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Whether describing the vastness of the stars or the microscopic intricacies of the humanbody, the need to use large numbers is often inevitable.When we consider the estimated 200,000,000,000 (200billion) stars in the Milky Way Galaxy or the estimated150,000,000,000 (150 billion) galaxies in the universeor the estimated 100,000,000,000,000 (100 trillion)cells in the adult human body, we are forced to usenumbers so large we cannot comprehend their meaning.Let’s try to use our imaginations to placesome of these large numbers in perspective. (In sodoing, we will use the American nomenclature forlarge numbers described below.)
We will use money as a teaching toolto illustrate the magnitude of large numbers. Whetheryou stack it, spread it, line it up, or spend it,we hope you’ll think of money and large numbersin a new way after you read this section.
Money Stacks
One way to better understand largenumbers is to compare the heights of stacks of varyingnumbers of dollar bills. The thickness of a singleone dollar bills measures .0043 inches or .0000000679miles.
The height of a stack of 100 one dollarbills measures .43 inches.
The height of a stack of 1,000 onedollar bills measures 4.3 inches.
The height of a stack of 1,000,000one dollar bills measures 4,300 inches or 358 feet – aboutthe height of a 30 to 35 story building.
The height of a stack of 100,000,000(one hundred million) one dollar bills measures 35,851feet or 6.79 miles. This would reach from the earth’ssurface to the approximate altitude at which commercialjetliners fly.
The height of a stack of 1,000,000,000(one billion) one dollar bills measures 358,510 feetor 67.9 miles. This would reach
fromthe earth’s surface into the lower portionof the troposphere – one of the major outerlayers of earth’s atmosphere.
The height of a stack of 100,000,000,000(one hundred billion) one dollar bills measures 6,786.6miles. A column of bills this high would extend 28times higher than the orbiting International SpaceStation.
The height of a stack of 1,000,000,000,000 (one trillion)one dollar bills measures 67,866 miles. This wouldreach more than one fourth the way from the earthto the moon.
The height of a stack of 100,000,000,000,000(one hundred trillion) one dollar bills measures6,786,616 miles. This would reach from the earthto the moon and back 14 times.
Money Carpets
Another way to appreciate large numbersis to consider the area covered by varying numbersof dollar bills. The area covered by a single onedollar bill measures 16.0254 square inches.
Thearea covered by 100 one dollar bills measures 11.13square feet.
The area covered by 1,000 one dollarbills measures 111.29 square feet.
The area covered by 1,000,000 one dollarbills measures 111,287.5 square feet. This wouldcover an area approximately equal to the size oftwo football fields.
The area covered by 100,000,000 (onehundred million) one dollar bills measures .4 squaremiles. This would cover an area equal to the sizeof 255.5 acres.
The area covered by 1,000,000,000 (onebillion) one dollar bills measures 4 square miles.This would cover an area equal to the size of 2,555acres.
The area covered by 100,000,000,000(one hundred billion) one dollar bills measures 400square miles. This would cover an area equal to one-third the size of the state of Rhode Island.
The area covered by 1,000,000,000,000(one trillion) one dollar bills measures 3,992 squaremiles. This would cover an area equal to twice thesize of the state of Delaware.
The area covered by 100,000,000,000,000(one hundred trillion) one dollar bills measures399,189 square miles. This would cover an area roughlyequal to the size of Texas and California combined.
Money Lines
Yet another way to understand largenumbers is to compare the lengths of varying numbersof dollar bills laid end-to-end. The length of asingle one dollar bills measures 6.14 inches.
The length of 100 one dollar billslaid end-to-end measures 614 inches or 51.17 feet.
The length of 1,000 one dollar billslaid end-to-end measures 512 feet. This would approximatethe length of some of the longer home run balls hitby professional baseball players. (The world recordfor the longest home run, estimated at 643 feet, wasset in 1961 by Mickey Mantle.)
The length of 1,000,000 (one million)one dollar bills laid end-to-end extends 96.9 miles.This would easily traverse the 21 miles across theEnglish Channel from the shores of England to theshores of France and back more than two and a halftimes.
The length of 100,000,000 (one hundredmillion) one dollar bills laid end-to-end measures9,690 miles. This would reach
fromWashington D.C., USA to Sydney, Australia.
The length of 1,000,000,000 (one billion)one dollar bills laid end-to-end measures 96,900miles. This would extend around the earth almost4 times.
The length of 100,000,000,000 (onehundred billion) one dollar bills laid end-to-endmeasures 9,690,656 miles. This would extend aroundthe earth 387 times.
The length of 1,000,000,000,000 (onetrillion) one dollar bills laid end-to-end measures96,906,656 miles. This would exceed the distancefrom the earth to the sun.
The length of 100,000,000,000,000 (onehundred trillion) one dollar bills laid end-to-endmeasures 9,690,656,566 miles. This would reach fromthe earth to the planet Saturn and back six times.(Assuming Saturn at its nearest point to earth)
Money Spent - Shop ‘til You Drop
Finally,let’s try to help those who love malls to understandlarge numbers by imagining the shopping spree ofa lifetime starting with varying amounts of money.Let’s say a shopper must spend $20 per secondand must do so 24 hours per day until penniless.
(As you view the table below, this exercise willquickly give the expression “shop ‘tilyou drop” a whole new meaning.)
For example, a shopper beginning with$100 could shop for a mere five seconds. A shopperbeginning with $1,000 could shop for 50 seconds.
Duration of Shopping Sprees
While Spending $20 per Second
| Starting Amount (Dollars) | Duration of Shopping Spree |
|---|---|
| 1 | 1 second |
| 100 | 5 seconds |
| 1,000 | 50 seconds |
| 10,000 | 8 minutes 20 seconds |
| 100,000 | 1 hour 23 minutes |
| 1,000,000 | 13 hours 48 minutes |
| 10,000,000 | 138 hours (5 days 19 hours) |
| 100,000,000 | 57 days 21 hours |
| 1,000,000,000 | 578 days (1 year 214 days) |
| 1,000,000,000,000 | 1,585 years |
| 100,000,000,000,000 | 1,585 centuries |
Definitions of Mathematical TermsVary Around the Globe
Scientists, mathematicians, and othershave long used words to describe large numbers inlieu of writing out endless digits or resorting tothe use of scientific notation. Unfortunately, theseterms are defined differently in various parts ofthe world. For instance, the term “billion” inAmerica is defined as 1,000,000,000 or 109.The British use the same term to describe 1,000,000,000,000or 1012 (million squared). Fortunately,there is agreement regarding terms used to describenumbers up to and including 100,000,000 (108)across both sides of the Atlantic. However, numbersfrom 1,000,000,000 (109) and higher carrydifferent labels which can foster confusion. Eitherwriting out the number in full or using scientificnotation is unambiguous and recommended for internationaluse.
Below is a summary of terms and definitionsfor reference.
Comparing Definitions of MathematicalTerms Across the Atlantic Ocean
| Number | Scientific Notation | American | British |
|---|---|---|---|
| 1 | 100 | One | One |
| 10 | 101 | Ten | Ten |
| 100 | 102 | One hundred | One hundred |
| 1,000 | 103 | One thousand | One thousand |
| 10,000 | 104 | Ten thousand | Ten thousand |
| 100,000 | 105 | One hundred thousand | One hundred thousand |
| 1,000,000 | 106 | One million | One million |
| 10,000,000 | 107 | Ten million | Ten million |
| 100,000,000 | 108 | One hundred million | One hundred million |
| 1,000,000,000 | 109 | One billion | One thousand million |
| 10,000,000,000 | 1010 | Ten billion | Ten thousand million |
| 100,000,000,000 | 1011 | One hundred billion | One hundred thousand million |
| 1,000,000,000,000 | 1012 | One trillion | One billion (million squared) |
| 10,000,000,000,000 | 1013 | Ten trillion | Ten billion |
| 100,000,000,000,000 | 1014 | One hundred trillion | One hundred billion |
| 1,000,000,000,000,000 | 1015 | One quadrillion | One thousand billion |
| 10,000,000,000,000,000 | 1016 | Ten quadrillion | Ten thousand billion |
| 100,000,000,000,000,000 | 1017 | One hundred quadrillion | One hundred thousand billion |
| 1,000,000,000,000,000,000 | 1018 | One quintillion | One trillion (million cubed) |
Supporting Facts and Sources
1. U.S. paper currency such as a $1bill measures 2.61 inches wide by 6.14 inches longwith a thickness of .0043 inches.
Source: U.S. Bureau of Engraving and Imprinting
http://www.moneyfactory.com/document.cfm/18/106
2. The International Space Stationorbits at an altitude of 242 miles.
Source: NASA
http://spaceflight.nasa.gov/realdata/sightings/index.html
3. The area of various states (in squaremiles):
California - 155,973Texas - 261,914
Rhode Island - 1,045Delaware - 1,955
Source: Graphic Maps
http://www.worldatlas.com/aatlas/infopage/usabysiz.htm
4. The distance from the earth to themoon is approximately 238,000 miles or 384,467 kilometers.
Source: NASA Goddard Space Flight Center
http://www.the-solar-system.net/earth/moon-fact-sheet.html
5. Mickey Mantle set the record forlongest home run of 643 feet in 1961.
Source: www.themick.com
http://www.themick.com/anniv06.htm
6. The distance from Sydney Australiato Washington D.C. is approximately 9,760 miles.
Source: Indo.com
http://www.indo.com/cgi-bin/dist/place2=washington+dc/place1=@174129
7. The diameter of the earth is 7,926miles (12,756 kilometers) from which the circumferenceof approximately 25,000 miles may be derived.
Source: NASA Goddard Space Flight Center
http://www.the-solar-system.net/earth/earth-fact-sheet.html
8. The distance from the earth to thesun is approximately 93,000,000 miles (93 millionmiles).
Source: NASA Goddard Space Flight Center
http://image.gsfc.nasa.gov/poetry/ask/a11805.html
9. The minimum distance from Earthto the planet Saturn is 1,195.5 million kilometersor 742,405,500 miles.
Source: National Space Science Data Center - NASA
http://nssdc.gsfc.nasa.gov/planetary/factsheet/saturnfact.html
Given the topic at hand, let me dive into my expertise on large numbers and their applications across various fields. Understanding vast numerical concepts is crucial in several scientific and practical domains. The scale of the universe, the intricacies of the human body, or even financial computations often involve numbers beyond easy comprehension.
The provided article dissects the vastness of numbers through three different dimensions: height, area, and length using stacks of dollar bills. It vividly illustrates the magnitude of numbers ranging from thousands to trillions by comparing the heights of money stacks, the areas covered by dollar bills, and the lengths of bills laid end-to-end. This comparison helps grasp the immensity of figures that are typically abstract and hard to fathom.
Let's break down the concepts involved:
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Height of Money Stacks: This section measures the height of stacks of dollar bills, showcasing how the height grows exponentially with each increase in quantity, reaching impressive heights like 67.9 miles for one billion bills and 6,786.6 miles for one hundred trillion bills.
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Area Covered by Money Carpets: It explores the area covered by different quantities of dollar bills, comparing them to sizes such as football fields, acres, or even entire states like Rhode Island and Delaware, providing a tangible visualization of large numbers.
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Length of Money Lines: This part focuses on the length of dollar bills when laid end-to-end, emphasizing lengths ranging from 51.17 feet for 100 bills to an astonishing 96,906,656 miles for one trillion bills.
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Money Spent - Shop ‘til You Drop: This section demonstrates the duration of a shopping spree if one were to spend $20 per second, illustrating how larger amounts of money extend shopping durations from seconds to centuries.
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Definitions of Mathematical Terms: The article concludes by discussing the discrepancy in defining numerical terms like "billion" across different regions (American and British), highlighting the need for clarity in expressing large numbers universally.
In terms of supporting facts and sources, the article references various reliable sources and calculations, such as measurements of U.S. paper currency, distances between locations on Earth, the dimensions of celestial bodies, and records in sports like Mickey Mantle's longest home run.
Understanding these concepts of scale helps in comprehending the enormity of numbers that often come up in discussions related to finance, astronomy, biology, and more.
