Why are all rational numbers not integers?

Numbers are mathematical values or numbers used to measure or compute quantities. It’s represented by numbers like 2, 4, 7, and so on. Numbers include integers, whole numbers, natural numbers, rational and irrational numbers, and so on. In the number system, there are several types of numbers, such as prime numbers, odd numbers, even numbers, rational numbers, whole numbers, and so on. These symbols and phrases can be used to express these symbols and words. Integers such as 40 and 65, when given as figures, can also be written as forty and sixty-five.

A number system sometimes referred to as a numeral system, is a fundamental foundation for representing numbers and figures. It is a unique way of representing numbers in arithmetic and algebraic frameworks.

Rational Numbers and Integers 

Rational numbers have the form p/q, where p and q are integers and q equals zero. Most people have difficulty distinguishing between fractions and rational numbers because of the underlying structure of numbers, the p/q form. When a rational number is divided, the result is in decimal form, which might be either ending or recurring. Examples of rational numbers are 3, 4, 5, and so on, which may be stated in fraction form as 3/1, 4/1, and 5/1.

Integers are a class of integers that include all positive counting numbers, zero, and all negative counting numbers that count from negative infinity to positive infinity. It does not include fractions and decimals. The set of integers is symbolized by the letter ‘Z.’ Z = Represents the set of integers. -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, … is the example.

The number from the set of negative and positive numbers, including zero, has no decimal or fractional portion. Integers include the numbers -8, -7, -5, 0, 1, 5, 8, 97, and 3,043.

Types of Integers

Integers are classified into two types,

• Positive Integers: A positive integer number is one that is larger than zero. For instance: 1, 2, 3, 4, 5, 6.
• Negative Integers: A negative integer number is one that is less than zero. For instance, -1, -2, -3, -4,… thus in this context, zero is defined as neither a negative nor a positive integer. It is an even number. -8, -7, -5, -4, -3, -2, -1, 0, 1, 2, 3,… are the examples.

Why are all rational numbers not integers?

Answer:

Rational numbers are not integers because as per their definition. Integers are a class of integers that include all positive counting numbers, zero, and all negative counting numbers that count from negative infinity to positive infinity. It does not include fraction and decimal. The set of integers can be represented as Z = …,-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5,…

Rational numbers have the form p/q, where p and q are integers and q equals zero. Most people have difficulty distinguishing between fractions and rational numbers because of the underlying structure of numbers, p/q form. They can be expressed in fraction and decimal form as 3/1, 4/1, and 5/1, 8.99, 0.90… 

Therefore Rational numbers includes all fractions and decimals whereas integers does not include fractions and decimals value only include set of positive counting numbers.

Hence Rational numbers are not integers. Examples of numbers which are rational as well as integer: 2, 3, 4, 56, 88, 89, …

Sample Questions

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

7.88, 6, 3/4, 1890, 65.8989

Answer:

Here 6 and 1890 are both rational and integer numbers as it can be written as 8/1, 1890/1.

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

Question 2: Identify integers from the below numbers?

75, 88.09, 4/9, 1898, -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 75, 1898, -4, -878 are integers.

Question 3:  Is 5.89 a rational number or an irrational number?

Answer:

Here, the given number, 5.89 can be expressed in the form of p/q as, 

5.89 = 589/100   

Hence, 5.89 is a irrational number.

Question 4: Determine whether 8.44848… is a rational number or an irrational number.

Answer:

Here, the given number 8.44848… is an irrational number as it has non terminating and non recurring digits. Observing the digits after the decimal, they are neither terminating nor recurring.

Question 5: Determine whether the product of √4 × √4 is rational or irrational?

Solution:

Given: √4 × √4 both are irrational numbers but it is not necessary that the product of two irrational number will be irrational.

Therefore √4 × √4 = √16

But here square root of 12 is 4… which is terminating after decimal.

Hence the product of √4 × √4 is rational.