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.

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.
  
   
• Associative Property: Subtraction is not associative because the grouping of integers being subtracted affects the result.
  
   
• Identity Property: Subtracting zero from any integer leaves the integer unchanged.
  
   
• Inverse Property: An integer plus its additive inverse always results in zero.
   

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

• Subtracting integers with different signs

• Subtracting a positive integer from another positive integer results in a positive integer if the minuend is larger.

• When subtracting a negative integer from another negative integer, the results can be positive or negative depending on their absolute values.

• Subtracting a negative integer from another positive integer results in adding the two numbers.
  
  
  The given chart below shows all the rules of subtracting integers:
  
   

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.
 

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.
 

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.
 

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

• Banking and Finances - For easy calculation in bank transactions or maintaining balance, 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 have they 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, indicating 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.