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Why Are Real Numbers a Field?
B A S I C   A L G E B R A
Linear Equations and Inequalities
Lesson 7

6.0 Students graph a linear equation and compute the x- and y- intercepts (e.g., graph 2x + 6y


Why are real numbers a field?
First we have to understand the meaning of a field

A field is a set F, with two operations: addition and multiplication

a + b: FF ® F
ab: FF ® F

The "" means to pair elements from these two groups and refers to whatever operation is involved, either addition or multiplication.

In other words, if we take 2 elements from the set of F and carry out the indicated operation, the result is another number which is an element of F. (So if we take 2 elements of the real number set F and add them together, the result is another real number F.

(Does the ray/arrow looking symbol read onto or does it mean something else?)

So, a field is a set F, with two operations: addition and multiplication

a + b: FF ® F
ab: FF ® F

satisfying the following properties:

commutative
a + b = b + a, ab = ba

associative
(a + b) + c = a + (b + c), (ab)c = a(bc)

distributive
a(b + c) = ab + ac

zero
There is a special element 0 Î F such that 0 + a = a Was the rectangular box the symbol or was it something else? Please clarify by telling me in words what the symbol said.

negative
For any a Î F there is an element -a such that a + (a) = 0

unit element
There is a special element 1Î F such that 1a = a

inverse
For any a ¹ F {0} there is an element a-1 such that (a)(a1) = 1