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The properties of decimal numbers include the properties of Closure, Associative, Commutative, Identity, and Inverse for the operations of addition and multiplication. Another common property is the Distributive property.
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The Closure property states that the result of the addition or multiplication of any two numbers in a set will produce another number in the same set. Since the sum of any two decimal numbers results in a decimal number, the property of Closure exists for the set of decimal numbers for the operation of addition. The Closure property also exists for the multiplication of decimal numbers. Symbolically, for any decimal number a and b,
a + b = decimal number and,
a x b = decimal number.
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The Associative property for the operation of addition states that the order in which we add three whole numbers does not matter since the result will always be the same sum. This property also works for the operation of multiplication. Therefore, for any decimal numbers a, b and c:
(a + b) + c = a + (b + c) associative property of addition and,
(a x b) x c = a x (b x c) associative property of multiplication.
The Associative property is useful for computations. For example, the sum of three decimal numbers 1.2, 1.4, and 1.6 is calculated by adding left to right following the order of operations: (1.2 + 1.4) + 1.6. However, it can also be found using the Associative property of addition: 1.2 + (1.4 + 1.6) where regrouping not only gives the same result but is slightly easier to compute. The results are the same because the sum is equivalent no matter how the numbers are grouped. |
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The Commutative property provides the opportunity to change the order of terms under the operations of addition and multiplication. The order in which we add two decimal numbers does not change the result. For example,
1.3 + 1.5 = 2.8 and 1.5 + 1.3 = 2.8
thus,
1.3 + 1.5 = 1.5 + 1.3
Likewise,
1.3(1.5) = 1.95 and 1.5(1.3) = 1.95
thus,
1.3(1.5) = 1.5(1.3).
This seems trivial but it is a powerful property for algebra that allows one to simplify expressions. For example, the expression
2.1x + 3 + 5.1x can be written as 2.1x + 5.1x + 3
since
2.1x + 3 + 5.1x = 2.1x + 5.1x + 3
by the Commutative property. Using the order of operations and combining like terms yields
7.2x + 3.
Symbolically, for any decimal numbers a and b,
a + b = b + a commutative property of addition and,
a(b) = b(a) commutative property of multiplication.
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The identity property of addition identifies the numbers that do not change the value of a decimal number under the operations of addition and multiplication. The additive identity is zero. For example, when you add zero to any decimal number, the result is the original decimal number,
1.4 + 0 = 1.4.
The additive identity is used later in Algebra 1 when completing the square.
The multiplicative identity is one. Any decimal number multiplied by one will result in the original decimal number with the exception of the decimal number zero. For example,
1.4 x 1 = 1.4.
Symbolically, for any decimal numbers a and b,
a + 0 = a additive identity and,
a(1) = a where a ≠ 0 multiplicative identity.
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The inverse property of addition states that the sum of the original decimal number and its inverse is zero. Finding an additive inverse is important when solving an equation. For example,
In this case the second line is applying the additive inverse property to isolate the variable x. The additive inverse of 2 is –2.
The inverse property of multiplication states that the product of the original decimal number and its inverse is one. Finding a multiplicative inverse is important when solving an equation. For example,
In this case the second line is applying the multiplicative inverse property to isolate the variable x. The multiplicative inverse of 2.5 is  .
Symbolically, for any decimal number a,
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The Distributive property is a useful tool for the multiplication of decimal numbers. In the real world we frequently use this property for mental multiplication. For example, when finding a standard tip of 15% for a $24 meal one might take
24(0.15) = 24 (0.1 + 0.05) = 2.4 + 1.2 = 3.6 or $3.60.
Symbolically, for any decimal numbers a and b,
a(b + c) = ab + ac.
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