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Real Numbers
properties of real numbers
14
Mathematics
Undergraduate 1
03/15/2017

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Cards

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