In addition and subtraction of integers, we will learn how to add and subtract integers with the same sign and different signs. We can also make use of the number line to add and subtract signed integers. There are certain rules for integers that have to be followed to perform operations on them.
Adding two positive integers results in positive integers, whereas adding two negative integers will result in the sum with a negative sign. But, the addition of two different signed integers will result in subtraction only and the sign of the result will be the same as the larger number has. See a few examples below:
- 2+2 = 4
- 2 + (-2) = 0
- -2 + (-2) = -4
- -2 – (-2) = 0
Addition and subtraction
Addition and subtraction are two primary arithmetic operations in Maths. Besides these two operations, multiplication and division are also two primary operations that we learn in basic Maths.
The addition represents the values added to the existing value. For example, a basket has two balls, and if we add more than 2 balls to it, there will be four balls in total. Similarly, if there are four balls in a basket and if we take out two balls out of it, then the basket is left with only two balls, which shows subtraction.
Addition and subtraction are not only used for integers but also rational numbers and irrational numbers. Therefore, both the operations are applicable for all real numbers and complex numbers. Also, the addition and subtraction algebraic expressions are done based on the same rules while performing algebraic operations.
Rules to Add and Subtract
Integers are a special group of numbers that are positive, negative and zero, which are not fractions. Rules for addition and subtraction are the same for all.
Negative Sign and Positive Sign
The integers which we add or subtract could be positive or negative. Hence, it is necessary to know the rules for positive and negative symbols.
Positive sign/symbol: (+)
Addition of Integers
The three main possibilities in the addition of integers are:
- Addition between two positive numbers
- Addition between two negative numbers
- Addition between a positive number and a negative number
Whenever a positive number and a negative number are added, the sign of the greater number will decide the operation and sign of the result. In the above example 10 + (-15) = -5 and (-10) + 15 =5; here, without sign 15 is greater than 10 hence, numbers will be subtracted and the answer will give the sign of the greater number.
We know that the multiplication of a negative sign and a positive sign will result in a negative sign, therefore if we write 10 + (-5), it means the ‘+’ sign here is multiplied by ‘-’ inside the bracket. Therefore, the result becomes 10 – 5 = 5.
Alternatively, to find the sum of a positive and a negative integer, take the absolute value (“absolute value” means to remove any negative sign of a number, and make the number positive) of each integer and then subtract these values. Take the above example, 10 + (-15); absolute value of 10 is 10 and -15 is 15.
⇒ 10 – 15 = -5
Thus, we can conclude the above table as follow:
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Note: The sum of an integer and its opposite is always zero. (For example, -5 + 5= 0)
Subtraction of Integers
Like in addition, the subtraction of integers also has three possibilities. They are:
- Subtraction between two positive numbers
- Subtraction between two negative numbers
- Subtraction between a positive number and a negative number
For ease of calculation, we need to renovate subtraction problems the addition problems. There are two steps to perform this and are given below.
- Convert the subtraction sign into an addition sign.
- After converting the sign, take the inverse of the number which comes after the sign.
Once the transformation is done, follow the rules of addition given above.
For example, finding the value of (-5) – (7)
Step 1: Change the subtraction sign into an addition sign
⇒ (-5) + (7)
Step 2: Take the inverse of the number which comes after the sign
⇒ –5 + (-7) (opposite of 7 is -7)
⇒ –5 + (-7) = -12 [Add and put the sign of greater number]
Properties Of Addition Of Integers
The addition properties for whole numbers are valid for integers.
Closure Property: The sum of any 2 integers results in an integer.
For instance, 12 + 3 = 15 and 15 is an integer.
In the same way, 17 + (- 20) = – 3 and -3 is an integer.
Commutative property: Even if the order of addition is changed, the total of any 2 integers is the same.
For instance, – 19 + 15 = 15 + (- 19) = – 4
Associative property: The grouping of the integers does not matter when the total of 3 or more integers is computed.
For example, – 13 + (- 15 + 16) = (- 13 + (- 15)) + 16 = – 12
Additive identity: When the sum of zero with any integer is taken, the resultant answer is an integer. The additive identity is the integer zero.
For instance, 0 + 15 = 15
Additive inverse: For each integer, when an integer is added to that integer results in 0. The two converse integers are termed additive inverse of one another.
Properties Of Subtraction Of Integers
Closure property: The difference between any two given integers results in an integer.
For instance, 13 – 17 = – 4 and – 4 is an integer. In the same way, – 5 – 8 = – 13 and – 13 is an integer.
Commutative property: The difference between any two given integers changes when the order is reversed.
For example, 6 – 3 = 3 but 3 – 6 = – 3.
So, 6 – 3 ≠ 3 – 6
Associative property: In the method of subtraction, there is a change in the result if the grouping of 3 or more integers changes.
For example, (80 – 30) – 60 = – 10 however [80 – (30 – 60)] = 110.
So, (80 – 30) – 60 ≠ [80 – (30 – 60)].
Multiplication of Integers
In addition and subtraction, the sign of the resulting integer depends on the sign of the largest value. For example, -7+4 = -3 but in the case of multiplication of integers, two signs are multiplied together.
| (+) × (+) = + | Plus x Plus = Plus |
| (+) x (-) = – | Plus x Minus = Minus |
| (-) × (+) = – | Minus x Plus = Minus |
| (-) × (-) = + | Minus x Minus = Plus |
Rules:
- When two positive integers are multiplied then the result is positive.
- When two negative integers are multiplied then also the result is positive.
- But when one positive and one negative integer is multiplied, then the result is negative.
- When there is no symbol, then the integer is positive.
Solved Examples
Example 1: Evaluate the following:
- (-5 )+ 9
- (-1) – ( -2)
Solution:
- (-5 )+ 9 = 4 [Subtract and put the sign of greater number]
- (-1) – ( -2)
⇒ (-1) + (-2) [Transform subtraction problems into addition problems]
⇒ (-1) + (2) [Subtract and put the sign of greater number]
Hence,
(-1) – ( -2) = 1
Example 2: Add -10 and -19.
Solution: -10 and -19 are both negative numbers. So if we add them, we get the sum in negative, such as;
(-10)+(-19) = -10-19 = -29
Example 3: Subtract -19 from -10.
Solution: (-10) – (-19)
Here, the two minus symbols will become plus. So,
-10 + 19 = 19 -10 = 9
Example 4: Evaluate 9 – 10 +(-5) + 6
Solution: First open the brackets.
9 – 10 -5 + 6
Add the positive and negative integers separately.
= 9 + 6 – 10 -5
= 15 – 15
= 0
&
Win
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