The Invariant Property of Subtraction
The invariant property of subtraction states that the difference between two numbers $ a-b $ remains unchanged if the same value $ c $ is added to or subtracted from both terms. $$ a - b = (a - c) - (b - c) $$ $$ a - b = (a + c) - (b + c) $$
In other words, when you subtract the same number \( c \) from both terms:
$$ a - b = (a - c) - (b - c) $$
This shows that if we subtract the same value \( c \) from both \( a \) and \( b \), their difference remains unaffected.
Similarly, if we add the same number \( c \) to both, the difference still stays the same.
$$ a - b = (a + c) - (b + c) $$
Why is this useful?
This property is helpful for simplifying calculations and solving equations where the difference between two numbers must remain constant, even when both terms are adjusted by the same amount.
Example
Let’s go through a numerical example to demonstrate the invariant property of subtraction.
Suppose you want to find the difference between \( a = 12 \) and \( b = 5 \) .
$$ 12 - 5 = 7 $$
The difference is 7.
Now, you can subtract 2 from both the minuend (12) and the subtrahend (5) to get the same result.
$$ (12-2) - (5-2) = 10-3 = 7 $$
This transforms the subtraction into $ 10 - 3 $, which still results in 7.
Why do this? By subtracting 2 from 12, you make the number easier to work with. Mental calculations become simpler when numbers are rounded. In other words, it’s often quicker to compute $ 10-3 $ rather than $ 12-5 $. The outcome remains the same.
Here’s another example.
Find the difference between \( a = 23 \) and \( b = 9 \) .
$$ 23 - 9 = 14 $$
The difference is 14.
Now, let’s apply the invariant property by adding 1 to both numbers.
$$ (23+1) - (9+1) = 24-10 = 14 $$
The difference remains 14, just as before.
In this case, you’ve rounded one of the numbers again. It’s faster to calculate $ 24-10 $ than $ 23-9 $. In both scenarios, the result is the same.
These examples clearly show that adding or subtracting the same number from both terms keeps the difference unchanged.
