Facts about Rational Numbers

As you might remember, a rational number is any number that can be written as a fraction $\frac{p}{q}$. Here the numerator and denominator are both integers, and $q \neq 0$. The other explanations mostly listed out specific examples of these numbers. Now I want to talk about a couple other cool things that happen. You might call these properties of rational numbers, because they are statements which are true for any such number.

Countability

Here we're going to talk about the countability of rational numbers. That is, we're going to answer the question of how many there are.

You might be inclined to try counting these numbers for yourself. So you're probably writing down as many of them as you can. Go ahead. I'll wait.

Before long you probably realized it would take forever to list all rational numbers. Don't get discouraged because that's actually true. You could list every rational number but it would never end. Such lists of numbers are called countable.

Believe it or not, there are other lists of countable numbers. The most obvious one might be the positive integers:

$$1, 2, 3, 4, 5, 6, \cdots$$

This set of numbers is sometimes called the natural numbers or the counting numbers. The negative integers

$$-1, -2, -3, -4, -5, -6, \cdots$$

are also countable. Any two lists of countable numbers are just as long as each other. That means the list of positive integers and the list of rational numbers are equally long. You could start writing out rationals

$$4, -\frac{5}{3}, 7, 1.3, -\frac{1}{12}, \cdots$$

but this is even harder than listing out the positive integers.

Density

Another important property of the rational numbers is something called density. Here density means something completely different than it does in physics or chemistry. Now, density refers to how closely packed numbers are on a number line. In other words, how much space is between two numbers on the number line? A set of numbers is called dense if the size of this space is really small.

It turns out the rational numbers are dense. But let's make this term a little bit more precise. It means that between any two rational numbers we can find another rational number. And that's despite the fact that these numbers are countable! This gives us a way of generating more rational numbers forever. On a number line, it means numbers on this list are very close together. Here's an example.

Example: Let's find a rational number between $\frac{3}{4}$ and $\frac{5}{6}$. Specifically, we're going to find a fraction. Our technique is going to involve adding fractions with unlike denominators so be sure to review that first.

Let's pick the number halfway between $\frac{3}{4}$ and $\frac{5}{6}$. This is what we call the average or the midpoint of the numbers. We guess that this might be rational since all the numbers in our problem statement are. These are $\frac{3}{4}$, $\frac{5}{6}$, and $\frac{1}{2}$ (the fraction meaning "halfway"). To find the average of $\frac{3}{4}$ and $\frac{5}{6}$, we add them together and then multiply by $\frac{1}{2}$. This looks like

$$\frac{1}{2}\left(\frac{3}{4} + \frac{5}{6}\right)$$

and we can use the order of operations to handle this. The order of operations, or PEMDAS for short, says we want to simplify everything in parentheses first. As we said before, we add the two fractions $\frac{3}{4} + \frac{5}{6}$ together first. Since the denominators are different, we must find the least common multiple of them. Then that will become our new denominator:

$$\begin{align} \frac{3}{4} + \frac{5}{6} &= \\ \frac{3 \times 3}{4 \times 3} + \frac{5 \times 2}{6 \times 2} &= \\ \frac{9}{12} + \frac{10}{12} &= \\ \frac{19}{12} \end{align}$$

Now, we multiply this fraction by $\frac{1}{2}$:

$$\frac{19}{12} \times \frac{1}{2} = \frac{19 \cdot 1}{12 \cdot 2} = \frac{19}{24}$$

This is a rational number because the numerator and denominator are integers. Also important is that the denominator is not zero.

Feel free to try this process for a few more pairs of numbers.