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How Big Is A Billion?
(Dictionary.com's definition - Merriam-Webster's definition)

If I gave you \$1,000 a day, seven days a week, how long would it take you to collect 1 billion dollars?  (Assume that you spend none of the money, and collect no interest on it.)

• answer: 2,737.85 years (2,737 years, 10 months, 7 days)
• This means that if the payments had begun on the first day of the common era (January 1, year 1 on the Gregorian calendar), it would be November 7, 2738 before the \$1 billion would accumulate.
• The calculation
1. 1,000,000,000 / 1,000 gives us the number of days.  In this case, 1 million.
2. Dividing 1 million by 365.25 (roughly the number of days per calendar year) gives us the number of years.

[A note on precision: Correctly using the formula above results in the "answers" on this page. However the number of days in a year is not precisely 365.25. While the length of the mean solar year is a matter of debate among astronomers, I gather that many now place it at 365.24218967 days (see http://www.hermetic.ch/cal_stud/cal_art.html#Variation for more information).  I have chosen to use 365.25 as it is an easier number with which to work. Also, the length of the year is constantly changing ever so slightly. Over time those changes add up significantly. Therefore, using 365.25 as a constant leads to answers that appear precise but are not. For example, using the Gregorian Calendar, I'm told that adding 1 million days to January 1, 0001 results in November 27, 2738 rather than the November 7, 2738 that the formula above generates. This is another example of the importance of teaching Process over Answers.]

• The last time I conducted this exercise, the students with whom I worked (a mixed age group of middle schoolers) arrived at a consensus answer of 4 years.  I at first thought I'd heard 40 years, but was corrected by the person who originally offered 4 years.  This result was not appreciably different from that arrived at by other groups of secondary students with whom I've worked.  The reason, I believe, is what mathematician and author John Allen Paulos has called "Innumeracy".  He begins his book (Innumeracy: Mathematical Illiteracy and its Consequences, Hill and Wang 1988) by defining it as "an inability to deal comfortably with the fundamental notions of number and chance".  The fact that the middle and high school students in my experience are so far off in their attempts to estimate answers to this sort of problem, proves to me that our school systems are failing to instill this fundamental grasp of number.
• By the way, to show the magnitude of the difference between one billion and one million, have your students figure out how long it would take to accumulate one million dollars following the same rules.

the answer: 2.73785 years (2 years, 8 months, 26 days)

To complicate things a bit (but once again show the importance of stressing processes rather than answers), the answer to the starting question shown above is correct as long as one is in the U.S. where one billion is defined as a thousand million.  In Britain, however, one billion is defined as 1,000,000 million (one million million, or what we in the U.S. call a trillion).  So in Britain, the analysis would look like this:

• answer: 2,737,850.787 years (2 million 737 thousand 850 years, 9 months, 13 days)
• This means that if the payments had begun on the first day of the common era (January 1, year 1 on the Gregorian calendar), it would be September 13, 2737850 before the \$1 billion would accumulate.  To put that date in some perspective, you might want to modify the "calendar" that illustrates events in human evolution found in the Classroomtools.com activity Putting Time in Perspective.  When I do so, I find the following:
 Scale 7,235,848 years = 365 days Event # of years ago Appears the Ape-Human split 7,235,848 January 1 12:00:00 AM Australopithecus Afarensis 6,235,848 February 20 10:38:20 AM Homo Ergaster 4,435,848 May 22 05:47:21 AM Homo Neanderthalis 2,935,848 August 5 09:44:52 PM Homo Sapiens 2,855,848 August 9 10:35:56 PM Writing is invented 2,740,848 August 15 05:49:20 PM Common Era begins 2,737,850 August 15 09:27:06 PM Today, May 18, 2002 2,735,848 August 15 11:52:32 PM 1 British billion accumulates 0 December 31 11:59:59 PM

Notice that on this calendar, almost all that we know of human history (commencing with the invention of writing approximately 5,000 years ago) occurs in just over six hours on August 15.  The nearly two and three quarter million years between today on the calendar and the final accumulation of one billion (British) dollars is nearly equivalent to the time span between the disappearance of Australophithecus Afarensis (the species represented by the "Lucy" skeleton discovered by Donald Johanson in the mid-70s) and today.  What might a species descended from us, given that amount of evolutionary time to develop, be like?  Of one thing we can be fairly certain - the dollar will no longer exist.
• The calculation
1. 1,000,000,000,000 / 1,000 gives us the number of days.  In this case, 1,000 million.
2. Dividing 1,000 million by 365.25 (roughly the number of days per calendar year) gives us the number of years.

If you like thinking about large numbers, don't miss Jim Holt's essay, Larger Than Life: Can numbers become too big?. (Unfortunately, Lingua Franca, the magazine that published Holt's essay, suspended publication at the end of 2001. His essay, and many other fine articles, disappeared from the web when they took down their site.  Thankfully, the Internet Archive saved it, as it has so much else.  Also, I assume that Holt's essay was published on paper, so perhaps you can find a copy in a library that subscribed to Lingua Franca.  I believe it appeared in the February 2001 edition (Volume 11, No. 1).)    You also might be interested in the notation for some very large numbers.  If so, have a look at this.

Books that help put the very large and very small into perspective

Web sites that help put the very large into a manageable perspective

Related Classroomtools.com activities