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 real numbers. They include such number as , , , , , , , and . The basic algebraic properties of the real numbers can be expressed in terms of the two fundamental operations of addition and multiplication.

Basic Algebraic Properties:
Let and denotes real numbers.

(1) The Commutative Properties
(a) (b)
The commutative properties says that the order in which we either add or multiplication real number doesn’t matter.

(2) The Associative Properties
(a) (b)
The associative properties tells us that the way real numbers 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 rewrite a sum as product:

(4) The Identity Properties
(a) (b)
We call the additive identity and the multiplicative identity for the real numbers.
(5) The Inverse Properties
(a) For each real number , there is real number , called the additive inverse of , such that (b) For each real number , there is a real number , called the multiplicative inverse of , such that Although the additive inverse of , namely , is usually called the negative of , you must be careful because isn’t necessarily a negative number. For instance, if...

...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 algebraicnumber) 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...

...RealNumberProperties
In this assignment we were asked to solve three expressions using the properties of realnumbers in order to do so. Each of the realnumberproperties are essential in solving algebraic expressions. Although you may not need to use all of them in the same expression to solve you will need to use at least one. In this paper I will demonstrate the use of the properties and show the steps needed to solve each part of an expression.
Understanding the properties of algebra is important because you must be able to use these properties in order to correctly put expressions and equations into their simplest form by moving terms around to solve or simplify the expression. Distribution is used to multiply across terms inside parentheses, allowing you to remove the parentheses for the problem. Moving terms to different locations is known as the commutative property. Combining like terms is referred to as the associative property.
2a(a-5)+4(a-5)
2a(a-5) + 4(a-5) This is the given expression, for problem number one.
2a^2-10a+4a-20 Using the distributive property I removed the parentheses.
2a^2-6a-20 I added the coefficients here in order to add the like terms.
2a^2-6a-20 This is the simplest form of this...

...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 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
-------------------------------------------------
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,...

...-------------------------------------------------
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...

...Equitable Conversion – once a contract is signed, equity regards the buyer as the owner of the property. The seller’s interest is looked at as personal property. The legal title of the property remains with the seller and is considered to in trust and the risk is on the seller. The right of possession follows the legal title; the seller is entitled to possession until closing.
Risk of Loss – there is a split of authority on risk of loss when a contract is signed, equity is passed to buyer through escrow and the risk of loss is on buyer. If property is destroyed before closing, the majority rule places the risk on the buyer. If the property is damaged or destroyed, the seller is to credit any monies from the insurance against the purchase price the buyer is required to pay.
Because Birdwell did not rescind the contract he will be required to pay the $90,000 because he did not consult an attorney and because the real estate agent put a new price on the property of $50,000. However, since the contract was silent at risk, the Uniform Vender and Purchaser Risk Act, Birdwell could request this option. However, neither party had insurance on the property.
Here, no one had insurance on the property. If property is destroyed and the seller has insurance, the seller will be required to reduce the sale price by the amount of...

...This week I’m asked to solve the following word problem in relation to a real world radical formula. Problem 103 on pages 605-606 states:
To be considered safe for ocean sailing, the capsize screening value C should be less than 2 (www.sailing.com). For a boat with a beam (or width) b in feet and displacement d in pounds, C is determined by the function: C=〖4d〗^(-1/3) b.
Find the capsize screening value for the Tartan 4100, which has a displacement of 23,245 pounds and a beam of 13.5 feet.
Solve this formula for d?
The accompanying graph shows C in terms of d for the Tartan 4100 (b=13.5). For what displacement is the Tartan 4100 safe for ocean sailing? (Dugopolski, 2012).
a) The first part of the problem requires that I substitute the variables with their given values. I need to find the value of C, which represents the capsize screening value. To do so, I need to replace d, the displacement value in pounds, with 23,245; and, also replace b, the beam’s width in feet, with 13.5. I do not need to convert the inches to feet using a decimal value because that was already done. By following the order of operations I first need to solve for the exponent before multiplying across. The radical exponent of -1/3 means that I have to apply the reciprocal of the cubed root of d and use that value within my multiplication.
C=4d^(-1/3) b Capsize formula
C=4(23245)^(-1/3) (13.5) Replace variables with given values
C=4(1/〖23245〗^(1/3) )(13.5) Convert...

...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...

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