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). Realnumbers can be thought of as points on an infinitely long line called the number line or real line, where the points corresponding to integers are equally spaced. Any realnumber can be determined by a possibly infinite decimal representation such as that of 8.632, where each consecutive digit is measured in units one tenth the size of the previous one. The real line can be thought of as a part of the complex plane, and correspondingly, complex numbers include realnumbers as a special case.
These descriptions of the realnumbers are not sufficiently rigorous by the modern standards of pure mathematics. The discovery of a suitably rigorous definition of the realnumbers — indeed, the realization that a better definition was needed — was one of the most important developments of 19th century mathematics. The currently...

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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, the integers, such as 5, express discrete rather than continuous quantities. Complex numbers include realnumbers as a special case. Realnumbers can be divided into rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two. A realnumber can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue indefinitely. The realnumbers are sometimes thought of as points on an infinitely long line called the number line or real line.
History
Vulgar fractions had been used by the Egyptians around 1000 BC; the Vedic "Sulba Sutras" ("The rules of chords") in, ca. 600 BC, include what may be the first 'use' of irrational numbers. The concept of irrationality was implicitly accepted by early Indian...

...The smallest integer that can be expressed as the sum of the cubes of two other 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 4th, 237,156,667 − 1, GIMPS, 2008 September 6, 11,185,272
5th, 232,582,657 − 1, GIMPS, 2006 September 4, 9,808,358 6th, 230,402,457 − 1, GIMPS, 2005 December 15, 9,152,052
7th, 225,964,951 − 1, GIMPS, 2005 February 18, 7,816,230 8th, 224,036,583 − 1, GIMPS, 2004 May 15, 7,235,733
9th, 220,996,011 − 1, GIMPS, 2003 November 17, 6,320,430 10th, 213,466,917 − 1, GIMPS, 2001 November 14, 4,053,946
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History
The following table lists the progression of the largest known prime number in ascending order. Here Mn= 2n − 1 is the Mersenne number with exponent n.[4]
Number, Digits, Year found
M127, 39, 1876 180×(M127)2 + 1, 79, 1951 M521, 157, 1952 M607, 183, 1952 M1279, 386, 1952 M2203, 664, 1952 M2281, 687, 1952 M3217, 969, 1957 M4423, 1332, 1961 M9689,...

...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 abbreviations for several multiplications by the same value). However, the division by a constant is allowed, because the multiplicative inverse of a non-zero constant is also a constant. For example, x2 − x/4 + 7 is a polynomial, but x2 − 4/x + 7x3/2 is an algebraic expression that is not a polynomial, because its second term involves a division by the variable x (the term 4/x), and also because its third term contains an exponent that is not a non-negative integer (3/2).
A polynomial function is a function which is defined by a polynomial. Sometimes, the term polynomial is reserved for the polynomials that are explicitly written as a sum (or difference) of terms involving only multiplications and exponentiation by non negative integer exponents. In this context, the other polynomials are called polynomial expressions. For example, is a polynomial expression that represents the same thing as the polynomial The term "polynomial", as an adjective, can also be used for quantities that can...

...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 RationalNumbers
6 is a rational number because it can be expressed as 6.0 and therefore it is a terminating decimal.
-7 ½ is a rational number because it can be expressed as -7.5 which is a terminating decimal.
Examples of Rational Numbers
Square root 25 is a rational number because it can be expressed as 5 or 5.0 and therefore it is a terminating decimal.
2.45 is a rational number because it is a repeating decimal.
Irrational Numbers
-An irrational number is a number that cannot be written as a fraction of two integers.
-Irrational numbers written as decimals are non-terminating and non-repeating.
Note: if a whole number is not a perfect square, then its square root is an irrational number.
Caution!
A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits....

...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, . . .
* The use of three dots at the end of the list is a common mathematical notation to indicate that the list keeps going forever.
At some point, the idea of “zero” came to be considered as a number. If the farmer does not have any sheep, then the number of sheep that the farmer owns is zero. We call the set of natural numbers plus the number zero the whole numbers.
Whole Numbers
Natural Numbers together with “zero”
0, 1, 2, 3, 4, 5, . . .
About the Number ZeroWhat is zero? Is it a number? How can the number of nothing be a number? Is zero nothing, or is it something?Well, before this starts to sound like a Zen koan, let’s look at how we use the numeral “0.” Arab and Indian scholars were the first to use zero to develop the place-value number system that we use today. When we write a number, we use only...

...
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 S = {-3, 0,[pic], [pic], e, , 4, 8…}, identify the set of
(a) natural numbers (b) whole numbers (c) integers
(d) rational numbers (e) irrational numbers (f) realnumbers
4. Express each of the numbers as a quotient [pic]
(a) 0.7777…… (b) 2.7181818….
5. Write each of the following inequalities in interval notation and show them on the realnumber line.
(a) 2 < x < 6 (b) (5 < x < (1
(c) (3 ( x ( 7 (d) (2 < x ( 0
(e) x < 3 (f) x ( (1
(g) x ( (2 (h) (3 ( x < 2
6. Show each of the following intervals on the realnumber line.
(a) [(2, 3] (b) ((4, 4)
(c) (((, 5] (d) [(1, ()
(e) ((3, 6] (f) [(2, 3)
(g) ((2, 0) ( (3, 6) (h) [(6, 2) ( [(3, 7)
2 Evaluate
(a) [pic] (b) 27[pic] (c) [pic] (d) [pic]
(e) (0.36)[pic] (f) (2.56)[pic] (g) [pic] (h)...

...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 psychologist David McClelland
This personality trait is characterized by an enduring and consistent concern with setting and meeting high standards of achievement. This need is influenced by internal drive for action (intrinsic motivation), and the pressure exerted by the expectations of others (extrinsic motivation). Measured by thematic appreciation tests, need for achievement motivates an individual to succeed in competition, and to excel in activities important to him or her.
Need for Achievement is related to the difficulty of tasks people choose to undertake. Those with low N-Ach may choose very easy tasks, in order to minimize risk of failure, or highly difficult tasks, such that a failure would not be embarrassing. Those with high N-Ach tend to choose moderately difficult tasks, feeling that they are challenging, but within reach.
People high in N-Ach are characterized by a tendency to seek challenges and a high degree of independence. Their most satisfying reward is the...

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