What Are Rational Numbers and What Are They For?

Rational numbers are those that are symbolized by fractional numbers and whole numbers, represented by fractions. A rational number is also any number that can be represented as the fraction of two integers with a denominator other than zero.

Rational numbers have no achievement, since between each rational number there are unlimited numbers that could only be recorded throughout eternity. These numbers are used to interpret measurements, in some cases it is more favorable to express a number in this way than to convert it to an exact decimal, due to the many decimals that could be obtained from its division.

What is the function of rational numbers in everyday life?

Rational numbers are found in everyday life when you have to divide things into equal parts, when representing a measure, and when expressing numbers in exact decimals. Some examples of how rational numbers are used in everyday life are:

• When shopping: Give me 3/4 kilo of tomato, please.

• When talking to someone: I will call you in 1/2 hour , or I have brought three 1 and a half liter bottles of water.

• When doing an activity or sport: We ran 3 and a half kilometers.

• When we have to calculate portions: We buy a 14-piece pizza and to distribute it among 12 people, we have to count to give each one a piece of the pizza, in this way 12/14 is being used and that is a rational number . Another example would be with the sheet of paper: 1/8 of the paper is 1 part that has been cut into 8 parts, if we used all 8 parts, then it would be 8/8, which is the complete sheet.

Classification of Rational Numbers

Rational numbers are classified into:

• Non-null rationals (Q *) : They are represented by an asterisk (*) above the letter Q. This set is made up of rational numbers except zero (0).

• Non-negative rational numbers (Q +) : They are represented by the plus sign (+) that goes just after the letter Q. This set is made up of positive rational numbers in addition to zero.

• Non-positive rational numbers (Q-) : They are represented by the minus sign (-) that goes just after the letter Q. This set is made up of negative rational numbers in addition to zero.

• Positive rationals (Q * +) : It is represented by the asterisk (*) and the plus sign (+) . This set is made up of all positive rational numbers, except zero, which is a neutral number.
• Negative rationals (Q * -) : Represented by the asterisk (*) and minus sign (-) . This set is made up of the negative rational numbers minus zero.

Types of rational numbers

The equivalent fractions. They are those fractions that, although they are written differently, represent the same quantity. The fraction 1/3, for example, implies that the number 1 is divided into 3, or in other words, 1 divided by 3. Two or more equivalent elements, as long as they are similar or equal.

The proper fractions. They are rational numbers in which the numerator is less than the denominator. A proper fraction represents a number less than one. A proper fraction will always represent less than a whole cake.

The improper fractions. They are rational numbers in which the numerator is greater than the denominator. Improper fractions can be rewritten as a mixed number, that is, an integer plus a transparency of its own. An improper fraction represents a number greater than one.

Representation of Rational Numbers on the number line
On the number line the rational numbers are represented between the spaces of the numbers on said line, some of which are occupied by points that represent rational numbers.

When any rational numbers are represented, one has to work on their canonical representative, and in this case we are presented with two cases:

• The denominator is greater than the numerator : the point on the real line that represents these types of fractions is always between 0 and 1 or between 0 and -1 if it is negative. To carry them out, the unit is divided into as many parts as indicated by the denominator and taken as many as indicated by the numerator.
• The numerator is greater than the denominator: First, the fractional number is decomposed into a number composed of an integer and a fractional part, separating the numerator into two addends, one that is the highest multiple of the possible denominator and the other the difference up to the numerator .

The first of the added is divided by the denominator, thus obtaining the integer part of the number. The integer part tells us, from which unit we should make the representation, and the fractional part is treated the same as the first case, only with the indicated unit.

How are the rational numbers ordered?

Rational numbers are ordered directly depending on whether they are fractions with different denominators or if they are fractions with the same numerator. From there, simply compare the numbers in the cases that are different.

• Fractions that have the same denominator : the larger fraction will be the one with a larger numerator. An example, the order from highest to lowest would be: 10/6> 5/6> 4/6.

• Fractions with the same numerator : In this case, the fraction with the higher denominator will be less. An example: 3/5> 3/12.

• Fractions with different denominator and numerator: The least common multiple technique is used. The result of 3/4, 2/5 and 3/6 would be the following: 3/4> 3/6> 2/5.

Rational number operations

Add and subtract rational numbers.

There are two different ways to add and subtract rational numbers. The first is when the addends have a different denominator, and the second is when the addends have the same denominators.

• When we solve the addition and subtraction of rational numbers with the same denominator, the same denominator is preserved, which is the number located in the lower part of the fraction and the numerators that are the numbers located in the upper part are added or subtracted of the fraction. Example:

2/5 + 9/5 = 2+9/5 = 11/5

• When the denominators are of different value, the common denominator is determined, which will be the least common multiple of the denominators. This common denominator is divided by each of the denominators, multiplying the quotient obtained by the corresponding numerator and the numerators of the equivalent fractions obtained are subtracted or added depending on the operation. Example:

5/4 – 1/6 = 15-2/12 = 13/12

m.c.m. (4, 6) = 12

Multiplication of rational numbers

To multiply rational numbers, the first thing to do is multiply the numerators of all the numbers, and then the result is used as the numerator. Then the denominators are multiplied and the resulting product is located as the denominator, regardless of whether the value is the same or different. That is, the numerators are multiplied with each other and the denominators with each other. Example:

4/3 x 2/6 = 4×2/3×6 = 8/18

The division of rational numbers

In the division of rational numbers, the numerator of the initial fraction is taken and multiplied by the denominator of the next fraction, the result obtained will be used as the numerator. Subsequently, the denominator of the initial fraction is taken and multiplied by the numerator of the next fraction, and this result is the one that will be used as the denominator of the new fraction. That is, the dividend is multiplied by the multiplicative inverse of the divisor.

2/3 / 5/3 = 2×3/3×5 = 6/15

Properties of rational numbers

The following properties apply to rational numbers:

• The commutativity property: It is a property that has the addition and multiplication, since the operation of these does not depend on the order in which the numbers are placed, that is, the order of the factors does not affect the result.

• The associativity property : It is a property in algebra and propositional logic that is fulfilled in this way, given any three or more elements of a certain set, they can all be associated with each other, this means that the way of grouping the factors does not vary the result of the multiplication.

• Distributive property : It is the reverse process of the associative property. If several addends have a common factor, we can transform the sum into a product by extracting that factor.

• Neutral element: It is a null quantity which if added to any rational number, the answer will be the same rational number.

• Inverse additive or opposite element: It is the property of rational numbers according to which there is a negative number that cancels the existence of another. In other words, when adding them together, the result is zero.
• Internal property: This applies that when multiplying rational numbers, the result is also a rational number. This property also applies to division.

Rational numbers in decimal base

Every whole number accepts an infinite decimal representation, this representation is unique if unlimited sequences of 9 are discarded. Using the decimal representation, every rational number can be manifested as a limited decimal number and vice versa. In this way, the decimal value of a rational number is simply the result of dividing the numerator by the denominator.

Rational numbers are defined by having a decimal writing, which can be:

• Exact or Finite: It is the decimal part that has a limited number of digits. Since the zeros to the right of the decimal point are not significant, they can be omitted, resulting in a terminal expression.

• Pure periodic: It is a rational number with fractional part characterized by having a period figures that are infinitely repeated in their decimal expansion. This period can consist of one or more digits. Example: 1/7 = 0.142857142857…

• Mixed periodic: not all decimal fragments repeat. Example: 1/60 = 0.01666…

Conclusion on rational numbers

Any number that can be written in the form n / m, where the value of m is not equal to zero, is called a rational number. In our daily work, we continually have to use certain quantities that cannot be expressed with whole numbers. Therefore, the rational numbers cover this and many other basic needs.

Rational numbers are not just the fractions that we have known since we are children, all people at some point in their daily life use rational numbers, either to measure quantities of a recipe, to divide portions, in short, there are many applications practices that can be done with them, that make things easier.