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Is Every Integer a Rational Number?

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  • Last Updated : 19 May, 2022

A number is a fundamental concept in mathematics. Numbers are used to counting, measure, maintain things in order, index, and so on. Natural numbers, whole numbers, rational and irrational numbers, integers, real numbers, complex numbers, even and odd numbers, and so on are examples of different sorts of numbers. We can use the fundamental arithmetic operations of numbers to calculate the resultant number. These numbers and words can be used to express these figures and words. A number system, often known as a numeral system, is a basic framework for representing numbers and figures. It is a separate approach to representing numbers in arithmetic and algebraic structures.

Integers 

Integers are a collection of numbers that include all positive counting numbers, zero, and all negative counting numbers that count from negative infinity to positive infinity. Fractions and decimals are not part of integers. The set of integers is symbolized by the letter ‘Z.’ Z = …-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5,… is the set of integers. The number from the set of negative and positive numbers, including zero, has no decimal or fractional portion. Integers include the numbers -9,-8, -7, -5, 0, 1, 5, 8, 97, and 20221, etc. There are two types of integers,

  1. Positive Integers: A positive integer number is one that is greater than zero. For instance: 1, 2, 3, 4,…
  2. Negative Integers: An integer number which is less than zero is known as negative integer. 

For instance, -1, -2, -3, -4, -5 , -6 … so on. Here in this context, zero is neither a negative nor a positive integer ,It is a whole number.

Rational numbers 

Rational numbers are of the type p/q, where p and q are integers and q ≠ 0. Most people have difficulty distinguishing between fractions and rational numbers due to the underlying structure of numbers, the p/q form. When a rational number is split, the result is in decimal form, which can be either ended or repeated. 5, 6, 7, and so on are instances of rational numbers since they can be written in fraction form as 5/1, 6/1, and 7/1.

Is Every Integer a Rational Number?

Answer: 

All integers are rational numbers because integers are the set of numbers that includes all positive counting numbers, zero, and all negative counting numbers that count from negative infinity to positive infinity. The collection does not include fractions or decimals.

However, all rational numbers are not integers since, as it is well known, rational numbers are of the type p/q, where p and q are integers and q ≠ 0. Most people have difficulty distinguishing between fractions and rational numbers due to the underlying structure of numbers, p/q form. They may be stated as fractions and decimals like 3/1, 4/1, and 5/1, 8.99, 0.90…

Integers do not include decimal or fractional values; instead, they are merely sets of counting numbers, whereas Rational numbers have both decimal and fractional values. Integer as well as rational number examples are -2, -1 , 1, 3, 4, 66, 88, 8900…

Sample Questions

Question 1: Identity from the below numbers which are both integer numbers and rational?

8.88, 9, 7/4, 174590, 65.222

Solution: 

Integers are a collection of numbers that include all positive counting numbers, zero, and all negative counting numbers that count from negative infinity to positive infinity. Fractions and decimals are not the part of integers .

Here, 9 and 174590 are both rational and integer numbers as it can be written as 8/1, 17890/1, and, 8.88, 7/4, 65.222 are only rational numbers.

Question 2: Identify the integers from the given number?

55, 558.09, 8/9, 188, – 7, -877

Solution:

Integers are a collection of numbers that include all positive counting numbers, zero, and all negative counting numbers that count from negative infinity to positive infinity. Fractions and decimals are not the part of integers. Hence, 55, 188, -7, -877 are integers.

Question 3: Give some Examples of Rational numbers?

Answer:

Rational numbers are of the type p/q, where p and q are integers and q ≠ 0. Most people have difficulty distinguishing between fractions and rational numbers due to the underlying structure of numbers, p/q form. 

Examples are 5,6,7,8,9 etc. They can be expressed in fraction and decimal form as 5/1, 6/1, and 7/1, 8.99, 0.90…

Question 4: Is 75 a rational number?

Answer:  

Rational numbers are of the type p/q, where p and q are integers and q ≠ 0. Most people have difficulty distinguishing between fractions and rational numbers due to the underlying structure of numbers, p/q form. 

Hence 75 here is a rational number as it can be expressed in form of p/q.

Question 5: What will be the number? If we subtract two integers?

Answer:

Let p and q are two integer, here p = 2 and q = 8.

Then p – q =  2 – 8 = -6 is an integer as well as rational number, because it can be written as -6/1.

Question 6: Is 96 a rational number or integer?

Answer:  

Rational numbers are of the type p/q, where p and q are integers and q ≠ 0. Most people have difficulty distinguishing between fractions and rational numbers due to the underlying structure of numbers, p/q form.

Hence 96 here is a rational number as it can be expressed in form of p/q and it is also an integer.

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