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Page 8

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: F×F ® F

ab: F×F ® 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: F×F ® F

ab: F×F ® 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 1*a* = *a*

inverse

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