RATIONAL NUMBERS: PROPERTIES, EXAMPLES AND OPERATIONS - MATHS - 2025

The rational numbers are all numbers can be obtained as the division of two integers. Examples of rational numbers are: 3/4, 8/5, -16/3 and those that appear in the following figure. In a rational number the quotient is indicated, being possible to do it later if required.

The figure represents any object, round for greater comfort. If we want to divide it into 2 equal parts, as in the right, we have two halves left and each one is worth 1/2.

Figure 1. Rational numbers are used to divide the whole into several parts. Source: Freesvg.

By dividing it into 4 equal parts, we will get 4 pieces and each one is worth 1/4, as in the image in the center. And if it has to be divided into 6 equal parts, each part would be worth 1/6, which we see in the image on the left.

Of course, we could also divide it into two non-equal parts, for example we could keep 3/4 parts and save 1/4 part. Other divisions are also possible, such as 4/6 parts and 2/6 parts. The important thing is that the sum of all the parts is 1.

In this way, it is evident that with rational numbers you can divide, count and distribute things like food, money, land and all kinds of objects in fractions. And so the number of operations that can be done with numbers is expanded.

Rational numbers can also be expressed in decimal form, as can be seen in the following examples:

1/2 = 0.5

1/3 = 0.3333…..

3/4 = 0.75

1/7 = 0.142857142857142857 ………

Later we will indicate how to go from one form to another with examples.

Properties of rational numbers

Rational numbers, whose set we will denote with the letter Q, have the following properties:

-Q includes natural numbers N and integers Z.

Taking into account that any number a can be expressed as the quotient between itself and 1, it is easy to see that among the rational numbers there are also natural numbers and integers.

Thus, the natural number 3 can be written as a fraction, and also -5:

3 = 3/1

-5 = -5/1 = 5 / -1 = - (5/1)

In this way, Q is a numerical set that includes a greater number of numbers, something very necessary, since the "round" numbers are not enough to describe all the possible operations to do.

-Rational numbers can be added, subtracted, multiplied and divided, the result of the operation being a rational number: 1/2 + 1/5 = 7/10; 1/2 - 1/5 = 3/10; (1/2) x (1/5) = 1/10; (1/2) ÷ (1/5) = 5/2.

-Between each pair of rational numbers, another rational number can always be found. In fact between two rational numbers there are infinite rational numbers.

For example, between the rationals 1/4 and 1/2 are the rationals 3/10, 7/20, 2/5 (and many more), which can be verified by expressing them as decimals.

-Any rational number can be expressed as: i) a whole number or ii) a limited (strict) or periodic decimal: 4/2 = 2; 1/4 = 0.25; 1/6 = 0.16666666 ……

-A same number can be represented by infinite equivalent fractions and all of them belong to Q. Let's see this group:

They all represent the decimal 0.428571…

-Of all the equivalent fractions that represent the same number, the irreducible fraction, the simplest of all, is the canonical representative of that number. The canonical representative of the example above is 3/7.

Figure 2.- The set Q of the rational numbers. Source: Wikimedia Commons. Uvm Eduardo Artur / CC BY-SA (https://creativecommons.org/licenses/by-sa/4.0).

Examples of rational numbers

-Proper fractions, those in which the numerator is less than the denominator:

-Improper fractions, whose numerator is greater than the denominator:

-Natural numbers and whole numbers:

-Equivalent fractions:

Decimal representation of a rational number

When the numerator is divided by the denominator, the decimal form of the rational number is found. For example:

2/5 = 0.4

3/8 = 0.375

1/9 = 0.11111…

6/11 = 0.545454…

In the first two examples, the number of decimal places is limited. This means that when the division is done, a remainder of 0 is finally obtained.

On the other hand, in the next two, the number of decimal places is infinite and that is why the ellipsis are placed. In the latter case there is a pattern in the decimals. In the case of the fraction 1/9, the number 1 is repeated indefinitely, while in 6/11 it is 54.

When this happens, the decimal is said to be periodic and is denoted by a caret like this:

Transform a decimal to a fraction

If it is a limited decimal, the comma is simply eliminated and the denominator becomes the unit followed by as many zeros as there are figures in the decimal. For example, to transform the decimal 1.26 into a fraction, write it like this:

1.26 = 126/100

Then the resulting fraction is simplified to the maximum:

126/100 = 63/50

If the decimal is unlimited, the period is first identified. Then these steps are followed to find the resulting fraction:

-The numerator is the subtraction between the number (without a comma or caret) and the part that does not have the caret.

-The denominator is an integer with as many 9 as there are figures under the circumflex, and as many 0 as there are figures in the decimal part that are not under the circumflex.

Let's follow this procedure to transform the decimal number 0.428428428… into a fraction.

-First, the period is identified, which is the sequence that is repeated: 428.

-Then the operation is done to subtract the number without a comma or accent: 0428 from the part that does not have a circumflex, which is 0. It remains like this 428 - 0 = 428.

-The denominator is constructed, knowing that under the circumflex there are 3 figures and all are under the circumflex. Therefore the denominator is 999.

-Finally the fraction is formed and simplified if possible:

0.428 = 428/999

It is not possible to simplify more.

Operations with rational numbers

- Add and subtract

Fractions with the same denominator

When the fractions have the same denominator, adding and / or subtracting them is very easy, because the numerators are simply added algebraically, leaving the same as the addends as the denominator of the result. Finally, if possible, it is simplified.

Example

Carry out the following algebraic addition and simplify the result:

The resulting fraction is already irreducible.

Fractions with different denominators

In this case, the addends are replaced by equivalent fractions with the same denominator and then the procedure already described is followed.

Example

Add algebraically the following rational numbers, simplifying the result:

The steps are:

-Determine the least common multiple (lcm) of the denominators 5, 8 and 3:

lcm (5,8,3) = 120

This will be the denominator of the resulting fraction without simplifying.

-For each fraction: divide the LCM by the denominator and multiply by the numerator. The result of this operation is placed, with its respective sign, in the numerator of the fraction. In this way, a fraction equivalent to the original is obtained, but with the LCM as the denominator.

For example, for the first fraction, the numerator is constructed like this: (120/5) x 4 = 96 and we get:

Proceed in the same way for the remaining fractions:

Finally, the equivalent fractions are replaced without forgetting their sign and the algebraic sum of the numerators is carried out:

(4/5) + (14/8) - (11/3) + 2 = (96/120) + (210/120) - (440/120) + (240/120) =

= (96 + 210-440 + 24) / 120 = -110 / 120 = -11/12

- Multiplication and division

Multiplication and division are done following the rules shown below:

Figure 3. Rules for multiplying and dividing rational numbers. Source: F. Zapata.

In any case, it is important to remember that multiplication is commutative, which means that the order of the factors does not alter the product. This does not happen with division, so care must be taken to respect the order between dividend and divisor.

Example 1

Carry out the following operations and simplify the result:

a) (5/3) x (8/15)

b) (-4/5) ÷ (2/9)

Answer to

(5/3) x (8/15) = (5 x 8) / (3 x 15) = 15/120 = 1/8

Answer b

(-4/5) ÷ (2/9) = (-4 x 9) / (5 x 2) = -36 / 10 = -18/5

Example 2

Luisa had $ 45. She spent a tenth of it buying a book and 2/5 of what was left on a t-shirt. How much money does Luisa have left? Express the result as an irreducible fraction.

Solution

The book cost (1/10) x $ 45 = 0.1 x $ 45 = $ 4.5

Therefore, Luisa was left with:

45 - 4.5 $ = 40.5 $

With that money Luisa went to the clothing store and bought the shirt, the price of which is:

(2/5) x $ 40.5 = $ 16.2

Now Luisa has in her portfolio:

40.5 - 16.2 $ = 24.3 $

To express it as a fraction it is written like this:

24.3 = 243/10

That is irreducible.

References

1. Baldor, A. 1986. Arithmetic. Editions and Distributions Codex.
2. Carena, M. 2019. Manual of Mathematics. National University of the Litoral.
3. Figuera, J. 2000. Mathematics 8. Ediciones Co-Bo.
4. Jiménez, R. 2008. Algebra. Prentice Hall.
5. The rational numbers. Recovered from: Cimanet.uoc.edu.
6. Rational numbers. Recovered from: webdelprofesor.ula.ve.