real numbers
real numbers (reals) The numbers that allow a numerical quantity to be assigned to every point on an infinite line or continuum. Real numbers are thus used to measure and calculate
exactly the sizes of any continuous line segments or quantities. The development of a number system that meets these requirements has proved to be a long and complex process that reached a conclusion only in the 19th century. Establishing theoretical foundations for mathematical developments such as the calculus have involved sorting out subtle, conflicting, and inconsistent ideas about the reals (such as infinitesimals). The set of reals is infinite and not countable: there does not exist a method of making finite representations or codings of real numbers. Research on the foundations of the continuum continues – for instance on computation with the reals and on the uses of infinitesimals.
The real numbers, like the
natural numbers, are one of the truly fundamental data types. Unlike the natural numbers, however, reals cannot be represented exactly in computations. They can be approximated to any degree of accuracy by
rational numbers.
A real number can be defined in several ways, for example as the limit of a sequence of rational numbers. A real
x is represented by a sequence
q(0),
q(1),… of rational numbers that approximates
x in the sense that for any degree of accuracy ϵ there exists some natural number
n such that for all
k >
n, |
q(
k) –
x| < ϵA real number is a
computable real number if there is an algorithm that allows us to compute an approximation to the number to any given degree of accuracy. A real
x is computable if (a) there is an algorithm that lists a sequence
q(0),
q(1),… of rational numbers that converges to
x, and (b) there is an algorithm that to any natural number
k finds a natural number
p(
k) such thatfor all
n >
p(
k), |
q(
n) –
x| < 2
–kMost of the real numbers that we know and use come from solving equations (e.g. the algebraic numbers) and evaluating equationally defined sequences (e.g. e and π) and are computable. However, most real numbers are noncomputable.
The approximations to real numbers used in computers must have finite representations or codings. In particular, there are gaps and separations between adjacent pairs of the real numbers that are represented (see
model numbers). The separation may be the same between all numbers (
fixed-point) or may vary and depend on the size of the adjacent values (
floating-point). Some programming languages ignore this difference, describing floating-point numbers as “real”. Calculations with real numbers on a computer must take account of these approximations.
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Giant recalls Shady Brook Farm turkeys. (Giant Food Inc.)
PR Newswire; 11/22/1988; 597 words
; GIANT RECALLS SHADY BROOK FARM TURKEYS WASHINGTON, Nov. 22 /PRNewswire/ -- Shady Brook Farms has advised Giant that some of the Shady Brook Fresh Whole Turkeys supplied to Giant may have been shipped to Giant under improper temperature controls...
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SPRING BROOK FARM OF LANGLADE COUNTY ORDERED TO PAY $27,500 FOR MANURE DISCHARGES
News Wire article from: US Fed News Service, Including US State News; 12/10/2007; 552 words
; ...Department of Justice, Spring Brook Farm LLC of the Town of Polar, Langlade...2002 and 2005. Spring Brook Farm operates a concentrated animal...manure annually. Spring Brook Farm's operation is regulated by...Spring Brook Creek. Spring Brook Farms has since ...
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SHADY BROOK FARMS SETS HOLIDAY SALES RECORD; FLOCKS NOT AFFECTED BY 'SPIKING MORTALITY
PR Newswire; 11/9/1995; 700+ words
; ...year's holiday sales, Shady Brook Farms, the country's leading...mortality has struck many turkey farms, raising the possibility of...for the holidays. Shady Brook Farms' flocks have not been affected...director of sales for Shady Brook Farms, "Spiking mortality has ...
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Shady Brook Farms spoilage problem confined to Baltimore-Washington metro region.
PR Newswire; 11/23/1988; 523 words
; SHADY BROOK FARMS SPOILAGE PROBLEM CONFINED TO BALTIMORE...Va., Nov. 23 /PRNewswire/ -- Shady Brook Farms issued the following statement...the Baltimore/ Washington area. Shady Brook Farms assures its customers and consumers...
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D.C. Area Giant Stores Recall 10,000 Turkeys; Shady Brook Farms Birds May Have Spoiled
Newspaper article from: The Washington Post; 11/23/1988; ; 619 words
; ...supermarkets recalled all 10,000 Shady Brook Farms Fresh Whole turkeys sold at Giant...the bad birds are. Giant and Shady Brook Farms have asked customers to return all turkeys labeled Shady Brook Fresh Whole turkeys to the store...
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Nation's Top Culinary Talent 'Talk Turkey' for the Holidays; Shady Brook Farms(R) Holiday Hotline Offers Expert Tips from the Country's Finest Chefs.
PR Newswire; 11/12/1998; 700+ words
; ...Nov. 12 /PRNewswire/ -- Shady Brook Farms is putting consumers in touch with...director of marketing for Shady Brook Farms. "Consumers look to professional...tradition of charitable giving, Shady Brook Farms has partnered with Second Harvest...
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Shady Brook Farms responds to New York Department of Agriculture actions.
PR Newswire; 9/26/1989; 504 words
; SHADY BROOK FARMS RESPONDS TO NEW YORK DEPARTMENT OF AGRICULTURE...We have never made any claims that Shady Brook Farms turkeys are kosher, or intended...been taken," Chenoweth stated. Shady Brook Farms has run a similar ad for the past...
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Celebrity Chefs Cook Up Fun for the Entire Family This Holiday Season; Shady Brook Farms Hotline Offers Satisfying Meal Planning Advice.
PR Newswire; 11/1/1999; 700+ words
; ...1 /PRNewswire/ -- This Holiday season Shady Brook Farms is offering culinary advice on how the whole family...says Jim Hoagland, director of marketing for Shady Brook Farms. Shady Brook Farms continues its partnership with Share Our Strength...
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Honeysuckle White(R) and Shady Brook Farms(R) Brands Carve Out Top Five Reasons to Eat Turkey Today.
PR Newswire; 7/10/2008; 700+ words
; ...The Honeysuckle White and Shady Brook Farms brands are reminding people that...the Honeysuckle White and Shady Brook Farms turkey brands have crafted...Honeysuckle White(R) or Shady Brook Farms(R) turkey breast cutlets...
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New Poultry Labeling Rules to Put Holiday Turkeys to The Fresh Test; Shady Brook Farms(R) to Lead Industry in Fresh Supply for Thanksgiving.
PR Newswire; 10/20/1998; 700+ words
; ...of fresh turkey products, Shady Brook Farms is already compliant with the revised...director of marketing for Shady Brook Farms. "Our fresh turkeys already...careful product monitoring. Shady Brook Farms Harrisonburg (VA) location...
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Brook Farm
Dictionary entry from: Dictionary of American History
BROOK FARM BROOK FARM. Founded in 1841 on 183 acres of land purchased from Charles Ellis in West Roxbury, Massachusetts, the Brook Farm Institute of Agriculture and Education was a utopian community organized...
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Dairy Mart Convenience Stores, Inc.
Book article from: International Directory of Company Histories
...began an aggressive plan of acquisition. In 1981 the company purchased Sunny-brook Farms and its 66 stores, followed in 1983 by the acquisition of Dutchland Farms ’ s 32 stores, and IPO with 143 stores. Still far from the goal, Dairy...
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Malil, Shelley 1964–
Book article from: Contemporary Theatre, Film and Television
...Hilton, 1998, United Airlines, 1998, Saturn, 1999, Long's Drugs, 2000, Budweiser, 2000 – 01, Shady Brook Farms, 2001, Dish Network, 2001, Microsoft, 2001, Mercedes Benz, 2002 and 2004, Intel, 2003, SBC, 2003, Boeing...
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Utopian Communities
Dictionary entry from: Dictionary of American History
...important of these communities was Brook Farm, established in West Roxbury, Massachusetts...American society, the creators of Brook Farm, chiefamong them George Ripley...most important components of life at Brook Farm. Emerson, Nathaniel Hawthorne...
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Dana, Charles A. (1819-1897)
Book article from: American Eras
...Dana (1819-1897) Newspaper editor Brook Farm. Born in Hinsdale, New Hampshire...September 1841 he joined the group at Brook Farm in West Roxbury, Massachusetts...intellectual-growth. Other residents at Brook Farm included the novelists Nathaniel...
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