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...

...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 realnumbers can be expressed in terms of the two fundamental operations of addition and multiplication.
Basic Algebraic Properties:
Let and denotes realnumbers.
(1) The Commutative Properties
(a) (b)
The commutative properties says that the order in which we either add or multiplication realnumber doesn’t matter.
(2) The Associative Properties
(a) (b)
The associative properties tells us that the way realnumbers are grouped when they are either added or multiplied doesn’t matter. Because of the associative properties, expressions such as and makes sense without parentheses.
(3) The Distributive Properties
(a) (b)
The distributive properties can be used to expand a product into a sum, such as or the other way around, to...

...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 · 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 realnumbers which is
bounded above has a least upper bound.
We will call properties (P1)–(P12), and anything that follows from them, elementary
arithmetic. These...

...
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 Body Surface of an adult (BSA), in engineering it can be used to measure voltage. These formulas are vital and important not only in algebra but also as we can see in our day-to-day lives. This assignment requires that we find the capsizing screening value for the Tartan 4100, solve the formula for variable of d, and find the displacement in which the Tartan 4100 is safe for ocean sailing. The problem is broken down into three parts. The utilization of formulas will be used.
The problem and work will be on the left hand side and a description will be to the right of the work describing the steps taken to solve this assignment. The following words will be bold throughout the assignment to indicate that there is a clear understanding of their definition and use: radical, root, and variable. The assignment requires solving problem 103 on page 605 of our reading material. With the given information we will solve three different parts using the information given and using radical...

...result just obtained (the first term of the eventual quotient). Write the result under the first two terms of the dividend (x2 · (x − 3) = x3 − 3x2).
3. Subtract the product just obtained from the appropriate terms of the original dividend (being careful that subtracting something having a minus sign is equivalent to adding something having a plus sign), and write the result underneath ((x3 − 12x2) − (x3 − 3x2) = −12x2 + 3x2 = −9x2) Then, "bring down" the next term from the dividend.
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).
-------------------------------------------------
Applications
Factoring polynomials
Sometimes one or more roots of a polynomial are known, perhaps having been found using the rational root theorem. If one root r of a polynomial P(x) of degree n is known then polynomial long division can be used to factor P(x) into the form (x - r)(Q(x)) where Q(x) is a polynomial of degree n–1. Q(x) is simply the quotient obtained from the division process; since r is known to be a root of P(x), it is known that the remainder must be zero.
Likewise, if more than one root is known, a linear factor (x – r) in one of them (r) can be divided out to obtain Q(x), and then a linear term in...

...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’ confidence, fluency and versatility are cared for through a strong emphasis on problem solving as an integral part of the learning provision within each topic. Skills in calculation are strengthened through solving a wide range of problems, exploiting links with work on measures and data handling and as much as meaningful application to cross-curricular themes and work in other subjects as possible.
We believe that practical, hands-on experience of using, comparing and calculating with numbers and quantities are of crucial importance in establishing the best mathematical start in the Early Years Foundation Stage and Key Stage 1.
Understanding of place value, fluency in mental methods, and good recall of number facts, such as multiplication tables and number bonds , are essential when applying written calculation methods for addition, subtraction, multiplication and division and thus great emphasis is placed on this throughout the school.
In the...

...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 Finding About How Algebra Affects Personal Life, Business, and Science 10
References ………………………………………………………………………….. 14
Introduction
Algebra, some of us fear it while some of us embrace it, algebra is not “arithmetic with letters” it is better described as a way of thinking. At its most fundamental level, arithmetic and algebra are two different forms of thinking about numerical issues. Many of these examples have been taken from our classroom discussions while others are examples I have discovered in my own research for this paper, several examples of each will be cited.
“Let’s start with arithmetic. This is essentially the use of the four numerical operations addition, subtraction, multiplication, and division to calculate numerical values of various things. It is the oldest part of mathematics, having its origins in Sumeria (primarily today’s Iraq) around 10,000 years ago. Sumerian society reached a stage of sophistication that led to the introduction of money...

...Basically I will be teaching u one aspect of algebra.
Algebra uses letters like a, band etc. algebra is like a puzzle. For example x(2) will be 2x.
There are many forms of algebra such as :
Introduction: The Ideas of Algebra
Lesson 0. Translating Words into Math Symbols
Lesson 1. Simple Operations
Lesson 2. Exponents and Powers
Lesson 3. Order of Operations
Lesson 4. Variables and Expressions
Lesson 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 Property
Lesson 8. Combining Like Terms
Lesson 9. Dividing Rational Numbers
Lesson 10. Formulas
Chapter Review
Chapter Test
Chapter 3: Solving Equations
Lesson 1. Solving Equations with Addition and Subtraction
Lesson 2. Solving Equations with Multiplication and Division
Lesson 3. Solving Multi-Step Equations
Lesson 4. Solving Equations with Variables on Both Sides of the Equation
Lesson 5. Equations with More than One Variable
Lesson 6. Formulas
Chapter Review
Chapter Test
Chapter 4: Proportions and Proportional Reasoning
Lesson 1. Proportions
Lesson...

...Zach Snider
Investigative Task:
SPREAD OF DISEASE – The Task
Disease can spread quickly without use of 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.
Equation: p(0)=10,000/(1+3^3) ~ 474.26
2. Assuming the disease does not present symptoms for 24 hours, how many people will have been infected after 3 hours? Show how you found your answer.
Equation: Substitute 3 hours for the time.
Answer: About 5,000 people would be infected in a mere 3 hours.
3. What is the maximum number of people who can become infected? (Note: e(3-t) will approach 0 for very large values of t).
The maximum number of people who could become infected is 10,000.
4. Explain why your answer for Question #3 is the maximum.
Because when you graph the equation you can see it only reaches 10,000.
5. The stadium needs to warn its guests about a rapid disease spread if it affects over 800 people. After how many minutes should the stadium inform its guests of the disease? Show how you found...