Axioms+for+Real+Numbers

__**Axioms for Real Numbers**__
 * Link For Axioms for Real Numbers**[|**http://www.wiziq.com/Content/Search.aspx?Tags=axioms%20for%20real%20numbers**]

__ ** VOCABULARY WORDS ** __

**Operations** - addition and multiplication are 2 operations used when working with real numbers.


 * Binary Operation**- assigns 2 real numbers to a third real number.


 * Sum**- the addition operation assigns 2 real numbers **a** and **b** to another real number, **a** + **b**.


 * Product**- multiplication assigns to any 2 real numbers a and b their product denoted by a * b or (a) b


 * Terms**- in the sum a + b, a and b are terms.


 * Factors**-in the product ab, a and b are factors.


 * Axioms-** statements that are assumed to be true**.**


 * Substitution Principle-** since a + b ab are unique, changing the numeral by which a number is named in an expression involving sums or products does not change the value of the expression.


 * Additive Inverse**- -a and 1/a

**Axioms of Additon and Multiplication in R** Let a, b, and c denote any real number ( a,b, and c is a real number)

a multiplied by 1 = a and 1 multiplied by a = a. || **Idenity Axiom for Multiplication** ||
 * 1. a + b is a unique real number || **Closure Axiom for Addition** ||
 * 2. (a+b) +c= a +(b+c) || **Associtative Axiom for Addition** ||
 * 3. a+b = b+a || **Communative Axiom for Addition** ||
 * 4. There exists an element 0 is a real number such that for each a is a real number, 0 + a = a and a+0 = a || **Identity Axiom for Addition** ||
 * 5. There exists an element -a is a real number for each a is a real number such that a+ (-a) =0 and (-a)+ a=0. || **Axiom of Additive Inverses** ||
 * 6. ab is a unique real number. || **Closure Axiom for Mulitplication** ||
 * 7. (ab)c =a (bc) || **Associative Axiom for Multiplication** ||
 * 8. ab = ba || **Communative Axiom for Multiplication** ||
 * 9. There exists an element 1 is a real number, 1 is not equal to 0, such that for each a is a real number,
 * 10. There exists an element 1/a is a real number for each nonzero a is a real number such that 1/a multiplied by a=1 || **Axiom of Multiplicative Inverses** ||
 * 11. a(b+c) = ab + ac and (b+c)a = ba +ca || **Distributive Axiom** ||

**Axioms of Equality**


 * //Let a,b, and c by any element of a real number// ||  ||
 * 1. a=a || **Reflexive Property** ||
 * 2. if a = b, then b = a || **Symmetric Property** ||
 * 3. If a = b amd b = c, then a = c || **Transitive Property** ||


 * Examples**

1. Using the communtative and associative axioms, determine the value of each expression. 16 + (-8) + 5 + (-11) + 9 ~ 16 +(-8) = 8 * solve the problem from left to right ~ 8 + 5 = 13 ~ 13 +(-11) = 2 ~ 2 + 9 = 11 = 11

2. (-4)(3)(5)(-2) * solve the problem from left to right ~ -4 * 3 = -12 ~ -12* 5 =- 60 ~ - 60 *- 2= 120 * 2 negative = a positive = 120

3. Complete each open sentence so that it is true for all real values of the variable. State the axiom or property that justifies each statement. 4 + ? = 0 4 + (-4) = 0 -4 satisfies the problem

**Quiz Questions** State the axiom or property that justifies each statement. Assume that each variable represents a real number.

1. If x = 5 and 5 = y, then x = y

2. 8 * 1/8 = 1

3. x + 5 is a real number.

Using the commutative and associative axioms, determine the value of each expression.

4. 16 + (-8) + 5 + (-11) + 9 5. (-3)(7)(-5)(-8)


 * 2 Non Media Related Hyperlinks**

- http://www.math.ucla.edu/~tao/resource/general/121.1.00s/axioms.html

- http://www.math.ucsb.edu/~moore/axiomsforreals.pdf


 * Sources Cited**

- "Axioms of the real line." __Www.math.ucla.edu__. UCLA. 4 Dec. 08 .

-Moore, John D. "Axioms for the Real Numbers." __Www.math.ucsb.edu__. 5 Oct. 08. UCSB. 4 Dec. 08 <[|http://http://www.math.ucsb.edu/~moore/axiomsforreals.pdf]>.