AXIOMS OF REAL NUMBERS
Field Axioms: there exist notions of addition and multiplication, and additive and multiplicative identities and inverses, so that: • (P1) (Associative law for addition): a + (b + c) = (a + b) + c • (P2) (Existence of additive identity): 9 0 : a + 0 = 0 + a = a • (P3) (Existence of additive inverse): a + (−a) = (−a) + a = 0 • (P4) (Commutative law for addition): a + b = b + a

• (P5) (Associative law for multiplication): a · (b · c) = (a · b) · c • (P6) (Existence of multiplicative identity): 9 1 6= 0 : a · 1 = 1 · a = a • (P7) (Existence of multiplicative inverse): a · a−1 = a−1 · a = 1 for a 6= 0 • (P8) (Commutative law for multiplication): a · b = b · a • (P9) (Distributive law): a · (b + c) = a · b + a · c

Order Axioms: there exists a subset of positive numbers P such that • (P10) (Trichotomy): exclusively either a 2 P or −a 2 P or a = 0. • (P11) (Closure under addition): a, b 2 P ) a + b 2 P
• (P12) (Closure under multiplication): a, b 2 P ) a · b 2 P Completeness Axiom: a least upper bound of a set A is a number x such that x _ y for all y 2 A, and such that if z is also an upper bound for A, then necessarily z _ x. • (P13) (Existence of least upper bounds): Every nonempty set A of real numbers which is bounded above has a least upper bound.

We will call properties (P1)–(P12), and anything that follows from them, elementary arithmetic. These properties imply, for example, that the real numbers contain the rational numbers as a subfield, and basic properties about the behavior of ‘>’ and ‘ on R. (That is, given any pair a, b then a > b is either true or false). It satisfies:

a) Trichotomy: For any a R exactly one of a > 0, a = 0, 0 > a is true. b) If a, b > 0 then a + b > 0 and a.b > 0
c) If a > b then a + c > b + c for any c
Something satisfying axioms I and II is called an ordered field. Examples

1. The field Q of rationals is an ordered field.
Proof
Define a/b > c/d provided that b, d > 0 and ad > bc in Z. One may...

...In mathematics, a realnumber is a value that represents a quantity along a continuous line. The realnumbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers such as √2 (1.41421356... the square root of two, an irrational algebraic number) and π (3.14159265..., a transcendental number). Real...

...-------------------------------------------------
Realnumber
In mathematics, a realnumber is a value that represents a quantity along a continuum, such as 5 (an integer), 3/4 (a rational number that is not an integer), 8.6 (a rational number expressed in decimal representation), and π (3.1415926535..., an irrational number). As a subset of the realnumbers,...

...integers in two different ways. 1729 = 93 + 103 = 13 + 123. (This was the subject of a very famous mathematical anecdote involving Srinivasa Ramanujan and G.H. Hardy, circa 1917. See A Mathematician's Apology by Hardy.
Rank, Prime number, Found by, Found date, Number of digits
1st, 257,885,161 − 1, GIMPS, 2013 January 25, 17,425,170 2nd, 243,112,609 − 1, GIMPS, 2008 August 23, 12,978,189
3rd, 242,643,801 − 1, GIMPS, 2009 April 12, 12,837,064...

...Polynomial
The graph of a polynomial function of degree 3
In mathematics, polynomials are the simplest class of mathematical expressions (apart from the numbers and expressions representing numbers). A polynomial is an expression constructed from variables (also called indeterminates) and constants (usually numbers, but not always), using only the operations of addition, subtraction, multiplication, and non-negative integer exponents (which are...

...RealNumbers
-RealNumbers are every number.
-Therefore, any number that you can find on the number line.
-RealNumbers have two categories, rational and irrational.
Rational Numbers
-Any number that can be expressed as a repeating or terminating decimal is classified as a rational number
Examples of Rational...

...THE REALNUMBER SYSTEM
The realnumber system evolved over time by expanding the notion of what we mean by the word “number.” At first, “number” meant something you could count, like how many sheep a farmer owns. These are called the natural numbers, or sometimes the counting numbers.
Natural Numbers
or “Counting Numbers”
1, 2, 3, 4, 5, . . .
*...

...TUTORIAL: NUMBER SYSTEM
1. Determine whether each statement is true or false
a) Every counting number is an integer
b) Zero is a counting number
c) Negative six is greater than negative three
d) Some of the integers is natural numbers
2. List the number describe and graph them on the number line
a) The counting number smaller than 6
b) The integer between -3 and 3
3. Given...

...Need for achievement
Need for achievement (N-Ach) refers to an individual's desire for significant accomplishment, mastering of skills, control, or high standards. The term was first used by Henry Murray and associated with a range of actions. These include: "intense, prolonged and repeated efforts to accomplish something difficult. To work with singleness of purpose towards a high and distant goal. To have the determination to win". The concept of NAch was subsequently popularized by the...

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