Operations on Real Numbers

What Are Real Numbers?

To understand real numbers, we first have to understand what rational and irrational numbers are. Rational numbers are ones that can be written in the form of p/q where p is the numerator and q is the denominator, and both p and q are integers. For example, 7 can be written as 7/1 so it is a rational number. Irrational numbers are numbers which cannot be written in p/q form. For example,√2 is an irrational number because √2 = 1.41421 .. and continues to infinity. Hence, it cannot be written as a fraction and is non terminating and non-recurring decimals. Rational and irrational numbers together form real numbers. You can also find a worksheet on real numbers at the end. 

What Are Mathematical Operations?

The four basic mathematical operations are addition (+), subtraction (-), multiplication (x) and division (/). We will now understand these operations on real numbers - both rational and irrational. The real numbers worksheet will help you understand this topic better.

Operations On Two Rational Numbers

Addition Of Two Rational Numbers

When two rational numbers are added, the result is a rational number. For example, 0.24 + 0.68 = 0.92. 0.92 can be written as 92/100 which is a ratio or in the p/q form. 

Subtraction Of Two Rational Numbers

When two rational numbers are subtracted, the result is a rational number. For example, 0.93-0.22 = 0.71 which can be written as 71/100.

Multiplication Of Two Rational Numbers

When two rational numbers are multiplied, the result is a rational number. For example, 0.5 multiplied by 185 is 92.5 which can be written as 925/10.

Division of Two Rational Numbers 

When a rational number is divided by another rational number, the result is a rational number. For example, 0.352 divided by 0.6 is 0.58 which can be written as 58/100.


Operations On Two Irrational Numbers
Addition of Two Irrational Numbers

When two irrational numbers are added, the result can be an irrational or a rational number. For example, √3 added to (√3) is 3.46 or 2√3 which can be written as 346/100 which is a rational number. However, when 2√5 is added to 5√3, we get a non-terminating and non-recurring decimal which is an irrational number. It is written as 2√5+5√3.

Subtraction of Two Irrational Numbers

Similarly, when two irrational numbers are subtracted, the result can be an irrational or a rational number. √2 is subtracted from √2, the answer is 0. When 4√5 is subtracted from 5√3, we get 5√3-4√5.

Multiplication of Two Irrational Numbers 

The product of two irrational numbers can be an irrational number or a rational number. For example, when √2 is multiplied with √2, we get 2 which is a rational number. However, when √2 is multiplied by √3, we get √6 which is an irrational number.

Division of Two Irrational Numbers

Similar to multiplication, we can get either an irrational number or a rational number as a result when an irrational number is divided by another. For example, when √2 is divided by √2, we get 1 which is a rational number. But when √2 is divided by √3, we get √2/√3, which is an irrational number. 


Operations On A Rational And An Irrational Number

Addition of an Irrational and a Rational Number

The sum of a rational and an irrational number is always irrational. For example, when 2 is added to 5√3, we get 2 + 5√3, which is a rational number.

Subtraction of an Irrational and a Rational Number

The difference of a rational and an irrational number is always irrational. For example, when we subtract 5√3 from 2, we get  2 - 5√3, which is an irrational number. 

Multiplication of an Irrational and a Rational Number

The product of a rational and an irrational number might be rational or irrational. For example, when 2 is multiplied with √2, we get 2√2 which is an irrational number, but when√12 is multiplied with √3, we get √36, or 6, which is a rational number. 

Division of an Irrational Number with a Rational Number

When a rational number is divided by an irrational number or vice versa, the quotient is always an irrational number. For example, when 8 is divided by √2, we get 8/√2 which is an irrational number. The answer can be further simplified to 4√2 which is also an irrational number.


Operations On Real Numbers Worksheet 

Example 1:

Solve:
(7√3) x (- 5√3)

Solution:

(7√3) x (- 5√3)

= 7 x -5 x √3 x √3

= -35 x 3 = -105

Example 2:

Solve:

(3√27 / 9√3)

Solution:

(3√27 / 9√3)

= 3√27 = √3x3x3 = 3 x 3√3 = 9√3

= 9√3/ 9√3 = 1

To understand the topic further, operations of real numbers worksheet might be of great help to students. 

FAQ (Frequently Asked Questions)

Q1. What are the Properties of Real Numbers?

A1.The properties of real numbers are applicable in addition and multiplication. When we add two real numbers, we have to keep in mind the
- Closure property: If a and b are real numbers, a+b is also a real number
- Associative property: a+(b+c) = (a+b)+c
- Commutative property: a+b = b+a
- Additive identity: a+0 = a
- Additive inverse: a + (-a) = 0

For Multiplication -
- Closure property: If a and b are real numbers, ab is also a real number
-  Associative property: a x (b x c) = (a x b) x c
- Commutative property: a x b = b x a
- Identity property: a x 1 = 1 x a = a
- Inverse property: a x 1/a = 1
Another property holds true for both multiplication and addition - the distributive property, which is the same in both cases and says a x (b + c) = (a x b) + (a x c) = ab + ac.

Q2. Are There Non-Real Numbers? What are They?

A2. The two kinds of numbers known to us are real numbers and complex or imaginary numbers. We can represent any given number y as y = a + ib, where a is the real number and ib is the complex part of the number. Like we mentioned earlier, real numbers are a combination of all rational and irrational numbers which include whole numbers, repeating decimals and non-repeating decimals. If we take the number 3 + √-5, 3 is the real part and √-5 is the complex part of the number.