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# Rational numbers

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## What is a rational number?

The rational numbers can be represented by a fraction a / b, where a is the numerator and b is the denominator that must be nonzero. Suppose a = 2 and b = 4:

2 ← numerator / 4 ← denominator

Each rational number can be represented with infinite equivalent fractions.

2 / 4
=
4 / 8
=
1 / 2
...

By dividing each fraction we would get the same result:

0.5

The rational numbers can be expressed in fraction or as decimal numbers.

## What are the rational numbers?

The set of rational numbers is represented mathematically by the letter "Q", the set of the whole numbers is contained in the set of the rational numbers.

Note: Every number can be represented as a fraction but that does not mean that the fraction is a rational number.

The rational numbers are classified into two groups:

• Restricted: They are the ones that in their decimal representation have a fixed number. For example:

Examples:

1/2 = 0.5
1/4 = 0.25
9/4 = 2.25
• Periodicals: They are those that in their decimal representation have an unlimited number, they are classified in:
Periodicals Pure: When the repetition of the number or group of numbers is from the first decimal, for example:

Examples:

1/3 = 0.3333... ← Periodical = 3
8/6 = 1.3333... ← Periodical = 3
1/7 = 0.142857142857142857... ← Periodical = 142857
Periodicals mixed: When the repetition of the number or group of numbers from the second or subsequent decimal. For example:

Examples:

1/60 = 0.01666... ← Periodical = 6
937/330 = 2.8393939393... ← Periodical = 93
7/6 = 1.16666... ← Periodical = 6

### Characteristics of rational numbers

• They are infinite.
• It can be expressed in fraction or in decimal form.
• Between two rational numbers there are infinite rational numbers.
• Rational numbers contain whole numbers, they contain natural numbers.

## How can I identify if a number is rational?

Every number in the form of a fraction is a rational number, but if the incognita is a decimal number then we must verify if it is a rational or irrational number. The following steps is a procedure to know if the number is rational and of which classification, having as example A, B and C in the form of fraction:

A)
6 / 7
B)
12 /
5
C)
125 /
66
1. We must convert the fraction to a decimal number, for this we must make the division:
A) 6/7 = 0.857142857142857142857142...
B) 12/5 = 2.4
C) 125/66 = 1.89393939393...
2. We identify if in the decimal part a number or group of numbers is repeated, where the number or group of numbers corresponds to the repetition period and determine to which classification it corresponds.
A) Periodical = 857142 ← Periodical Pure.
B) Restricted
C) Periodical = 93 ← Periodical Mixed.

Now suppose that the unknown is given as a decimal number, therefore we now have as an example D, E and F in the form of a decimal number:

D) 0.27838383838383...
E) 123.142857142857...
F) 6.01001000100001...
1. The simplest is to verify if there is repetition of the numbers and in this way identify if it is a rational or irrational number.
D) Rational, Periodical = 83, Periodical Mixed.
E) Rational, Periodical = 142857, Periodical Pure.
F) Irrational.

When identifying that there is a period, in other words there is repetition every certain number we say that it is a rational number, but if there is no defined period it is said to be an irrational number.

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