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There are five basic properties in Algebra. They are commutative, associative, distributive, identity and inverse. |
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Commutative Property |
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If the result of an operation remains unchanged, with a change in the order of operands, then the operation is said to be commutative.
Let a and b be two operands.
Commutative law over Addition: a + b = b + a
Commutative law over Multiplication: a * b = b * a |
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Associative Property |
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If the order of performing multiple operations is not important, then the operation is said to be associative.
Associative law over Addition: (a + b) + c = a + (b + c)
Associative law over Multiplication: (a * b) * c = a * (b * c) |
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Distributive Property |
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Distributive property includes both the addition and multiplication of real numbers.
a * (b + c) = a * b + a * c.
Here addition is said to be distributive over multiplication. |
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Identity Property |
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If the value of the operand(s) remains unchanged even after performing an operation, then the operation is said to have identity property.
The additive identity is ‘0,’ i.e. a + 0 = 0 + a = a
The multiplicative identity is ‘1,’ i.e. a * 1 = 1 * a = a |
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Inverse Property |
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The value which gives the additive identity when added to the original number is called the additive inverse. The additive inverse is the negative of the value.
a + (- a) = 0.
The value which gives the multiplicative identity when multiplied with the original number is called the multiplicative inverse. The multiplicative inverse is 1/a for a real number a.
a * 1/a=1 |
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Example |
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The algebraic properties make calculations simple. Let us see the following example which proves it.
Evaluating 5 * 206 directly takes time, definitely, but applying distributive property here can lessen the calculation time and also the effort.
5 * (200 + 6) = 5 * 200 + 5 * 6
= 1000 + 30
= 1030 |
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Try this problem |
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1. Describe the 5 basic properties of algebra. |
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The five basic properties of algebra are commutative, associative, distributive, identity and inverse. |
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If the result of an operation remains unchanged, with a change in the order of operands (such as a and b), then the operation is said to be commutative. Thus, a+b=b+a and a*b=b*a. |
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If the order of performing multiple operations is not important, then the operation is said to be associative. For example, (a+b)+c=a+(b+c) and (a*b)*c=a*(b*c). |
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The distributive property includes both the addition and multiplication of real numbers, such as a*(b+c)=a*b+a*c. |
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If the value of the operand(s) remains unchanged even after performing an operation, then the operation is said to have identity property. For example, if the additive identity is ‘0,’ then a+0=0+a=a. Also, if the multiplicative identity is ‘1,’ then a*1=1*a=a. |
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The value which gives the additive identity when added to the original number is called the additive inverse. The additive inverse is the negative of the value: for example a+(-a)=0. |
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The value which gives the multiplicative identity when multiplied with the original number is called the multiplicative inverse. The multiplicative inverse is 1/a for a real number a: for example a * 1/a=1. |
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