The additive identity property says that a number does not change when zero is added to that number.

\(2.5+0=2.5\)

Examples: 

\((1)\;4.2+0=4.2\\ (2)\;8.9+0=8.9\\ (3)\;11.52+0=11.52\)

Multiplicative identity

The multiplicative identity property says that a number does not change when \(1\) is multiplied to that number.

Examples: 

\(2.5×1=2.5\)

\(4.2×1=4.2\)

\(6.8×1=6.8\)

• The commutative property says that the numbers can be added or multiplied in any order, and we still get the same result.

For addition: \(1.2+2.5=2.5+1.2\)

For multiplication: \(1.2×2.5=2.5×1.2\)

• For example: Suppose we are adding \(1.6\) and \(1.8\), then

\(1.6+1.8=3.4\)

or

\(1.8+1.6=3.4\)

Thus, the answer is still the same.

Examples:

\((1)\;2.14×1.28=1.28×2.14\\ (2)\;3.15+7.45=7.45+3.15\\ (3)\;10.2×5.5=5.5×10.2\)

Associative property of addition:

• The associative property says that it does not matter how we group the numbers when we add more than two numbers, the sum remains the same.
• The grouping of the addends (numbers being added) does not change the sum.

\((1.2+2.3)+3.4=1.2+(2.3+3.4)\)

Associative property of multiplication:

• The associative property says that it does not matter how we group the numbers when we multiply more than two numbers, the product remains the same. 
• The grouping of the factors (numbers being multiplied) does not change the product.

\((1.2×2.3)3.4=1.2(2.3×3.4)\)

• For example:

\((3.2+5.6)+4.1=3.2+(5.6+4.1)\)

\(8.8+4.1=3.2+9.7\)

\(12.9=12.9\)

In both the groupings, the sum of the addition does not change.

Distributive property over addition:

The distributive property over addition states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products together.

\(4.2(3.1+1.2)=(4.2×3.1)+(4.2×1.2)\)

\(4.2×4.3=13.02+5.04\)

\(18.06=18.06\)

Distributive property over subtraction:

Either we find out the difference first and then multiply, or we first multiply with each number and then subtract, the result remains the same.

\(4.2(3.1-1.2)=(4.2×3.1)-(4.2×1.2)\)

\(4.2×1.9=13.02-5.04\)

\(7.98=7.98\)

• Note : In case, if we change the order of subtraction, then the result also changes.

\(4.2(3.1-1.2)\neq(4.2×1.2)-(4.2×3.1)\)

An implicit expression can be solved as an explicit expression with the help of properties.

• A decimal number can be written as a combination of the whole and the decimal part using the addition operation. 

\(1.5=1+0.5\)

• Using the distributive property, we can multiply the numbers separately.

\(1.5(2.5+3.5)=1.5(2.5)+1.5(3.5)\)

• Using the associative property, we can change the grouping of the numbers.

\(1.5+(2.5+3.5)=(1.5+2.5)+3.5\)

• Using the commutative property, we can change the order of the numbers.

\(1.5+2.5=2.5+1.5\)

For example: Solving \(800×25.5\)

• The decimal number \(25.5\) can be written as :

\((25+.5)\)

• Rewriting the expression gives, 

\(800(25+.5)\)

• Using the distributive property, we multiply \(800\) to each of the addends and then adding the products together.

\(800(25)+800(.5)\)

\(20000+400\)

\(=20,400\)