Integers are a set of numbers that include all the whole numbers (positive numbers), their negative counterparts (negative whole numbers), and zero.
Operations of integers refer to the basic arithmetic processes that can be performed on integers. The primary operations of integers are :
Addition of Integers
Arithmetic addition is one of the natural operations of integers in mathematics and can be done on integers using the number line.
That is, when adding two integers, we step ahead on the number line for positive integers and step backward for the negative integers. The result of the addition operation of two integers is the number that is obtained when a certain number has been added to the respective integer.
Positive Numbers
- When adding two positive integers, we move to the right on the number line by the sum of the two numbers. For example, to add 3 + 5, we start at 0 and move 3 steps to the right (to 3), then 5 more steps to the right (to 8). Therefore, 5 + 6 = 11.
Negative Numbers
- When adding two negative integers, we move to the left on the number line by the sum of the two numbers. For example, to add (-3) + (-5), we start at 0 and move 3 steps to the left (to -3), then 5 more steps to the left (to -8). Therefore, -3 + -5 = -8.
Subtraction of Integers
Subtraction operation is the reverse of addition operations of integers and can be solved using the number line with integers. Therefore when performing subtraction of 2 integers we add for the left on the number line for positive integers and right for the negative integers. The number that is obtained by taking the corresponding steps is called the difference between two integers.
Positive Numbers
- When subtracting a positive integer from another positive integer, we move to the left on the number line by the value of the subtrahend (the number being subtracted). For example, to subtract 5 - 3, we start at 5 and move 3 steps to the left (to 2). Therefore, 7 - 3 = 4.
Negative Numbers
- When subtracting a negative integer from a positive integer, we move to the right on the number line by the value of the negative integer. For example, to subtract 5 - (-3), we start at 5 and move 3 steps to the right (to 8). Therefore, 5 - (-3) = 8.
Multiplication of Integers
Multiplication is a repeated addition operation of an integer that can be performed on integers. The product of two integers is the result of adding one integer a certain number of times, as specified by the other integer.
The rules for multiplying integers are as follows:
- A positive integer if multiplied by a positive integer will only give another positive integer.
- When two negative integers are multiplied, the product obtained is that of a positive integer.
- Multiplying a positive integer with a negative integer yields a negative integer.
Examples:
Division of Integers
Division is special since it is the opposite operation of the multiplication operation of integers. Division, also known as quotient, it indicates the number of times that the number being divided is contained in the other number being divided.
The rules for dividing integers are as follows:
- Whenever a positive integer is divided by another positive integer, the result so obtained can be either a positive integer or a fraction.
- A positive integer when divided by a negative integer yields a negative integer or fraction.
- If we divide a negative integer by a positive integer it yields either a negative integer or a fraction.
- Division carried out between two negative integers yields a positive integer or a fraction.
Examples:
Rules for Integer Operations
The rules of operations with integers are as follows:
- If the signs of the two integers are the same, then the sum equals the sum of the two absolute values of the two integers together with the sign that connects the two integers.
- When two integers to be added have different signs, the sum will be equal to the difference in the two numbers, using the absolute value and the sign of the number with the largest absolute value.
- Difference between the two integers is equivalent to the sum of the first integer and the additive inverse of the second integer.
- Any two integers multiplied together will yield a positive integer if both of the integers have the same sign.
- Multiplication operation of two integers results in a negative value if the two integers are of opposite signs.
- Quotient of two integers is positive if both integers share the same sign and negative if the two integers are of different signs.
- Additive identity zero is also true which states that the addition of zero to an integer will not alter its value.
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Solved Examples of Integer Operations
Question 1: Find the sum of 7 and 9.
To find the sum, we move 7 steps to the right on the number line and then 9 more steps to the right.
This gives us a total of 16 steps to the right, so the sum is 16.
7 + 9 = 16
Question 2: Find the difference between -8 and -3.
To find the difference, we move 8 steps to the left on the number line and then 3 steps to the right.
This gives us a total of 5 steps to the left, so the difference is 5.
(-8) - (-3) = -8 + 3 = 5
Question 3: Find the product of 4 and -6.
To find the product, we multiply the absolute values of the numbers (4 × 6 = 24) and then apply the sign of the product.
Since one of the numbers is negative, the product is also negative.
4 × -6 = -24
Question 4: Find the quotient of 12 and 4.
To find the quotient, we divide the dividend (12) by the divisor (4).
Since both numbers are positive, the quotient is also positive.
12 ÷ 4 = 3
Practice Questions on Operations of Integers
1. Find the sum of -5 and -7.
2. Find the difference between 9 and 5.
3. Find the product of -2 and 8.
4. Find the quotient of -15 and -3.