What are the Properties of Operations in Arithmetic?

Arithmetic probably has the longest history during the time. It is a method of calculation that is been in use from ancient times for normal calculations like measurements, labeling, and all sorts of day-to-day calculations to obtain definite values. The term got originated from the Greek word "arithmos" which simply means numbers.

Arithmetic is the elementary branch of mathematics that specifically deals with the study of numbers and properties of traditional operations like addition, subtraction, multiplication, and division.

Besides the traditional operations of addition, subtraction, multiplication, and division arithmetic also include advanced computing of percentage, logarithm, exponentiation and square roots, etc. Arithmetic is a branch of mathematics concerned with numerals and their traditional operations.

Basic Operations of Arithmetic

Arithmetic has four basic operations that are used to perform calculations as per the statement:

• Addition
• Subtraction
• Multiplication
• Division

Addition (+)

The simple definition for addition will be that it is an operation to combine two or more values or numbers into a single value. The process of adding n numbers of value is called summation.

0 is said to be the identity element of addition as while adding 0 to any value it gives the same result. For example, if we add 0 to 11 the result would be the same that is 11.

0 + 11 = 11

And, the inverse element includes the addition of the opposite value. The result of adding inverse elements will be an identity element that is 0. For example, if we add 11 with its opposite value -11, then the result would be

11 + (-11) = 0

Subtraction(-)

Subtraction is the arithmetic operation that computes the difference between two values (i.e. minuend minus the subtrahend).In the condition where the minuend is greater than the subtrahend, the difference is positive. It is the inverse of addition.

6 – 2 = 4

While, if the subtrahend is greater than minuend the difference between them will be negative.

2 – 6 = -4

Multiplication(×)

The two values involved in the operation of multiplication are known as multiplicand and multiplier. It combines two values that is multiplicand and multiplier to give a single product.

The product of two values supposedly p and q is expressed in p.q or p × q form.

8 × 5 = 40

Division(÷)

The division is the operation that computes the quotient of two numbers. It is the inverse of multiplication. The two values involved in it are known as dividends by the divisor and if the quotient is more than 1 if the dividend is greater than the divisor the result would be a positive number.  

15 ÷ 3 = 5

Properties of Arithmetic

The main properties of arithmetic are: 

• Commutative property 
• Associative property
• Distributive property
• Identity element property
• Inverse element 

Commutative Property

It states that the operation of addition on the number does not matter what is the order, it will give us the same result even after swapping or reversing their position. 

Or we can say that the placement of adding numbers can be changed but it will give the same results.

This property is valid for addition and multiplication not for subtraction and division.

x + y = y + x

Example: If we add 3 in 2 or add 2 in 3 results will be same

3 + 2 = 5 = 2 + 3

Associative Property

This property states that when three or more numbers are added (or multiplied) or the sum(or product) is the same regardless of the grouping of the addends (or multiplicands). 

The addition or multiplication in which order the operations are performed does not matter as long as the sequence of the numbers is not changed. This is defined as the associative property.

That is, rearranging the numbers in such a manner that will not change their value.

(x + y) +  z = x + (y + z) and (x.y).z =  x.(y.z) 

Example: (4 + 5) + 6 = 4 + (5 + 6)                                                    (4 × 5) × 6 = 4 × (5 × 6)

                              15 = 15                                                                              120 = 120 

As you can see even after changing their order, it gives the same result in both the operations adding as well as multiplication

Distributive Property

This property helps us to simplify the multiplication of a number by a sum or difference. It distributes the expression as it simplifies the calculation.

x × (y + z)  = x × y + x × z and x × (y - z) = x × y - x × z   

Example: Simplify 4 × (5 + 6)  

                            = 4 × 5 + 4 × 6 

                            = 20 + 24

                            = 44

It applies same for the subtraction also

Identity Element Property

This is an element that leaves other elements unchanged when combined with them. The identity element for the addition operation is 0 and for multiplication is 1.

For addition, x + 0 = x and for multiplication x.0 = 0               

Example: For addition, if x = 5

                                    x + 0 = 5 + 0 = 5

and for multiplication if x = 5 

                                     x.0 = 5.0 = 0

Inverse Element

The reciprocal for a number "a", denoted by 1/a, is a number which when multiplied by "a" yields the multiplicative identity 1.

The multiplicative inverse of a fraction: a/b is b/a  

The additive inverse of a number "a"  is the number that when added to "a", gives result zero. This number is also known as the additive inverse or opposite (number), sign change, and negation. 

Or we can say for a real number, it reverses its sign from positive number to negative and negative number to positive. Zero is itself additive inverse.

Example: Reciprocal of 6 is 1/6 and additive inverse of 6 is -6

Sample Questions

Question 1: The sum of two numbers is 100, and their difference is 30. Find the numbers.

Solution: 

Let the numbers be a and b. Now, as per the situation,

a + b = 100……………………(i)

and a – b = 30………………(ii)

We can write, a = 100-b, from equation (i),

now put the value of a in equation (ii), we get,

100 – b – b = 30

100 - 2b = 30

2b =100 - 30 

2b = 70

b = 70/2 

b = 35

and a = 100 – b

= 100 - 35

a = 65

Therefore, the two numbers are 65 and 35.

Question 2: Solve 45 + 2(27 ÷ 3) - 5

Solution:     

45 + 2(27 ÷ 3) – 5

⇒ 45 + 2(9) – 5

⇒ 45 + 18 – 5

⇒  63  – 5 = 58

Question 3: Find the value of a in the given equation 2a – 15 = 3.

Solution:

According to the equation,

=> 2a – 15 = 3

=> 2a = 15 + 3

=> 2a = 18

a = 9

Therefore, the value of a is 9.