In mathematics, a real number is a value that represents a quantity along a continuous line. The real numbers 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 numbers 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 real number 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 real numbers as a special case. These descriptions of the real numbers are not sufficiently rigorous by the modern standards of pure mathematics. The discovery of a suitably rigorous definition of the real numbers — indeed, the realization that a better definition was needed — was one of the most important developments of 19th century mathematics. The currently standard axiomatic definition is that real numbers form the unique complete totally ordered field (R,+,·,<), up to isomorphism,[1] Whereas popular constructive definitions of real numbers include declaring them as equivalence classes of Cauchy sequences of rational numbers, Dedekind cuts, or certain infinite "decimal representations", together with precise interpretations for the arithmetic operations and the order relation. These definitions are equivalent in the realm of classical mathematics. Definition

The real number system can be defined axiomatically up to an isomorphism, which is described below. There are also many ways to construct "the" real number system, for example, starting from natural numbers, then defining rational numbers algebraically, and finally defining real numbers as equivalence classes of their Cauchy sequences or as Dedekind cuts, which are certain subsets of rational numbers. Another possibility is to start from some rigorous axiomatization of Euclidean geometry (Hilbert, Tarski etc.) and then define the real number system geometrically. From the structuralist point of view all these constructions are on equal footing. Axiomatic approach

Let R denote the set of all real numbers. Then:
* The set R is a field, meaning that addition and multiplication are defined and have the usual properties. * The field R is ordered, meaning that there is a total order ≥ such that, for all real numbers x, y and z: * if x ≥ y then x + z ≥ y + z;

* if x ≥ 0 and y ≥ 0 then xy ≥ 0.
* The order is Dedekind-complete; that is, every non-empty subset S of R with an upper bound in R has a least upper bound (also called supremum) in R. The last property is what differentiates the reals from the rationals. For example, the set of rationals with square less than 2 has a rational upper bound (e.g., 1.5) but no rational least upper bound, because the square root of 2 is not rational. The real numbers are uniquely specified by the above properties. More precisely, given any two Dedekind-complete ordered fields R1 and R2, there exists a unique field isomorphism from R1 to R2, allowing us to think of them as essentially the same mathematical object. For another axiomatization of R, see Tarski's axiomatization of the reals. Construction from the rational numbers

The real numbers can be constructed as a completion of the rational numbers in such a way that a sequence defined by a decimal or binary expansion like (3, 3.1, 3.14, 3.141, 3.1415,...) converges to a unique real number. For details and other constructions of real numbers, see construction of the real numbers.

Basic properties
A real number may be either rational or irrational; either algebraic or transcendental; and either positive, negative, or zero. Real numbers are used to...

...Basic Algebraic Properties of RealNumbers
The numbers used to measure real-world quantities such as length, area, volume, speed, electrical charges, probability of rain, room temperature, gross national products, growth rates, and so forth, are called realnumbers. They include such number as , , , , , , , and .
The basic algebraic properties of the real...

...AXIOMS OF REALNUMBERS
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...

...
Real World Radical Formulas
Janeth Mendiola
MAT222: Intermediate Algebra
Instructor Lalla Thompson
March 21, 2014
Real World Radical Formulas
Radical formulas are used in the real world in the fields such as finance, medicine, engineering, and physics to name a few. In the finance department they use it to find the interest, depreciation and compound interest. In medicine it can be used to calculate the...

... 4. Repeat the previous three steps, except this time use the two terms that have just been written as the dividend.
5. Repeat step 4. This time, there is nothing to "pull down".
The polynomial above the bar is the quotient q(x), and the number left over (−123) is the remainder r(x).
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Applications
Factoring polynomials
Sometimes one or more roots of a polynomial are known, perhaps having been found using...

...means for investigating ideas and developing thinking skills. I share the school ethos that recognises mathematics as an important tool in everyday life and strives to provide children with the necessary skills to tackle a range of practical tasks and real- life problems.
At school, we all aim to provide a challenging and enjoyable curriculum for all children that helps build their confidence in mathematics and enables them to work with increasing accuracy.
Children’s’...

...Algebra in the Real World and Everyday Life
Hal Hagood
u07a2
Table of Contents
Page Number
Table of Contents …………………………………………………………………… 2
Introduction ………………………………………………………………………… 3
Ways That Algebra Affects Business or Science ………………………………….. 5
How Algebraic Concepts Can Solve Everyday Problems in Life …………………. 6
Ways Algebra Can Solve Everyday Problems in Business or Science ……………. 9
A Surprising...

...5. Working With Negative Numbers
Lesson 6. Solving Equations Using Properties of Mathematics
Chapter Review
Chapter Test
Chapter 2: Working with Rational Numbers
Lesson 1. Integers and the Number Line
Lesson 2. Absolute Value
Lesson 3. Rational Numbers
Lesson 4. Adding Rational Numbers
Lesson 5. Subtracting Rational Numbers
Lesson 6. Multiplying Rational Numbers
Lesson 7. Distributive...

...universal precautions. Suppose the spread of a direct contact disease in a stadium is modeled by the exponential equation P(t) = 10,000/(1 + e3-t) where P(t) is the total number of people infected after t hours. (Use the estimate for e (2.718) or the graphing calculator for e in your calculations.)
1. Estimate the initial number of people infected with the disease. Show how you found your answer.
Answer: A total of 474 people would be initially infected....