What are Rational Numbers?
The number system includes different types of numbers for example prime numbers, odd numbers, even numbers, rational numbers, whole numbers, etc. These numbers are used in different arithmetic operations like addition, subtraction, multiplication, division, percentage, etc to determine a definite value. These numbers can be expressed in the form of figures as well as words accordingly.
Numbers are used in various arithmetic operations as addition, subtraction, multiplication, etc which are applicable in daily businesses and trading activities. Numerals or numbers are the mathematical values used for, counting, measurements, labeling or recognizing time, and for many other activities. Numbers are generally also generally known as numerals.
Numbers are the mathematical or arithmetic figures used for the purpose of counting, measuring, and other arithmetic calculations. Some examples of numbers are integers, whole numbers, natural numbers, rational and irrational numbers, etc.
A Number system or numeral system is defined as a standardized system to express numbers. It is the unique way of representation in which numbers are represented in arithmetic and algebraic structure.
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 set of numbers counting 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 by
- is given N=1,2,3,4,5,6,7,……………
- Whole numbers: Whole numbers are the set of natural 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 of whole numbers is given by W=0,1,2,3,4,5,………………
- Integers: Integers are the set of numbers including all the positive natural 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 is given by 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.
- The 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’.
What are rational numbers?
The numbers which can be expressed as fractions of two integers and can be written as a positive number, negative number, prime, and even zero is called rational numbers.
It can be expressed as p/q, where q ≠0
For example, 2/3 is a rational number that expresses that 2 integers are divided by 3 integers.
Rational numbers can be expressed as fractions, decimals, and even zero. As all the numbers with a non zero denominator which can be written in p/q form is a rational number.
Let’s look into different expressions of rational numbers
In fraction Form
A rational number is a ratio of two integers which can be written in the form of p/q where q is not equal to zero. Hence, any fraction with a non-zero denominator is a rational number.
For an example:
2/5 is a rational number where 2 is an integer being divided by a non-zero integer that is 5.
In Decimal Form
A rational number can be also written in the decimal form if the decimal value is definite or has repeating digits after the decimal point.
0.3 is a rational number. As the value 0.3 can be further expressed in the form of ratio or fraction as p/q
0.3 = 3/10
which is a ratio of two definite integers.
0 is a rational number, how?
0 is also included into rational number as it has a non zero denominator. If we express 0 in the form of p/q
where 0 is a integer and divided by integer 1.
Properties of Rational Number
Generally, there are four properties of rational numbers, that is,
- Closure property
- Commutative Property
- Associative Property
- Destructive property
These properties under different operations are discussed below:
Addition: The sum of two rational numbers is always a rational number. For example, 2 and 3 are rational numbers, and 2+3=5, which is also a rational number.
Subtraction: The difference of two rational numbers is always a rational number. For example, 5 and 2 are rational numbers and their difference is 5-2=3, which is also a rational number.
Multiplication: The product of two rational numbers is always a rational number. For example: 2 and 3 are rational numbers and their product is 2×3=6, which is also a rational number.
Division: The division of two rational numbers is also a rational number unless the denominator of the ratio is not equal to zero. For the example p and q are two rational numbers p/q=r, r will be a rational number if q is not equal to zero.
Addition: The sum of two rational numbers can be expressed in any order. For example: If p and q are two rational numbers and p+q=q+p
Subtraction: Subtraction does not show the commutative property as the difference of two rational p and q is p ≠q then, p-q≠q-p whereas, if p=q then, p-q=q-p
Multiplication: The product of two rational numbers can be expressed in any order. For example: if p and q are two rational numbers pxq=qxp
Division: Division also does not shows the commutative property as if p and q are two rational numbers then, p/q≠q/p
Addition: The sum of three rational numbers can be expressed in any order. For example: if p, q, and r are three rational numbers then, (p+q)+r= p+ (q+r)
Subtraction: The difference of three rational numbers p, q, and r is expressed as (p-q)-r=p-(q-r)=0
Multiplication: The product of three rational numbers can be grouped in any order. If p, q, and r are the three rational numbers then, px(qxr)=(pxq)xr
Division: If p, q, and r are three rational numbers. Then their expression will be (p÷q)÷r≠p÷(q÷r)
Destructive property shows that the product of the sums or differences of the rational numbers is equal to the sum or difference of their products. This order is expressed as p.(q+r)= p.q+q.r
When a negative rational number is added to the number to make its value 0 is additive inverse property. For, rational number p, (-P) is additive inverse. For example: 2+(-2)=0.
The product of any rational number and 1 is the rational number itself. If p is a rational number its product with 1 will give px1=p. Hence,
Question 1: What are the rational numbers between 3 and 5?
The rational numbers between 3 and 5 are 31/10, 32/10, 33/10, 34/10, 35/10, 36/10,…………..,49/10.
To find out a set of rational numbers between two numbers suppose A and B we need to express numbers A and B in rational numbers.
Lets express 3 and 5 as rational numbers as
Hence, the rational numbers between 3 and 5 are 30/10 and 50/10 are 31/10, 32/10, 33/10, 34/10, 35/10, 36/10, 37/10, 38/10, 39/10, 40/10, …………..49/10.
Question 2: What are the five rational numbers between 0 and 1?
The five rational numbers between 0 and 1 are 12, 21, 34, 41, and 51.
Question 3: Is 2.6 a rational number?
Yes, 2.6 is a Rational Number. As rational numbers can be expressed as decimals values as well as fractions. The number can also be written as 26/10 which is the ratio of two integers.
Take a look at the below proof.
The number 2.6 can be represented as shown below:
This can be further broken down as,
The number 13/5 is the ratio of two integers that are 13 integers divided by 5 integers and expressed in fraction form (as p/q where q is not equal to 0).
Question 4: Is 4.5 a rational number?
Yes, the number 4.5 is a rational number. Since rational numbers can also be expressed as decimals with repeating digits after the decimal point.
Take a look at the proof given below:
The given number 4.5 can be expressed as
This can be further broken down as
The number 9/2 is the ratio of two integers that are 9 integers divided by integer 2.
Question 5: Is 0 a rational number?
Yes, 0 is a rational number because it has a non-zero denominator. Since the number 0 can also be written as 0/1.
Take a look at the below proof.
The number 0 can be represented as shown below:
From the above expression, we can conclude that the number 0 can be expressed in the form of p/q where q is not equal to zero