Subtraction of Integers: Definition, Rules, Properties, Examples

Subtraction of Integers: You are already familiar with whole-number addition and subtraction. Are you aware that whole numbers are included in integers? Whole numbers and their negatives are included in integers. An integer is any number on a number line that does not have a fractional part. But, like whole numbers, can integers be added or subtracted?

For example, suppose the temperature in your city was \(3\) degrees Celsius, and it drops by \(8\) degrees Celsius. What is the temperature in your city right now? Integer addition and subtraction are two operations that we use to increase or decrease the value of integers. Let us study more about these two fundamental integer operations.

Subtraction of Integers

Natural numbers, their negatives, and zero are all examples of integers. A complete entity is an integer. Integers are numbers with no fractional element that can be positive, negative, or zero (no decimals). Integers, like whole numbers, can be subtracted. Integer subtraction refers to performing subtraction operations on two or more integers using the subtraction operators. It is critical to understand an integer’s absolute value before diving deeper into the idea.

The absolute value of an integer is the number’s distance from \(0\) on a number line. Because distance is a scalar quantity, it has no direction. It is always a good thing.

Subtraction of Integers with Examples

Subtraction usually refers to lowering the value. However, when dealing with integers, subtraction may increase or decrease in the value of the provided number. When we subtract a negative integer from another integer, the value increases, and when we subtract a positive integer, the value decreases.
Please take a look at the samples below and notice the operation we are applying on integers. A worker descends three stairs from the \({{\rm{4}}^{{\rm{th}}}}\) the step he is now working on:

Also, the temperature drops by \(3\) degrees Fahrenheit from \(-1\) degrees Fahrenheit: \(\left( { – 1 – 3 =  – 4} \right)\)

We employ the concept of integer subtraction in the examples above. When subtracting a positive number from a given number, we must go to the left or negative side of the number line to indicate the subtraction of integers. When we remove a negative number from a given number, on the other hand, we move towards the right side or positive side.

Let’s look at how we can subtract \(-3\) from \(5\) 

Thus, we get \(5 – ( – 3) = 5 + 3 = 8\).

As a result, we can subtract \(-3\) from \(5\) by adding the negative (or additive inverse) of \(-3.\) As mentioned below, this is the rule for subtracting integers.

Subtraction of Integers Rule

Addition and subtraction are inverse operations, as you must know. As a result, any subtraction problem can be represented as a problem of addition. Let us look at a few examples to see how this is done.

\(5 – 8 = 5 + ( – 8)\)

\(9 – 4 = 9 + ( – 4)\)

\( – 7 – 5 = – 7 + ( – 5)\)

We must put the subtraction sign inside the bracket and add the addition operator between the two terms when creating any subtraction issue. This is one method of resolving subtraction problems.

Let us also study the rules of subtraction to make our calculations easier when working with integers.

Rule 1: Two positive numbers are subtracted.

That is, \(( + x) – ( + y) = x – y\)

When subtracting two positive numbers, we take the difference between their absolute values and attach the sign of the larger number to the result.

Example: \(5 – 7 = – 2\) and \(9 – 3 = 6\)

Rule 2: A positive and a negative number are subtracted.

That is, \(x – ( – y) = x + y\)

\(( – x) – y = – (x + y)\)

We add the absolute values of both integers and append the sign of the minuend to the response when subtracting a positive and a negative number.

Example: \(4 – ( – 7) = 4 + 7 = 11\) and \(( – 3) – 6 = – 9\)

Rule 3: Two negative numbers are subtracted.

That is, \(( – x) – ( – y) = – (x – y)\)

We only need to remember one rule when subtracting two negative numbers: if a negative sign appears outside the bracket, the sign of the phrase inside the bracket is changed. After that, we must subtract the absolute values of both numbers and attach the revised sign of the higher number to the solution.

Example: \(( – 3) – ( – 5) = 2\) and \(( – 13) – ( – 8) = – 13 + 8 = – 5\)

Subtraction of Integers Properties

Closure property: The subtraction of any two integers is an integer, i. e., for any two integers \(a\) and \(b,\,a – b\) is an integer.

Commutative property: Subtraction of integers is not commutative, i.e., for any two integers \(a\) and \(b,\,a – b = b – a\) 

For example,

\(2 – 3 \ne 3 – 2\) and \( – 7 – ( – 3) \ne – 3 – ( – 7)\) and so on.

Identity property: We have \(a – 0 = a\) for any integer \(a\), indicating that \(0\) is the correct identity for subtraction.

It’s important to note that \(0 – a \ne a\) is not the left identity.

If \(a,\,b\) and \(c\) are integers and \(a > b,\,a – c > b – c\) is true.  We learned that integer addition is both commutative and associative. As a result of these two features of integer addition and subtraction, we can now use the following methods to find the values of expressions, including various terms with plus and minus signs:
Step 1: Obtain the expression whose value is to be determined.
Step 2: Put all terms containing plus signs together and add them.
Step 3: Put all terms containing minus signs and add them.
Step 4: Find the difference in the absolute values of the two sums obtained in Steps \(1\) and \(3\).
Step 5: Assign to the result of step \(4\), the sign of the sum having a larger absolute value.

Subtraction of Integers Formula

Add two integers with the same sign, add their absolute values and use the same sign for the result as the given integers. To combine two integers with different signs, we subtract their absolute values (in the sequence larger number minus smaller number) and apply the sign of the larger number to the result.

Subtraction is done in the same way as addition, except we use the rule \(a – b = a + ( – b)\) Integer addition and subtraction formulas are as follows:

1. \(( + ) + ( + ) = + \)
2. \(( – ) + ( – ) = \,- \)
3. \(( + ) + ( – ) = + \)  (A positive number has a greater absolute value.)
4. \(( – ) + ( + ) =\, – \)  (A negative number has a larger absolute value.)

Solved Examples – Subtraction of Integers

Q.1. One particular day, the temperature in Delhi was \({13^{\rm{o}}}{\rm{C}}\) at \(10\,{\rm{a}}.{\rm{m}}\), but by midnight, it had dropped to \({6^{\rm{o}}}{\rm{C}}\). The temperature in Chennai was \({18^{\rm{o}}}{\rm{C}}\) at \(10\,{\rm{a}}.{\rm{m}}\). On the same day, but it dropped to \({10^{\rm{o}}}{\rm{C}}\) by midnight. Which of the two falls is greater?
Ans: We have,
The temperature in Delhi in the fall \( = {13^{\rm{o}}}{\rm{C}} – {6^{\rm{o}}}{\rm{C}} = {7^{\rm{o}}}{\rm{C}}\) 
The temperature in Chennai in the fall \( = {18^{\rm{o}}}{\rm{C}} – {10^{\rm{o}}}{\rm{C}} = {8^{\rm{o}}}{\rm{C}}\) 
Clearly, \({8^{\rm{o}}}{\rm{C}} > {7^{\rm{o}}}{\rm{C}}\).
Hence, the fall in temperature of Chennai is greater.

Q.2. Find the value of \( – 12 + ( – 98) – ( – 84) + ( – 7)\)
Ans: We have \( – 12 + ( – 98) – ( – 84) + ( – 7)\)
\( = – 12 – 98 + 84 – 7\)
\( = ( – 12 – 98 – 7) + 84\)
\( = – 117 + 84\)
\( = – (117 – 84) = – 33\)
Hence, the value of the given expression is \( – 33\).

Q.3. Subtract the first integer from the second in \( – 225,\, – 135\).
Ans: We have, \( – 225,\, – 135\)
We need to subtract the first integer from the second.
That is, \( – 135 – ( – 225)\)
\( = – 135 + 225 = 90\)
Hence, the difference between the given integers is \(90\).

Q.4. Subtract the sum of \(-1250\) and \(1138\) from the sum of \(1136\) and \(-1272\)
Ans: The sum of \(-1250\) and \(1138\) is \( – 1250 + 1138 = – 112\)
And the sum of \(1136\) and \(-1272\) is \(1136 + ( – 1272) = 1136 – 1272 = – 136\)
Now, we have to subtract \(-112\) from \(-136\).
That is, \( – 136 – ( – 112) = – 136 + 112 = – 24\)
Hence, the difference in the numbers is \( – 24\).

Q.5. Subtract the sum of \(-5020\) and \(2320\) from \(-709\).
Ans: The sum of \(-5020\) and \(2320\) is \( – 5020 + 2320 = – 2700\).
We have to subtract \(-2700\) from \(-709\).
That is, \( – 709 – ( – 2700) = – 709 + 2700 = 1991\)
Hence, the difference in the numbers is \(1991\).

Summary

In this article, we learnt about subtraction of integers definition, subtraction of integers with examples, subtraction of integers rule, subtraction of integers properties, subtraction of integers formula, solved examples on subtraction of integers and FAQs on subtraction of integers.

The learning outcome of this article is how to simplify integers with the same and different signs and how integers are used in real life like to compute temperature differences in a day.

FAQs

Q.1. What is the definition of subtraction of integers?
Ans: Integer subtraction refers to performing subtraction operations on two or more integers using the subtraction operators.
Example: \( – 9 – ( – 6) = – 3\)

Q.2. What is an example of subtracting integers?
Ans: The following is the example to subtract two integers.
Example: Subtract \(-39\) from \(66\).
Solution: We have to subtract \(-39\) from \(66\).
That is, \(66 – ( – 39) = 66 + 39 = 105\).

Q.3. What is the formula for subtracting integers?
Ans: Subtraction is done in the same way as addition, except we use the rule \(a – b = a + ( – b)\) Integer addition and subtraction formulas are as follows:
1. \(( + ) + ( + ) = + \)
2. \(( – ) + ( – ) = \,- \)
3. \(( + ) + ( – ) = + \)  (A positive number has a greater absolute value.)
4. \(( – ) + ( + ) =\, – \)  (A negative number has a larger absolute value.)

Q.4. How do you subtract integers?
Ans: The following are steps to subtract integers:
1. Keep the first number for now (known as the minuend).
2. Second, switch from subtraction to addition as the procedure.
3. Get the opposite sign of the second number as the third step (known as the subtrahend).
4. Finally, perform a standard addition of integers.

Q.5. What are the rules of integers of addition and subtraction?
Ans: The rules for addition and subtraction of integers are as follows:
1. Subtract the two numbers and sign the larger number if the two numbers have different signs, such as positive and negative.
2. If two numbers have the same sign, either positive or negative, add them together to get the common sign.

Learn About Addition and Subtraction Of Integers

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