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Arithmetic operations or mathematical operations are the basic operators applied to numbers or variables which have great significance in daily life. The arithmetic operators are addition, subtraction, multiplication and division. The arithmetic operators are applied between two or more numbers or quantities. This study of operations and numbers is further used in the field of geometry, data handling and algebra. In everyday life, the calculation of money spent, measuring of length and many more problems are solved with the help of four basic arithmetic operators.
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Key Terms: Arithmetic, Addition, Subtraction, Multiplication, Division, Basic, Associative, Commutative, Distributive
Addition Operator
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Addition operator with the symbol ‘+’ is one of the basic operators used to calculate the total of two or more numbers or quantities. The operator can be applied to real numbers, complex numbers, fractions and decimals. The associative and commutative law holds good for the addition operator which means that the order of the quantity added does not affect the result.
For example, suppose Amit goes to market and spends 25 Rs on bread and 50 Rs on chocolates. The total amount of rupees spent by Amit is 25 + 50 Rs which gives the total of 75 Rs.
Subtraction Operator
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Subtraction operator with the symbol ‘−‘ is used to find the difference between the quantities. It is the inverse operation of the addition. But like the addition operator, the associative and commutative does not hold goods for the subtraction operation i.e., order of the numbers affect the result. It is basically used to find the left out from the total if some amount is taken away.
For example, suppose Amit has 100 Rs before he goes to market. Then he spent a total of 75 Rs on bread and chocolate. So, the amount of money left with him is 100 – 75 Rs = 25 Rs.
Also Read: Addition and Subtraction of Integers
Multiplication Operator
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Multiplication operator with symbol ‘×’ is used when the same number or quantity is added multiple times. Let the number a is added for n times then the sum of the equation is
a + a + .... n times = a × n
Both the laws i.e., commutative and associative law is true for the multiplication operator. It reduces the time to calculate the number repeatedly.
For example, suppose Amit spends 10 Rs for three days on the same chocolate. So, the total he spent on chocolate is 10 × 3 = 30 Rs.
Division Operator
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Division operator is the inverse of the multiplication operator. It is denoted by the symbol ‘÷’ and also by ‘/’. Also, the equation a/b represents that a is divided by b where b ≠ 0. If b = 0 then the result does not exist. Here, a is called the dividend and b is known as the divisor. Division operator does not follow the associative and commutative law. It is generally used to calculate the value of a single item from the given total. Moreover, the result is in the form of a quotient and a remainder.
For example, suppose Amit spent a total of 40 Rs on 4 chocolates. Then, the cost of 1 chocolate is 40 ÷ 4 = 10 Rs.
Also Read: Multiplication and Division of Integers
Basic arithmetic properties for real numbers
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There are three basic arithmetic properties for real numbers:
- Commutative Property: For the real numbers X and Y, the commutative property says that:
X * Y = X * Y
Where * is an operation
For example, the addition of real numbers 5 and 7
5 + 7 = 7 + 5
12 = 12
Hence, the commutative property holds good for addition. Also, it is true for multiplication.
- Associative Property: For the real numbers X, Y and Z, the associative property says that:
X * (Y * Z) = (X * Y) * Z
Where * is an operation
For example, the multiplication of 3, 4 and 5
3 × (4 × 5) = (3 × 4) × 5
3 × 20 = 12 × 5
60 = 60
The property holds good for addition and multiplication operators.
- Distributive Property: According to the distributive property, for any real numbers X, Y and Z
X × (Y + Z) = (X × Y) + (X × Z)
For example, for real numbers 5. 6 and 3
5 × (6 + 3) = (5 × 6) + (5 × 3)
5 × 9 = 30 + 15
45 = 45
Things to Remember
- There are four basic arithmetic operations.
- Subtraction is the inverse of addition.
- Division is the inverse of multiplication.
- In division, the divisor cannot be zero.
- Commutative law holds good for addition and multiplication.
- Addition of two negative integers gives a negative integer.
Also Read: What is Binary operations?
Sample Questions
Ques: Use the addition operator between: (5 Marks)
45 and 67
23 and 100
Ans: Addition operator denoted by ‘+’ used between two numbers a and b is represented by a + b and give the result as sum of them.
- 45 and 67
Using the addition operator between 45 and 67
45 + 67 = 112
Hence, the sum of 45 and 67 is 112.
- 23 and 100
Using the addition operator between 23 and 100
23 + 100 = 123
Hence, the sum of 23 and 100 is 123.
Ques: Use the subtraction operator between: (5 Marks)
52 and 38
49 and 80
Ans: Subtraction operator denoted by ‘-’ used between two numbers a and b is represented by a - b and tells difference between two numbers.
- 52 and 38
Using the subtraction operator between 52 and 38
- - 38 = 14
Hence, the difference between 52 and 38 is 14.
- 49 and 80
Using the subtraction operator between 49 and 80
49 – 80 = -31
Hence, the difference between 49 and 80 is -31.
Ques: Use the multiplication operator between: (5 Marks)
15 and 12
21 and 33
Ans: Multiplication operator denoted by ‘×’ used between two numbers a and b is represented by a × b and give the result as product of them.
- 15 and 12
Using the multiplication operator between 15 and 12
15 × 12 = 180
Hence, the product of 15 and 12 is 180.
- 21 and 33
Using the multiplication operator between 21 and 33
21 × 33 = 693
Hence, the product of 21 and 33 is 693.
Ques: Use the division operator between: (5 Marks)
456 and 24
352 and 16
Ans: Division operator denoted by ‘÷’ used between two numbers a and b is represented by a ÷ b and gives the result in the form of quotient and remainder.
- 456 and 24
Using the division operator between 474 and 24
456 ÷ 24 = 19
Hence, the quotient is 19 and the remainder is 0 when 456 is divided by 24.
- 352 and 16
Using the division operator between 352 and 16
352 ÷ 16 = 22
Hence, the quotient is 22 and remainder is 0 when 352 is divided by 16.
Ques: What are the rules to follow while using the addition and subtraction operator? (5 Marks)
Ans: The following are the addition and subtraction rules for integers:
- When the addition operator is applied to two positive integers, it gives the value as a positive integer only.
- When the addition operator is applied to two negative integers, it gives the value as a negative integer only.
- While adding positive and negative integers, subtract the integers and use the sign of the largest integer number.
- Subtraction of a bigger number from smaller gives the negative value.
Ques: What are the rules to follow while using multiplication and division operators? (5 Marks)
Ans: The following are the multiplication and division rules for integers.
- The product and division of two positive integers is a positive integer.
- Multiplication and division operator used between two negative integers also gives positive integers.
- The division and product of integers with different signs results in the negative integer.
Ques: Find the value of the operation: (5 Marks)
Add 23 and 40 and then subtract 20 from the sum.
Solve 20 + 20 + 20 + 20 + 20
Ans: All the four basic arithmetic operators which are addition, subtraction, multiplication and division are applied between two or more numbers.
- Add 23 and 40 and then subtract 20 from the sum
Let Sum denote the addition of 23 and 40
Sum = 23 + 40
Sum = 63
Then, subtracting 20 from Sum gives the value ‘Val’
Val = Sum – 20
Val = 63 – 20
Val = 43
- Solve 20 + 20 + 20 + 20 + 20
This can be solved by two methods.
Method 1: Adding 20 + 20 + 20 + 20 + 20
Val = 20 + 20 + 20 + 20 + 20
Val = 40 + 40 + 20
Val = 40 + 60
Val = 100
Method 2: Multiplying 20 with 5
Val = 20 × 5
Val = 100
Ques: Why does the commutative property not hold good for subtraction? (5 Marks)
Ans: For the real numbers X and Y, the commutative property says that:
A * B = B * A
Where * is an operation
For subtraction, it does not hold good because of different signs of the resultant. For example,
Subtraction of 5 and 3
5 – 3 = 2
But, 3 – 5 = -2
Therefore, 5 – 3 ≠ 3 – 5
Ques: How are arithmetic operations on real numbers different from rational numbers? (3 Marks)
Ans: Rational numbers are in the form of p/q where q cannot be zero. Arithmetic operations on rational numbers are quite similar to the real numbers but for the addition and subtraction operators, rational numbers are operated differently like least common factor (LCM) is calculated to make denominators same. Otherwise, the operations remain the same in rational numbers as in real numbers.
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