Properties of Subtraction
8 Minutes
Properties of subtraction for students: see which addition rules apply to subtracting integers. A short maths lesson in English.
- Testing the closure property with integer subtraction
- Climbing a mountain to see why commutativity fails
- Explaining why associativity and identity do not work
What this lesson teaches
Integer subtraction obeys only the closure property and fails to follow the commutative, associative, and identity properties that apply to addition. While the difference between any two integers is always an integer, reversing the order of the numbers or altering how they are grouped leads to completely different results.
Closure and commutativity
The closure property states that performing an operation on any two integers must produce an integer. When subtracting minus 3 from minus 2, the answer is 1, which is an integer, proving that closure works for subtraction. The commutative property states that changing the order of numbers should give the same result, which works for addition where 10 plus 2 equals 2 plus 10. In subtraction, 3 minus 2 gives 1, but reversing the numbers to 2 minus 3 gives minus 1, meaning commutativity fails.
Associative property
The associative property deals with how numbers are grouped when combining three or more values. In addition, grouping minus 3, 4, and 9 in any order gives 10, just like mixing coffee powder, sugar, and milk yields coffee regardless of the sequence. In subtraction, grouping matters completely, just like the steps required when preparing biryani. Subtracting 4 from minus 3 and then subtracting 9 gives minus 16, whereas subtracting the result of 4 minus 9 from minus 3 gives positive 2.
Identity property
- In addition, adding zero to any number leaves the value unchanged, such as 0 plus 5 and 5 plus 0 both equaling 5.
- In subtraction, the order in which zero is used produces two entirely different numbers.
- Taking 5 minus 0 means climbing 5 steps and remaining there, giving an answer of 5.
- Taking 0 minus 5 means turning around from zero and descending 5 steps, giving minus 5.
- Because 5 minus 0 does not equal 0 minus 5, the identity property does not apply to subtraction.
Real-life example from the lesson
- A person took a loan of 10,000 rupees, paid back 7,000 rupees, and later paid another 3,000 rupees.
- Calculating the balance properly by subtracting 7,000 from 10,000 leaves 3,000, and subtracting the next 3,000 leaves zero.
- A math error happened by treating subtraction as associative and grouping the two repayments together incorrectly.
- Subtracting 3,000 from 7,000 gave 4,000, and subtracting that from 10,000 falsely showed an unpaid debt of 6,000 rupees.