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Designer Decimals

Ivars Peterson

4 November 2006

Calculate 100/89. You get the decimal expansion 1.1235955056 . . .

Look closely, and you'll see that this fraction generates the first five Fibonacci numbers (1, 1, 2, 3, and 5) before blurring into other digits. Recall that, starting with 1 and 1, each successive Fibonacci number is the sum of the two previous Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and so on.

Calculate 10000/9899. This time, you get 1.0102030508132134559046368 . . .

This fraction generates the first 10 Fibonacci numbers (using two digits per number). Going further, the fraction 1000000/998999 generates the first 15 Fibonacci numbers (using three digits per number).

For the full article:

http://sciencenews.org/article.....thtrek.asp

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Questions to explore further this topic:

What are fractions?

http://www.kidsolr.com/math/fractions.html

http://www.bbc.co.uk/skillswis.....roduction/

http://library.thinkquest.org/.....ctions.htm

http://www.321know.com/fra16_x2.htm

Help with fractions?

http://mathforum.org/paths/fra.....hstud.html

http://www.jamesbrennan.org/al.....tions1.htm

http://www.lessontutor.com/sv1.html

Fractions: Basic Idea (Video)

http://www.mathcentre.ac.uk/st.....ources/569

Adding and Subtracting Fractions: Video

http://www.mathcentre.ac.uk/st.....ources/570

Multiplying and Dividing Fractions: Video

http://www.mathcentre.ac.uk/st.....ources/571

Visual fractions

http://www.visualfractions.com/

Shapes and fractions

http://math.rice.edu/~lanius/Patterns/

FlashCards for fractions

http://www.aplusmath.com/Flash.....-mult.html

What are continued fractions?

http://www.mcs.surrey.ac.uk/Pe.....INTRO.html

Mixed numbers

http://www.visualfractions.com/MixedFraction.html

What are equivalent fractions?

http://www.themathpage.com/ARI.....ctions.htm

Fractions in algebra

http://www.wtamu.edu/academic/.....ctions.htm

What are decimals?

http://www.lessontutor.com/lw_decimals.html

http://www.cimt.plymouth.ac.uk.....iscon4.pdf

http://www.mathsrevision.net/gcse/pages.php?page=8

http://www.mathsisfun.com/decimals.html

Adding, subtracting, multiplying and dividing decimals

http://www.mccc.edu/~kelld/decff.htm

What is the meaning of percent?

http://www.mathgoodies.com/les.....rcent.html

Fractions and percents

http://www.mathgoodies.com/les.....cents.html

Lesson on special numbers

http://www.paete.org/forums/viewtopic.php?t=2676

GAMES

http://www.shodor.org/interact.....ionFinder/

http://www.shodor.org/interact.....ctionFour/

http://www.bbc.co.uk/skillswis.....vity.shtml

http://www.mathgoodies.com/les....._vol4.html

http://www.k111.k12.il.us/King/math.htm

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Week of June 30, 2007; Vol. 171, No. 26

A Golden Sales Pitch

Julie J. Rehmeyer

Can mathematics sell blue jeans? One company is hoping so.

The ProportionofBlu is a Los Angeles–based vendor of blue jeans designed to incorporate the celebrated golden ratio. The golden ratio is approximately 1.618:1, and it's defined as the ratio a : b such that a/b = (a + b) / a. Many have claimed that the golden ratio has divine, mystical, or highly aesthetic properties.

The company says that the ratio was used to design details such as the curve of the front pocket, the proportions of the rear pocket, and the ratio of the hip stitching to the inseam of the jeans. "This ratio is found throughout nature and has been recognized as a fundamental component of all things that man has found aesthetically pleasing," says a ProportionofBlu press release.

For the full article:

http://sciencenews.org/article.....thtrek.asp

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University of Chicago Press Journals

12 September 2007

When 2 plus 2 doesn't equal 4: How consumers miscalculate sale prices

Quick: You’re walking by a store window and you see a sign that says, “20% off the original price plus an additional 25% off the already reduced sale price.” So, how much is the discount" Consumers often mistakenly think the total discount is 45% off the original price when, in fact, the true discount is 40%. A thought-provoking new study from the October issue of the Journal of Consumer Research explores why consumers frequently think a double discount is a better deal than a single discount of the same total magnitude.

“Retailers frequently use the strategy of double discounts for their regular promotions or to induce customers to open a credit card account with them. Such errors in peoples’ judgments of the net effect of multiple price discounts . . . have implications for a variety of marketing settings including advertising, promotion, pricing, and public policy,” write Haipeng (Allan) Chen (University of Miami) and Akshay R. Rao (University of Minnesota).

Prior studies have shown that even math teachers frequently have trouble calculating percentages. In the first experiment, the researchers found that 59 percent of the respondents – students at a large university – erroneously added the two percentages to calculate the overall discount. Only 26 percent of students got the answer right.

As not everyone was prone to the miscalculation effect, the researchers then sought to identify the situations that help counter calculation error. They first incentivized one group of participants, offering $2 for correct answers. The rate of computational error was 26 percent, compared to 44 percent for students who were not offered money for correct answers.

Participants were also less prone to computational error when the problems were easier or the results seemingly illogical. Students presented with a base price of $100 from which to calculate the double discount were less likely to make an error than students presented with a base price of $80. Similarly, the researchers asked participants to calculate either a double percentage decrease of 70% and 45% or a price increase of the same percentages. Those who mistakenly added the figures in the decrease condition were confronted with a decrease of 115%, and were more likely to make errors than those confronted with a 115% increase (explained as a change in gasoline prices).

“Since this computational error can potentially influence peoples’ judgment in a variety of settings, the economic impact of such errors on consumer welfare may be substantial,” Chen and Rao write.

The researchers also point to policy implications of computational error outside of consumer settings, such as when a 70% increase followed by a 60% decrease in statewide test scores is considered positive, even though the net effect on test scores is a 32% decrease.

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Haipeng (Allan) Chen and Akshay R. Rao, “When Two and Two is Not Equal to Four: Errors in Processing Multiple Percentage Changes.” Journal of Consumer Research: October 2007.

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