HS PUMP. Spring 2010 CSUN Math. Big Numbers

Transcription

1 1. You have fiftee secods. Usig stadard math otatio, Eglish words, or both, write the biggest umber you ca thik of i the box below. Be precise eough for ay perso with adecuate math skills to determie exactly what umber you have writte. 2. We use expoetial otatio for large umbers. Thus there are about 300, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000 atoms i the observable uiverse, which we abbreviate by Neither the Egyptias or the Romas could have writte such umber. Today we have ames for may large umbers: millio for 10 6, billio (a thousad millio) for 10 9, vigitillio for We also have googol for The term was coied i 1938 by 9-year-old Milto Sirotta, ephew of Edward Kaser. Kaser the exteded the term to the larger googolplex, which is February 23,

2 3. Archimedes was perhaps the first to devise a system to really write big umbers. I his treatise Sad Reckoer he write: There are some, kig Gelo, who thik that the umber of the sad is ifiite i multitude; ad I mea by the sad ot oly that which exists about Syracuse ad the rest of Sicily but also that which is foud i every regio whether ihabited or uihabited. Agai there are some who, without regardig it as ifiite, yet thik that o umber has bee amed which is great eough to exceed its magitude. Ad it is clear that they who hold this view, if they imagied a mass made up of sad i other respects as large as the mass of the Earth, icludig i it all the seas ad the hollows of the Earth filled up to a height equal to that of the highest of the moutais, would be may times further still from recogizig that ay umber could be expressed which exceeded the multitude of the sad so take. But I will try to show you by meas of geometrical proofs, which you will be able to follow, that, of the umbers amed by me ad give i the work which I set to Zeuxippus, some exceed ot oly the umber of the mass of sad equal i magitude to the Earth filled up i the way described, but also that of the mass equal i magitude to the uiverse. The largest umber that existed i aciet Greek arithmetic was a myriad or te thousad. Archimedes itroduced a ew umber myriad myriad, a hudred millio, which he called octade or uit of the secod class. Octades octades (or te millio billios) is called a uit of the third class, ad so o. To estimate the umbers of grais of sad i the visible Uiverse of that time, Archimedes had to estimate the size of the uiverse. This he took as a big sphere. The he compared the size of that sphere with the size of a grai of sad ad arrived at the followig coclusio: It is evidet that the umber of grais of sad that could be cotaied i a space as large as that bouded the the stellar sphere as estimated by Aristrachus, is ot greater that oe thousad myriads of uits of the eight class. (a) What is this umber i moder scietific otatio? February 23,

3 4. As the leged goes, Kig Shirham of Idia wated to reward his grad vizier Sissa Be Dahir for ivetig ad presetig to him the game of chess. The grad vizier seemed modest i his desires: Majesty, he said to the kig, give a grai of rice to put o the first square of this chessboard, ad two grais to put o the secod square, ad four grais to put o the third, ad eight grais to put o the fourth, ad so, doublig the umber for each succeedig square, give eough to cover all sisty four square o the board. (a) How may grais of rice would have bee required to fullfil the grad vizier request? (b) How much is that i weight? February 23,

4 5. The largest umber that you ca write with oe digit is 9. The largest umber that you ca write with two digits (ad stadard math symbols) is ot 99, but 9 9. What is the largest umber that you ca write with three digits? 6. What is the largest umber that you ca write with three digits 2? You have several possibilities: 222, 2 22, 22 2, Which is largest? 7. What is the largest umber that you ca write with three digits 3? 8. What is the largest umber that you ca write with three digits 4? February 23,

5 9. The geral problem is: if a is oe of the digits 1, 2, 3, 4, 5, 6, 7, 8, 9, what is the largest umber that you ca write with three digits a? 10. What is the largest umber that you ca write with four digits 1? February 23,

6 11. What is the largest umber that you ca write with four digits 2? There are eight possible combiatios of four 2: 2222, 222 2, 22 22, 2 222, 22 22, 2 222, 2 222, Clearly, 2222 is the smallest. To fid the largest, do the followig: (a) Compare ad (b) Compare ad (c) The largest of 2222, 222 2, 22 22, ad is (d) Compare this umber to each of the four umbers 22 22, 2 222, 2 222, February 23,

7 { }} { 12. We use m or m to abbreviate m + m + + m. We also use m to abridge m or { }} { m m... m. Calculators prit mˆ for raisig m to the power. Old computer pritouts used m to the same effect. This suggest the followig arrow otatio, itroduced by Doald Kuth i 1972: 13. Compute the followig (a) 2 2 m abbreviates } mm {{ m} m abbreviates m m m } {{ } m abbreviates m m m } {{ } m abbreviates m m m } {{ } (b) 3 2 (c) m 2 February 23,

8 14. The Ackerma sequece is the sequece of umbers A(1) = 1 1, A(2) = 2 2, A(3) = 3 3, A(4) = 4 4,... Compute: (a) A(1) (b) A(2) (c) A(3) (d) A(4) Literature [1] Joh Coway ad Richard Guy, The Book of Numbers, Spriger-Verlag, New York, [2] Edward Kaser ad James Newma, Mathematics ad the Imagiatio, Pegui Books, [3] George Gamov, Oe, Two, Three... Ifiity, Dover Publicatios, February 23,