Term
| a field is closed under addition |
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Definition
| For all a, b that exist in set R, a + b is an element of R |
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Term
| a field is closed under multiplication |
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Definition
| For all a, b that exist in R, a · b is an element of R |
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Term
| addition is associative in a field |
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Definition
| for all a, b, c that exist in R, a + (b + c) = (a + b) + c |
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Term
| multiplication is associative in a field |
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Definition
| for all a, b, c that exist in R, a · (b · c) = (a · b) · c |
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Term
| addition is commutative in a field |
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Definition
| for all a, b that exist in R, a + b = b + a |
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Term
| multiplication is commutative in a field |
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Definition
| for all a, b that exist in R, a · b = b · a |
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Term
| 0 is the additive identity in a field |
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Definition
| there exists 0 as an element of R such that for all a in R, 0 + a = a + 0 = a |
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Term
| 1 is the multiplicative identity of a field |
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Definition
| there exists 1 as an element of R such that for all a in R, 1 · a = a · 1 = a |
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Term
| (-a) is the additive inverse of a in a field |
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Definition
| there exists (-a) as an element of R such that for all a in R, (-a) + a = a + (-a) = 0 |
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Term
| (1/a) is the multiplicative inverse of a in a field |
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Definition
| there exists (1/a) as an element of R such that for all a in R, (1/a) · a = a · (1/a) = 1 |
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Term
| multiplication is distributive in a field |
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Definition
| for all a, b, c that exist in R, a · (b + c) = a · b + a · c |
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Term
| inequalities with positive elements can be multiplied in an ordered field |
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Definition
| for all a, b, c that exist in R such that a > b and c > 0, a · c > b · c |
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Term
| inequalities can be added in an ordered field |
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Definition
| for all a, b, c that exist in R such that a > b, a + c > b + c |
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Term
| inequality is transitive in an ordered field |
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Definition
| for all a, b, c that exist in R such that a > b and b > c, a > c |
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