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Subtraction of Integers: Properties, Rules with Solved Examples

subtract integers

Subtracting integers is the method of discovering the difference between two numbers. The original value may increase or decrease depending on whether the numbers are positive, negative, or a combination. Subtracting integers involves finding the difference between integers with the same or different signs. In this article, we will learn more about subtracting integers.

Properties of Subtraction of Integers

Let us understand the properties of subtracting integers.

1. Closure property: The difference between any two integers is always an integer.

Example: 10−17=−7, and −7 is an integer. Likewise, −5−8=−13 and −13 is an integers.

2. Commutative property: If the order of two numbers is reversed, the difference between them changes.

For instance: 6−3=3 but 3−6=−3. Thus, 6−3≠3−6.

3. Associative property: When subtracting three or more numbers, the result changes depending on how the three or more integers are organized.

Example: 

(80−30)−60=−10 but [80−(30−60)]=110. 

Thus, (80−30)−60≠[80−(30−60)].

Adding and Subtracting Integers Rules

To subtract two numbers, certain rules must be observed. Integers are whole numbers with no fractional components. It includes positive, zero, and negative integers.

The following are the rules for subtracting integers:

  • When we subtract 0 from any number, the result is the integer itself. Example: 4 – 0 = 4
  • We may discover the additive inverse or the opposite of any number by subtracting it from 0. Example: 0 – 4 = -4 
  • Integer subtraction is accomplished by altering the sign of the subtrahend. Following this, if both integers have the same sign, we sum the absolute values and insert the common sign. If the signs of the numbers are different, we find the difference between them and insert the sign of the larger number in the result.

The table below will help you understand how to subtract integers with examples.

rules for subtracting integers

Subtracting Integers with the Same Sign

We subtract two integers with the same sign by subtracting their absolute values and adding the common sign to the result. The absolute value of a number is the number’s positive value. For example, the absolute value of 4 is 4, the absolute value of -4 is 4, and so on. We modify the sign of the subtrahend when subtracting numbers.

For example: -2 -(-5), maybe expressed as -2 + 5. Now, the absolute value of 5 is 5, and the absolute value of -2 is 2. 

Because 5 > 2, the answer will have the same sign as 5, which is positive. 

As a result,

 -2 -(-5) =  -2 + 5 = 3.

Some other examples of subtracting integers with the same sign are:

  • (-1) – (-6) = -1 + 6 = 5
  • 3 – 8 = -5
  • 24 – 17 = 7

Subtracting Integers with Different Signs

When subtracting two numbers with different signs, the sign of the subtrahend is changed. Then, if both integers become positive, the outcome will be positive; if both integers become negative, the result will be negative. For example, if we wish to remove -9 from 5, we will change the sign of 9 and then add the numbers. As a result, 5 – (-9) = 5 + 9 = 14.

This may also be understood using another way where the absolute values are added and the sign of the minuend is attached to the result. For example, if we wish to remove -9 from 5, we must first determine their absolute values. 

Absolute value of -9 = 9

Absolute value of 5 = 5. 

Now calculate the total of these absolute numbers

9 + 5 = 14. Because 5 is the minuend in this case with a positive sign, the response sign will also be positive. 

As a result, 5 – (-9) = 14.

Subtracting Integers on a Number Line

The following rules govern the subtraction of integers on a number line:

  • Every truth that can be represented as a subtraction fact can also be written as an addition fact.
  • Moving towards the right side (or the positive side) of the number line will result in the addition of a positive number.
  • To add a negative integer, move to the left (or negative) side of the number line.
  • Any of the given integers can be used as the starting point for traveling along the number line.

Let’s look at how to subtract integers on a number line. 

Step 1. Select a number line scale. 

We need to decide if we wish to plot numbers in multiples of 1, 5, 10, 50, and so on, based on the specified integers. For example, to simplify subtracting 10 from -30, we might use a scale of 10 on the number line. However, if we need to remove -2 from 7, we may use a scale beginning with 1. 

Step 2. Find any of the integers on the number line.

Ideally, you should try to locate an integer with a larger absolute value. For example, if we need to subtract 4 from 29, we should identify 29 on the line first and then take 4 leaps to the left, rather than locating -4 and then jumping 29 times.

Step 3. Add or subtract 

Now add or subtract the second integer to the previous step’s number by making hops to the left or right depending on whether the number is positive or negative.

To further comprehend this, consider the following example.

Example: Subtraction of -4 from -7

Solution: Let us take the following steps to subtract integers on a number line:

Step 1: Write the equation as -7 – (-4). Make a number line with a scale of 1 to 10 on it.

Step 2: Express -7 – (-4) as an addition expression by altering the negative sign of the subtrahend to positive. We obtain -7 + 4 as a result.

Step 3: Begin at 0 and work your way to -7 by jumping 7 times to the left.

Step 4: From -7, make four hops to the right, as we were adding four to -7.

As a result, -3 is the correct answer.

Solved Examples for Addition and Subtraction of Integers 

Example 1: Evaluate the following:

  1. (-5 )+  9
  2. (-1) – ( -2)

Solution:

  1. (-5 )+ 9 = 4  [Subtract and put the sign of greater number]
  2. (-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: Both -10 and -19 are negative numbers. So, if we combine them, we obtain a negative sum, such as;

(-10)+(-19) = -10-19 = -29

Example 3: Subtract -21 from -10.

Solution: (-10) – (-21)

Here, the two minus symbols will become plus. So,

-10 +21 = 21 -10 = 11

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

Example 5: A plane is flying at 3000 feet above sea level. It is directly above a submarine drifting 700 feet below sea level at one point. Calculate the vertical distance between two numbers using the concept of subtraction of integers.

Solution: The plane is traveling at a height of 3000 feet. The submarine’s depth is -700 feet (Negative, as it is below sea level). We will utilize the subtraction of two numbers operation to compute the vertical distance between them:

3000 – (-700) feet

= 3000 + 700 feet 

= 3700 feet

As a result, their vertical distance is 3700 feet.

Example 6: Calculate (2-3) using a number line.

Solution: On a number line, we shall begin with +2 because that is our minuend. Then we must move three steps to the left, as we are reducing the value of two by three. This is how we get to (-1), our solution.

Example 7: A worker steps down the ladder by 2 steps from the 5th step he is working on. Use the concept of subtracting integers using the number line and find out which step he is on. 

Solution: On a number line, we shall begin with +5 because that is our minuend. Then we must move two steps to the left, as we are reducing the value of 5 by 2. This is how we get to (+3), our solution.  

Example 8: In a grocery shop, what temperature difference would a consumer notice when they move from the vegetable department, which is set to 20 degrees Celsius, to the other section, which is set to – 20 degrees Celsius?

Solution: The temperature in the vegetable area is 20 degrees Celsius.

The temperature in the opposite region is –20 degrees Celsius.

As a result, the temperature differential = [ –20C–(–20C) ] 

= 20C+20C

= 40 degrees Celsius.

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