Invariant Property
The invariant property describes mathematical operations whose outcome remains unchanged when their terms are modified in the same way on both sides.
The invariant property appears in several areas of elementary mathematics.
It applies to subtraction, division, fractions, and equations.
Note. The invariant property does not apply to addition or multiplication.
Invariant property of subtraction
In a subtraction, adding or subtracting the same number from both the minuend and the subtrahend does not change the difference. $$ a-b = (a+c)-(b+c) = (a-c)-(b-c) $$
Example
The difference between 22 and 10 is 12.
$$ 22 - 10 = 12 $$
If I add 5 to both the minuend (22+5) and the subtrahend (10+5), the result stays the same.
$$ (22+5) - (10+5) = $$
$$ 27 - 15 = 12 $$
Note. The invariant property does not hold for addition. $$ 22+10 \ne (22+5)+(10+5) \ne (22-5)+(10-5) $$ $$ 32 \ne 27+15 \ne 17+5 $$ $$ 32 \ne 42 \ne 22 $$
Example 2
$$ 22 - 10 = 12 $$
If I subtract 30 from both the minuend (22-30) and the subtrahend (10-30), the difference is still the same.
$$ (22-30) - (10-30) = $$
$$ -8 - (-20) = $$
$$ -8 + 20 = 12 $$
Invariant property of division
In a division, multiplying or dividing both the dividend and the divisor by the same nonzero number (c≠0) does not change the quotient. $$ a:b = (a \cdot c):(b \cdot c) = (a : c):(b : c) $$
Example
The quotient of 20 divided by 10 is 2.
$$ 20:10 = 2 $$
If I multiply both the dividend (20) and the divisor (10) by 3, the quotient remains unchanged.
$$ (20 \cdot 3):(10 \cdot 3) = $$
$$ 60:30 = 2 $$
Example 2
$$ 20:10 = 2 $$
If I divide both the dividend (20) and the divisor (10) by 5, I again obtain the same quotient.
$$ (20 : 5):(10 : 5) = $$
$$ 4:2 = 2 $$
Note. The invariant property does not apply to multiplication. $$ 20 \cdot 5 \ne (20 \cdot 5) \cdot (5 \cdot 5) \ne (20 : 5) \cdot (5 : 5) $$ $$ 100 \ne 100 \cdot 25 \ne 4 \cdot 1 $$ $$ 100 \ne 2500 \ne 4 $$
Invariant property of fractions
In a fraction, multiplying or dividing both the numerator and the denominator by the same nonzero number (c≠0) produces an equivalent fraction. As a result, the value of the fraction does not change. $$ \frac{a}{b} = \frac{a \cdot c}{b \cdot c} = \frac{ac}{bc} $$
Example
Consider the fraction 4/5.
$$ \frac{4}{5} $$
If I multiply both the numerator (4) and the denominator (5) by 3, I obtain a fraction equivalent to the original one.
$$ \frac{4 \cdot 3}{5 \cdot 3} $$
$$ \frac{12}{15} $$
The two fractions are equivalent.
$$ \frac{4}{5} = \frac{12}{15} $$
Note. This can be verified by computing the cross product of the two fractions. $$ 4 \cdot 15 = 5 \cdot 12 $$ $$ 60 = 60 $$ The equality is confirmed. Therefore, the two fractions are equivalent.
Invariant property of equations
In an equation, applying the same operation to both the left-hand side and the right-hand side preserves the equality.
Example
This equation is satisfied for x=3, since 3+7=10.
$$ x + 7 = 10 $$
I subtract 7 from both the left-hand side and the right-hand side.
$$ x + 7 - 7 = 10 - 7 $$
$$ x = 3 $$
The result is unchanged.
And the same reasoning applies in general.
