Field Axioms 2

Example

100

Using variables to stand for numbers, write an example of the following axiom:

Additive Inverse

a + (-a) = 0

200

Using variables to stand for numbers, write an example of the following axiom:

Multiplicative Inverse

a * 1/a = 1

300

Using variables to stand for numbers, write an example of the following axiom:

Trichotomy

If x and y are real numbers, then one of the following is true,

y < x   OR   y > x    OR   y = x

400

Using variables to stand for numbers, write an example of the following axiom:

Transitivity for Equality

If x = y and y = z, then x = z

500

Using variables to stand for numbers, write an example of the following axiom:

Transitivity for Order

If x > y and y > z, then x > z

Name the Inverse

100

What is the additive inverse of 

3/4

-3/4

200

What is the multiplication inverse of 3/4

4/3

300

What is the additive inverse of -4

4

400

What is the multiplicative inverse of -4

-1/4

500

What is the multiplicative inverse of 

-5/6

-6/5

Example

100

Give an example that shows:

Subtraction is NOT a commutative operation

5 - 7 = -2 and 7 - 5 = 2

200

Give an example that shows:

{negative numbers} is NOT closed under multiplication

(-3) (-8) = 24

24 is not a negative number

300

Give an example that shows:

{digits} is NOT closed under addition

7 + 9 = 16

16 is not a digit, but a double digit number

400

Give an example that shows:

Subtraction is NOT an associative operation

10 - (3-2) = 9

(10-3) - 2 = 5

500

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Is it A Field

100

Is the following set of numbers a field?  If not, list the axiom or axioms that cause the axiom to be excluded:

{rational numbers}

It is a field

200

Is the following set of numbers a field?  If not, list the axiom or axioms that cause the axiom to be excluded:

{integers}

Not a field

the Multiplicative inverse of any integer would be a a fraction.  A fraction is not an integer.

300

Is the following set of numbers a field?  If not, list the axiom or axioms that cause the axiom to be excluded:

{positive numbers}

Not a field

The additive inverse would be a negative number

The additive identity is zero, zero is not a positive number

400

Is the following set of numbers a field?  If not, list the axiom or axioms that cause the axiom to be excluded:

{non-negative numbers}

Not a field

The additive inverse would be a negative number

The additive identity would be zero, which is neither negative or non negative

500

Is the following set of numbers a field?  If not, list the axiom or axioms that cause the axiom to be excluded:

{odd numbers}

Not a field

The multiplicative inverse would be a fraction

Evenness and Oddness is only defined for integers

Define

100

What is a variable

a letter that stand for an unspecified number

200

What is an expression

a collection of variables and constants connected by operation signs (+, -, *, /) which stands for a number 

300

What is an equation

Two expressions connected by an equal sign (=)

400

Define Simplify

To put into simpler terms to make the expression easier to work with

500

Define Solve or Evaluate

Finding a numerical solution to an equation