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Are all integers rational numbers?

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  • Last Updated : 08 Jun, 2022
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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 a unique way of representing numbers in arithmetic and algebraic structure.

Number

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 of sets include natural numbers, whole numbers, integers, decimal numbers, etc. Let’s learn about them in more detail,

  • 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’.

Are all integers rational numbers?

Answer: 

First, let’s learn about Rational numbers and Integers,

  • Rational number: 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
  • 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,… The number with no decimal or fractional part from the set of negative and positive numbers, including zero. Examples of integers are: -8, -7, -5, 0, 1, 5, 8, 97, and 3,043.
  1. Positive Integers: The integer number is positive if it is greater than zero. Example: 1, 2, 3, 4,…
  2. Negative Integers: The integer number is negative if it is less than zero. Example: -1, -2, -3, -4,… and here Zero is defined as neither negative nor positive integer. It is a whole number. Z = {… -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, …}

As per both the definition of Integer and Rational numbers,

All integers are rational numbers because 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.  

But All rational numbers are not integer because as it is known that 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.

They can be expressed in fraction and decimal form as 3/1, 4/1, and 5/1, 8.99, 0.90…

Whereas 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,…

Integers does not include decimal or fractional value, they only include sets of counting numbers whereas Rational number include decimal as well as fractional value. Thats why, all integers are rational number.

Examples of numbers which are  integer  as well as rational: 1 , 3 ,4 ,66 , 88 8900 …

Similar Problems

Question 1: Identity these numbers which are both integer numbers and rational?

7.88, 8, 5/4, 17890, 66.8989

Solution:

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

And, 7.88, 5/4, 66.8989 are only rational numbers.

Question 2: Identify integers?

45, 78.09, 5/9, 18898, -4, -878

Solution:

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.

Hence, 45, 18898, -4, -878 are integers 

Question 3: Examples of Rational numbers?

Answer:

All rational numbers are not integer because as known 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. 

Examples are 3, 4, 7, 9, 5. They can be expressed in fraction and decimal form as 3/1, 4/1, and 5/1, 8.99, 0.90…

Question  4: Is 45 a rational number?

Answer:  

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.

Hence 45 here is a rational number as it can be expressed in form of p/q.

Question 5: If we add two integers what will be the number?

Answer:

Let a and b are two integer , here a = 2 and b = 7 

Then a + b =  2 + 7 = 9 is an integer as well as rational number

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