Rational numbers

Rational numbers are all numbers that can be written as a quotient of two whole numbers, i.e. in the form of a fraction.

Definition

Rational numbers are therefore all numbers that can be written in the form

$$\frac{a}{b}\qquad a,b\in\mathbb{Z}, b\ne0.$$

The condition that both numbers are integers has an obvious meaning, because if you chose a number that is not rational, for example π, then if you constructed a fraction like this:

$$\frac{\pi}{1},$$

you would get back the number π, which is not rational. Therefore, both the denominator and numerator must be integers. And since we are not allowed to divide by zero, the denominator must be different from zero.

We can also say that rational numbers are numbers with finite decimal expansion except when some part repeats periodically. So examples of rational numbers, first in fraction form:

$$\frac13,\qquad -\frac59,\qquad \frac{13}{29},\qquad \frac{4}{1}, \ldots$$

and then numbers in decimal form: 0,1; −5; 14,5; −12,93; 0,33333...=$0,\overline{3}$

Periodic notation

In the previous example, we had the number 0,3333... This is the decimal representation of the fraction 1/3 and it is a number with infinite decimal expansion. Yet it is rational because the development is periodic. Periodic development means that from a certain part of the number onwards, the same sequence of numbers is repeated over and over again. In our case, the number three has been repeated ad infinitum. But it can repeat indefinitely, for example, a sequence of numbers 12345, it will still be a number with a period, a rational number.

So any number with a finite decimal expansion is a rational number. Any number with infinite decimal expansion where some part repeats periodically is also a rational number. A number with infinite decimal expansion where no part repeats periodically is an irrational number.

The period is indicated either by three dots or, more commonly, by a line over the numbers that repeat periodically. One number may repeat periodically, but more than one number may repeat periodically. The period may start at the beginning of the decimal part or at any time later. These are all periodic numbers:

$$\begin{array}{rclcl} 7/3&=&2{,}33333\ldots&=&2,\overline{3}\\ 16/11&=&1{,}454545\ldots&=&1,\overline{45}\\ 11/6&=&1{,}833333\ldots&=&1{,}8\overline{3} \end{array}$$

Markings and meaning

We mark rational numbers using the letter Q with double arcs: ℚ. It is from the English "quotient", Czech "quotient", indicating the result after division.

Rational numbers are used to denote a part of a whole. Typically, for example, "half" or "tenth". These are designations of the whole that can be expressed in rational numbers as a fraction 1/2 and 1/10. In these cases, the denominator denotes the whole and the numerator denotes a part of the whole. If the numerator equals the denominator, then it means we have the whole. These numbers are then usually converted to percentages.

Properties

Rational numbers are an infinite countable set.

Rational numbers are closed under the operations of addition, subtraction, multiplication and division. This means that when we divide two rational numbers, we get a rational number again. This is a change from integers that were not closed on the division operation.
Rational numbers contain all integers. If all rational numbers are expressible by the fraction a/b, then it is enough to b = 1 and after dividing a/1 = a we always get back the value of the numerator.
If we take two rational numbers a and b for which a < b holds , we can always find another rational number q, for which it holds: a < q < b. In other words, between any two rational numbers we can always find some other rational number. A stronger version is also true: between any two rational numbers there are infinitely many other rational numbers.
Thus, a consequence of the previous theorem is that there is no smallest positive (or largest negative) rational number. We can use a proof by contradiction to do this. Let m be the smallest positive rational number. Then we may not be able to find a smaller positive rational number. However, according to the previous theorem, 0 < m holds and so does 0 < s < m. We have found a number s, which is greater than zero (is positive) and is smaller than m. Which is a dispute with the fact that m is the smallest positive rational number.

Operations with rational numbers

All common and basic operations are described in the article Fractions.

Converting a periodic number to a fraction

Since a periodic number is also a rational number, it must be converted to a fraction. We will show the procedure to do this. To begin with, consider a periodic number a = 0,333... Write

$$a=0,\overline{3}$$

Now we will perform the classic equivalent equation adjustment and multiply the whole equation by ten:

$$10a=3,\overline{3}$$

On the right hand side, the 3's have moved to ones, leaving an infinite number of 3's still after the decimal point. We now subtract a from the equation , thus subtracting 0,333..., getting rid of the infinite expansion.

$$9a=3$$

This step may have been a bit complicated, so I'll break it down. We did this operation:

$$\begin{array}{ccccccc} &3&,&3&3&3&\ldots\\ -&0&,&3&3&3&\ldots\\ =&3&,&0&0&0&\ldots \end{array}$$

We just divide the top equation 9a = 3 by nine to get the result:

$$a=\frac{3}{9}=\frac13$$

Let's try one more example:

$$a=1{,}454545\ldots=1,\overline{45}$$

The procedure will be slightly different. In the last example, we had a period of length one. To be more precise, the period could be as long as we wanted because the same numbers kept repeating, but it was convenient to have it be one long. At the moment we have a period of length two. Again, we could take a period of length six, but it doesn't suit us. To really subtract the whole period, we have to multiply by 100 this time:

$$100a=145,\overline{45}$$

Subtract one a:

$$99a=144$$

And divide by 99:

$$a=\frac{144}{99}=\frac{16}{11}$$

If you try dividing these numbers on a calculator, you'll get just 1,454545... Most calculators round off, though, so it may give you a slightly different result.

Interesting thing with 0.9999...

Just for fun, let's try to convert a periodic number to a fraction

$$a=0,\overline{9}$$

Multiply by ten:

$$10a=9,\overline{9}$$

Subtract a:

$$9a=9$$

Divide by nine:

$$a=1$$

We see that we got one, so the number 0,999... is equal to 1.