 | Term | Definition |
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Closure Property for Addition |
a + b is a unique real number |
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Associative Property for Addition |
(a+b)+c=a+(b+c) |
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Commutative Property for Addition |
a+b=b+a |
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Identity Property for Addition |
There exists an element -a E R, for each a E R, such that: a+0=a & 0+a=a |
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Inverse Property of Addition |
a + (-a)=0 |
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Closure Property for Multiplication |
ab is a unique real number |
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Associative Property for Multiplication |
(ab)c=a(bc) |
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Commutative Property for Multiplication |
ab=ba |
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Identity Property for Multiplication |
There exists an element 1 € R, such that for each a € R: a•1=a & 1•a=a |
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Inverse Property of Multiplication |
There exists an element 1 ⁄a € R, for each nonzero a € R such that: 1/a•a=a & a•1/a=1 |
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Distributive Property for Mult. over Add. |
a(b+c)=ab+ac & (b+c)a=ba+ca |
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Substitution Principle |
Since a+b and ab are unique, changing the numeral by which a number is named in an expression involving sums or products does not change the value of the expression. *This principle is used as a last resort for any basic math that does not involve either Identity Properties, either Inverse Properties, Multiplication Properties of Zero or -1 |
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Reflexive Property of Equality |
a=a |
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Symmetric Property of Equality |
if a=b then b=a |
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Transitive Property of Equality |
if a=b and b=c, then a=c |
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Addition Property of Equality |
a+c = b+c; c+a = c+b |
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Multiplication Property of Equality |
ac=bc; ca=cb |
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Division Property of Equality |
a/c = b/c (provided c ≠ 0) |
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Multiplication Property of Zero |
For all real numbers a, a•0=0 and 0•a=0 |
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Multiplication Property of -1 |
For all real numbers a, a•(-1)=-a and (-1)•a=-a |
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Property of the Opposite of a Sum |
For all real numbers a and b, -(a+b)=(-a)+(-b) That is, the opposite of a sum of real numbers is the sum of the opposites of the numbers. |
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Properties of Opposites in Products |
For all real numbers a and b, (-a)b=-ab, a(-b)=-ab, (-a)(-b)=ab |
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Subtraction Rule |
For all real numbers a and b, a-b=a+(-b) |
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Division Rule |
For all real numbers a and all nonzero real numbers b , a-b=a+(-b) |
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Subtraction Property of Equality |
a-c = b-c |
