The properties of numbers can help us to rewrite, simplify and understand the expressions. Let's classify them!
| Properties of Equality
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| Property
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Symbols
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Any quantity is equal to itself.
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For any number a, a=a.
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If one quantity equals a second quantity, then the second quantity equals the first.
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For any numbers a and b, if a=b, then b=c.
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If one quantity equals a second quantity and the second quantity equals a third quantity, then the first quantity equals the third quantity.
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For any numbers a,b and c, if a=b and b=c, then a=c.
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| Substitution Property
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A quantity may be substituted for its equal in any expression.
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If a=b, then a may be replaced by b in any expression.
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There are also special properties associated to operations. The following properties are for the addition.
| Addition Properties
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| Property
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Words
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Symbols
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For any number a, the sum of a and 0 is a.
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a+0=0+a=a
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A number and its opposite are additive inverses of each other.
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a+(- a) = 0
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The following properties are associated with multiplication.
| Multiplication Properties
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| Property
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Words
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Symbols
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| Multiplicative Identity
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For any number a, the product of a and 1 is a.
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a * 1 =a, 1 * a =a
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| Multiplicative Property of Zero
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For any number a, the product of a and 0 is 0.
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a * 0 =0, 0 * a =0
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| Multiplicative Inverse
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For every number ab, where a,b≠0, there is exactly one number ba such that the product of ab and ba is 1.
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a/b * b/a =1, b/a * a/b =1
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Finally, we have two special properties that can be used for addition and multiplication.
| Property
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Words
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Symbols
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| Commutative Property
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The order in which you add or multiply numbers does not change their sum or product.
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For any numbers a and b, a+b=b+a and a * b = b* a.
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| Associative Property
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The way you group three or more numbers when adding or multiplying does not change their sum or product.
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For any numbers a, b, and c, (a+b) +c =a+ (b+c) and (ab )c = a(bc).
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