Subtracting Integers

Integers are the set of whole numbers, including positive, negative, and zero. Fractions and decimals do not belong to integers. It is represented as {..., -3, -2, -1, 0, 1, 2, 3,...} and extends in both directions on the number line. Integers are used for counting, measuring temperatures, handling financial transactions, and determining elevations above or below sea level.

For example, the temperature on a winter day is \(5 ^\circ c\), which is a negative integer, and the temperature on a sunny day is \(30 ^\circ C\), which is a positive integer.

Subtraction of integers involves two numbers: the minuend (from which another number is subtracted) and the subtrahend (the number being subtracted). There are several important properties related to the subtraction of integers, and they are as follows:

• Closure Property: Difference between any two integers always results in an integer.
  
  For example: \(8 - 5 = 3\)
  
  \(-7 - 4 = -11 \)
  
  \(-10 - (-3) = -7 \)
   
• Associative Property: Subtraction is not associative because the grouping of integers being subtracted affects the result.
  
  For example: \({(10 - 4) - 2} = {6 - 2 }= {4}\)
  
  \({10 - (4 - 2)} = {10 - 2} = {8} \)
   
• Identity Property: Subtracting zero from any integer leaves the integer unchanged.
  
  For example, \({{12 - 0 = 12}} \)
  
  \( {{-9 - 0 = -9}} \)
  
  \( {{0 - 0 = 0}}\)
   
• Inverse Property: An integer plus its additive inverse always results in zero.
  
  For example, \({{7 + (-7) = 0}}\)
  
  \({{-15 + 15 = 0}} \)
  
  \({{3 + (-3) = 0}}\)

Subtraction is one of the most basic mathematical operations. To subtract integers, we follow these steps:

Step 1: Identify the minuend and the subtrahend

Step 2: If both integers have the same sign, subtract their absolute values and assign the same sign to the result.

Step 3: If the integers have different signs, subtract their absolute values. The result takes the sign of the greater integer with absolute value.

Like all mathematical operations, even subtraction has a set of rules to be followed. In fact,
there are various rules for subtracting integers, and some of them are mentioned below.

• Subtracting integers with the same sign
  
  Example, \(15 -7 = 8\)
  
  \((-12) - (-5) = -12 + 5 = -7\)

• Subtracting integers with different signs
  
  Example, \(-8 - 5 = -8 + (-5) = -13\)
  
  \(7 - (-3) = 7 + 3 = 10 \)

• Subtracting a positive integer from another positive integer results in a positive integer if the minuend is larger.
  
  Example, \(20 - 7 = 13\)
  
  \(5 - 23 = -18 \)

• When subtracting a negative integer from another negative integer, the results can be positive or negative depending on their absolute values.
  
  Example, \((-5) - (-10) = -5 + 10 = 5 \)
  
  \((-12) - (-7) = -12 + 7 = -5  \)

• Subtracting a negative integer from another positive integer results in adding the two numbers.
  
  Example, \(6 - (-4) = 6 + 4 = 10\)
  
  \(15 - (-10) = 15 + 10 = 25 \)

           The given chart below shows all the rules of subtracting integers:

 

Subtracting Integers with the Same Sign
 

While subtracting integers with same sign, following steps are - 

Step 1: Before subtraction, identify the sign of the integers. If they have the same sign, proceed to the next step.

Step 2: Subtract the absolute values of the integers without considering the signs.

Step 3: If the absolute value of minuend is larger, then the result will have the same sign as the minuend, otherwise, it takes the opposite sign.

For example, \(-18 - (-10)\)

Here, both the integers are negative. 

Minuend = -18

Subtrahend = -10

Subtracting the absolute values

\(|18| - |10| = 18 - 10 \)

\(= 8 \)

Here, the absolute value of the minuend is larger than that of the subtrahend. So, the result takes the sign of the minuend, which is negative. 


So, \(-18 - (-10) = -8 \)

Subtracting Integers with Different Signs
 

The following steps are to be followed for subtracting integers with different signs (-) 

Step 1: Before subtraction, identify the signs of the integers. If they have different signs, proceed to the next step.

Step 2: Subtract the smaller absolute value from the larger one, ignoring the signs in this step.

Step 3: Assign the sign of the number with the greater absolute value to the result.Example, \(9 - (-4)\)

Here, minuend = 9

Subtrahend = -4

\(9 - (-4) = 9 + 4 \)

\(= 13 \)


Subtracting Integers on Number Line
 

We must follow the following steps to subtract integers on the number line:

Step 1: First locate the first integer on the number line.

Step 2: When subtracting a positive integer, move left; when subtracting a negative integer, move right.

Step 3: Move the number of steps equal to the absolute value of the integer being subtracted.

Step 4: The final position is where you end up on the number line after subtraction.

Example, \(-4 - (-6)\)

Here, minuend = -4

Subtrahend = -6

\(-4 - (-6) = -4 + 6 \)

Subtracting the absolute values: 

\(6 - 4 = 2 \)

So, \(-4 - (-6) = 2\)

Mastering subtraction of integers becomes easier when students understand the rules and follow simple tips and tricks. These tips will also help you with adding and subtracting integers, since both processes are related.
 

• Students can use the sign chart. When subtracting a positive number, move the number left. When subtracting a negative number, move the decimal point to the right.
   
• Use a number line to visualize the movement left or right. For example, \(4 - (-3)\), start at 4 and subtract -3 by moving right, and it lands on 7, so \(4 - (-3) = 7\). 
   
• Teachers can use color-coded number lines, like red for negatives and blue for positives. 
   
• Parents can use games like integer flashcards or number line jumps to help their children understand the concept of integers and their operations.
   
• Remember that when subtracting a negative number: \(a - (-b) = a + b\). For example, \(5 - (-2) = 5 + 2 = 7\). 
   
• After solving subtraction, check the answer using the opposite operation. For example, if \(9 - (-5) = 14\), check it by adding \(-5 + 14\), \(-5 + 14 = 9\). So the answer is correct.

The subtraction of integers has numerous real-life applications across various fields. Let us explore how subtracting integers is used in different areas:

• Banking and Finances - For easy calculation in bank transactions or maintaining balance, the basic subtraction operation is used. For example, if you have $500 in your account and withdraw $700, then \(500 – 700 = -200\). This shows that you now have a negative balance of $200 in your account.
  
   
• Elevation and Depth - While trekking or walking, trekkers or hikers use the distance mentioned in the milestones to understand how much distance is left or how much height they have climbed up or down. For example, if a hiker is at 500 m above sea level and the elevation change is \(500 – 800 = −300\) meters, it indicates the hiker is now 300 meters below sea level.
  
   
• Sports Scoring and Penalties - For sports like football, golf, basketball, points are added or subtracted on scoring or penalty, respectively. For example, if a team has 14 points but loses 7 due to a penalty, their new score is \(14 - 7 = 7\). 

• Temperature Change - Weather reports often show daily temperature differences. For example, if yesterday’s temperature was \(12^\circ C\), and today it is \(-3 ^\circ C\), then the change is \(12 − (−3) = 15^\circ C\), showing a 15°C drop in temperature.

• Travel Distance - Drivers or travelers use subtraction to find the remaining distance. For example, if the total distance is 250 km, and you have covered 160 km, the distance left is \(250 − 160 = 90km\).