# Why are all whole numbers rational?

• Last Updated : 03 Sep, 2021

The number system includes different types of numbers for example prime numbers, odd numbers, even numbers, rational numbers, whole numbers, etc. These numbers can be expressed in the form of figures as well as words accordingly. For example, the numbers like 40 and 65 expressed in the form of figures can also be written as forty and sixty-five. A Number system or numeral system is defined as an elementary system to express numbers and figures. It is the unique way of representing of numbers in arithmetic and algebraic structure.

Numbers are used in various arithmetic values applicable to carry out various arithmetic operations like addition, subtraction, multiplication, etc which are applicable in daily lives for the purpose of calculation. The value of a number is determined by the digit, its place value in the number, and the base of the number system. Numbers generally are also known as numerals are the mathematical values used for counting, measurements, labeling, and measuring fundamental quantities.

Numbers are the mathematical values or figures used for the purpose of measuring or calculating quantities. It is represented by numerals as 2, 4, 7, etc. Some examples of numbers are integers, whole numbers, natural numbers, rational and irrational numbers, etc.

### Types Of Numbers

There are different types of numbers categorized into sets by the number system. The types are described below,

• Natural numbers: Natural numbers are the positive numbers that count from 1 to infinity. The set of natural numbers is represented by ‘N’. It is the numbers we generally use for counting. The set of natural numbers can be represented as N = 1, 2, 3, 4, 5, 6, 7,…
• Whole numbers: Whole numbers are positive numbers including zero, which counts from 0 to infinity. Whole numbers do not include fractions or decimals. The set of whole numbers is represented by ‘W’. The set can be represented as W = 0, 1, 2, 3, 4, 5,…
• Integers: Integers are the set of numbers including all the positive counting numbers, zero as well as all negative counting numbers which count from negative infinity to positive infinity. The set doesn’t include fractions and decimals. The set of integers is denoted by ‘Z’. The set of integers can be represented as Z = …..,-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5,…
• Decimal numbers: Any numeral value that consists of a decimal point is a decimal number. It can be expressed as 2.5, 0.567, etc.
• Real number: Real numbers are the set numbers that do not include any imaginary value. It includes all the positive integers, negative integers, fractions, and decimal values. It is generally denoted by ‘R’.
• Complex number: Complex numbers are a set of numbers that include imaginary numbers. It can be expressed as a+bi where “a” and “b” are real numbers. It is denoted by ‘C’.
• Rational numbers: Rational numbers are the numbers that can be expressed as the ratio of two integers. It includes all the integers and can be expressed in terms of fractions or decimals. It is denoted by ‘Q’.
• Irrational numbers: Irrational numbers are numbers that cannot be expressed in fractions or ratios of integers. It can be written in decimals and have endless non-repeating digits after the decimal point. It is denoted by ‘P’.

### Why are all whole numbers rational?

In order to understand this question, it is important to first understand what are whole numbers and what are rational numbers, lets see in more detail what are whole numbers first,

The subset of numbers constituent of zero and all positive integers are whole numbers. The whole number counts from zero to infinity. These numbers are used for day-to-day calculation, mostly for the measurement of fundamental quantities. Whole numbers are the only constituent of natural numbers including zero. The subset is given by {0, 1, 2, 3, 4, 5,….},

The set does not include fractions, decimals, and negative integers.

Examples of Whole Numbers

Positive integers are also known as counting numbers including zero are parts of whole numbers, such as 0,1, 2, 3, 4, 5, etc, excluding negative integers, fractions, and decimals. 10, 11, 22, 100, 1000, etc all are examples of whole numbers.

Rational numbers are of the form p/q, where p and q are integers and q ≠ 0. Because of the underlying structure of numbers, p/q form, most individuals find it difficult to distinguish between fractions and rational numbers. When a rational number is divided, the output is in decimal form, which can be either ending or repeating. 3, 4, 5, and so on are some examples of rational numbers as they can be expressed in fraction form as 3/1, 4/1, and 5/1

Whole number is a constituent set of zero and all positive numbers like 0, 1, 2, 3, 10, 15,….includes only positive integers.It does not include negative integer, decimal or fractional value.

Rational numbers are of the form p/q, where p and q are integers and q ≠ 0. Because of the underlying structure of numbers, p/q form, most individuals find it difficult to distinguish between fractions and rational numbers.

When a rational number is divided, the output is in decimal form, which can be either ending or repeating. 3, 4, 5, and so on are some examples of rational numbers as they can be expressed in fraction form as 3/1, 4/1, and 5/1

Here set of whole number like 2 , 3 , 4 we can write this number as a fraction of 1 like 2/1 , 3/1 , 4/1 .. in ratio of two integers..

So thats why all whole numbers are rational. But its not valid as vice versa because every rational number is not whole number , as whole number doesn’t include decimal.

### Sample Problems

Question: identity from these numbers which are both rational and whole numbers?

7.88, 8, 3/4, 17890, 65.8989

Solution:

Here 8 and 17890 are both rational and whole numbers as it can be written as 8/1, 17890/1.

And 7.88 , 3/4 , 65.8989 are only rational numbers .

Question 2: identify whole numbers?

45, 68.09, 4/9, 16898

Solution:

Whole number is a constituent set of zero and all positive numbers like 0, 1, 2, 3, 10, 15,….includes only positive integers.

It does not include negative integer, decimal or fractional value. Here 45 and 16898 are the two whole numbers

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