Subtracting Integers

Subtraction of integers is a fundamental operation in mathematics that involves finding the difference between two integers. It is important to know that how to subtract integers is important for various mathematical applications including algebra, geometry, and calculus.

Let's know more about Subtracting Integers, how to subtract integers of the same sign, and of different signs, along with the properties of subtraction of integers.

Subtracting Integers in Math

Subtracting integers involves finding the difference between two numbers, taking into account their signs. It's the process of finding how much one number is less than or greater than another on the number line.

Subtracting integers involves finding the difference between two integers, considering their signs. When subtracting integers, it's essential to understand the concept of positive and negative numbers and how they interact with each other.

To perform integer subtraction, invert the sign of the number being subtracted (the subtrahend). When subtracting numbers that share the same sign, simply add their absolute values together and retain the shared sign in the answer. On the other hand, if the numbers have different signs, calculate the absolute difference between them and assign the result the sign of the number with the larger absolute value.

Subtracting Integers Definition

Subtraction of integers involves finding the difference between two integers where one integer is being taken away from another. Subtracting integers means operating subtraction on positive and negative whole numbers.

Subtrahend and Minuend

Subtrahend is the number being subtracted while the minuend is the number from which another number is subtracted.

In the expression 'a - b' where a is the minuend and b is the subtrahend, the result represents the difference between the two numbers. In other words, the number from which another number is subtracted is called the minuend, and the number being subtracted is called the subtrahend.

How to Subtract Integers

To subtract integers, subtract the numbers as usual and apply the rules for signs mentioned above. Rules for Subtracting Integers are listed below:

• Step 1: To subtract integers, follow these steps:

• Step 2: Identify the minuend (the number from which another number is subtracted) and the subtrahend (the number being subtracted).

• Step 3: If the signs of both integers are the same, subtract their absolute values and assign the same sign to the result.

• Step 4: If the signs are different, subtract their absolute values, and the result takes the sign of the integer with the greater absolute value.

The result obtained is the difference between the two integers.

Subtracting Integers Rules Chart

Below is the Rules Chart for Subtracting Integers.

Integer’s Sign	Operation	Sign After Operation	Example
+ - +	Subtract	+	8 - 2 = 6
+ - -	Add	+	8 - (-2) = 10
+ - -	Add	+	2 -(-8) = 10
+ - +	Subtract	-	2 - 8 = -6
- - -	Subtract	-	-8 -(-2) = -6
- - -	Subtract	+	-2 - (-8) = 6
- - +	Add	-	-2 - 8 = -10
- - +	Add	-	-8 - 2 = -10

Rules on Subtracting Integers

Various rules of operation are,

• Subtracting integers with the same sign: Subtract the absolute values and keep the sign of the larger number.

• Subtracting integers with different signs: Treat it as adding the opposite. Subtract the absolute values and take the sign of the number with the greater absolute value.

• Subtracting a positive integer from another positive integer results in a positive integer.

• Subtracting a negative integer from another negative integer results in a positive integer.

• Subtracting a negative integer from a positive integer results in adding the two numbers.

• When subtracting integers with different signs, subtract the absolute values and keep the sign of the number with the greater absolute value. Example: (-7) - 3 = -10

• When subtracting integers with the same sign, subtract the absolute values and keep the common sign. Example: (-4) - (-2) = -2

• Subtracting from Zero: Subtracting an integer from zero is straightforward, it retains the sign of the integer being subtracted. Example: 0 - (-5) = 5

Positive - Positive	Positive - Negative	Negative - Positive	Negative - Negative
Positive	Negative	Negative	Positive
Negative	Positive	Negative	Positive

Subtracting Integers with Same Sign

When subtracting integers with the same sign, the process is straightforward. Simply subtract the absolute values of the numbers and keep the sign of the larger number.

Here are the steps for subtracting integers with the same sign :

Step 1: Identify the sign of the integers you're subtracting. If they have the same sign, proceed to the next step.

Step 2: Subtract the absolute values of the integers. Ignore the signs while performing subtraction.

Step 3: Assign the common sign to the result. Since the integers have the same sign, the result will also have that sign.

Example: Subtracting (-7) - (-3):

Both integers have a negative sign, so they have the same sign.

Subtract the absolute values:

|-7| - |-3| = 7 - 3 = 4

Result takes the negative sign, so the answer is -4.

Subtracting Integers with Different Signs

When subtracting integers with different signs, it's akin to adding the opposite. Subtract the absolute values and take the sign of the number with the greater absolute value.

Here are the steps for subtracting integers with different signs:

Step 1: Determine the signs of the integers you're subtracting. If they have different signs, proceed to the next step.

Step 2: Subtract the absolute values of the integers. Ignore the signs while performing subtraction.

Step 3: Assign the sign of the number with the greater absolute value to the result.

Example: Subtracting (-7) - 3:

Signs of the integers are different.

Subtract the absolute values:

|-7| - |3| = 7 - 3 = 4

Since 7 has the greater absolute value and it's negative, the result takes the negative sign so the answer is -4.

Subtracting Integers on Number Line

Subtracting integers on a number line involves moving to the left (for subtraction) or to the right (for addition) from a starting point. When subtracting integers, move to the left if subtracting a positive integer and to the right if subtracting a negative integer. On a number line, it is thus the subtracting of integers involves moving to the left for positive integers and to the right for negative integers.

To subtract integers on a number line, follow these steps:

Step 1: Start at the initial position: Locate the first integer on the number line.

Step 2: Move to the left or right: Depending on whether you are subtracting a positive or negative integer, move in the opposite direction along the number line.

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

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

For example, subtracting -3 from 5:

Step 1: Start at 5 on the number line.

Step 2: Since we are subtracting a negative integer (-3), move to the right (positive direction).

Step 3: Move 3 steps to the right.

Step 4: End up at 8 on the number line, which is the result of 5 - (-3) = 8

Properties Of Subtraction Of Integers

When subtracting integers, several properties hold:

• Closure property: Difference between any two integers is always an integer.
• Associative property: Subtraction is associative, meaning the grouping of integers being subtracted does not affect the result. For example, (a - b) - c = a - (b - c).
• Identity property: Subtracting zero from any integer leaves the integer unchanged.
• Inverse property: The difference between an integer and its additive inverse (its negative) is always zero.
• Subtraction of integers is not commutative.
• Subtraction of integers is associative.
• Subtraction of integers is distributive over addition.

Must Read:

• Rational Numbers
• Integers
• Arithmetic Operations
• What is the rule for subtracting negative numbers?

Subtracting Integers Examples

Example 1: Joe had $50 in his wallet. He spent $25 on groceries. What is the balance in his wallet now?

Solution:

Joe had $50 initially

He spent $25 on groceries

To find the balance, subtract the amount spent from the initial amount:

$50 - $25 = $25

Example 2: A submarine descended 300 meters below sea level. Later, it ascended 150 meters. How far below sea level is the submarine now?

Solution:

Submarine descended 300 meters below sea level.

It then ascended 150 meters.

To find the current depth, subtract the ascent from the descent:

300 meters - 150 meters = 150 meters below sea level.

Example 3: In a football game, a team lost 15 yards on one play and gained 20 yards on the next. What is the net change in yards for the team?

Solution:

Team lost 15 yards and gained 20 yards.

To find the net change, subtract the lost yards from the gained yards:

20 yards - 15 yards

= +5 yards

Example 4: Sarah has a debt of $200. She paid back $50. What is her remaining debt?

Solution:

Sarah had a debt of $200.

She paid back $50.

To find remaining debt, subtract the amount paid back from the initial debt:

$200 - $50 = $150

Example 5: A temperature drops by 8 degrees Celsius overnight and then rises by 5 degrees Celsius during the day. What is the net change in temperature?

Solution:

Temperature dropped by 8 degrees Celsius and then rose by 5 degrees Celsius.

To find the net change, subtract the decrease from the increase:

5 °C - 8 °C

= -3 °C

Important Maths Related Links:

• Study Mathematics
• Nonsingular Matrix
• Sets in Mathematics
• Conjunction in Discrete Mathematics
• Volume of a Frustum of a Cone
• Subsets
• Difference between Permutations and Combinations
• How to Calculate LCM
• Properties of Square Numbers
• Formula to Calculate Pi

Subtracting Integers Worksheet

Problem 1. A lift was on the 5th floor. It descended 3 floors and then ascended 7 floors. Which floor is the lift on now?

Problem 2. Mary has a collection of 15 books. She gave away 7 books to her friend. How many books does Mary have left?

Problem 3. In a quiz, a student scored -5 points in one round and then scored 10 points in the next round. What is the student's total score?

Problem 4. A company had a profit of $500 last month but incurred a loss of $300 this month. What is the net profit or loss for the company over these two months?

Problem 5. A car travelled 50 kilometres north and then turned around and travelled 30 kilometres south. How far is the car from its starting point?