Big numbers AKA Grains of rice on a chess board
Some people think that when the number of digits exceeds a calculator's capacity, we have to resort to paper and pencil computation. It is not so. We show below how easy it is to use a scientific calculator (the TI-34 II; you may also use a TI-83/84) to deal with some problems involving big numbers.
There is an old story about an ignorant king who promised to pay the inventor of chess for his invention. The inventor asked for one grain for the first square on the chess board, two for the second, four for the third, eight for the fourth, and so on. How many grains did he demand?
The chess board has 64 squares. So he asked for
1 + 2 + 22 + 23 + ... + 263 grains of rice.
But 1 + 2 + 22 + 23 + ... + 263 = 264 -1, because
let s = 1 + 2 + 22 + 23
+ ... + 263
2s = 2 + 22 + 23 + ... + 263 + 264
2s - s = 264 - 1
The task is to write down this number, 264 - 1, as a decimal without rounding.
Solution.
264 - 1 = (232)2 - 1
Compute,
2^32→A 4294967296.
(Thus, 264 - 1 = 429492*1E10 + 2*42949*67296*1E5 + 672962 - 1, because
(x + y)2 = x2 + 2*x*y + y2.)
Store the two parts of the previous result in A and B. Compute and write down the answers, appending 10 and 5 zeros to the first two numbers. The answers should be aligned for addition. Put a comma after every fifth digit to avoid errors.
fPart(A,1E5)→B 67296.
iPart(A,1E5)→A 42949.
A^2
2AB
B^2-1
(Notice that we do not reenter any values. Typing in multi-digit numbers is prone to errors and should be avoided whenever possible.)
Add by hand,
18446,16601,00000,00000
57805,91808,00000
45287,51615
18446,74407,37095,51615
How to read it?
18 quintillion 446 quadrillion 744 trillion 73 billion 709 million 551 thousand 615.
An additional task
You may ask students to count the number of grains of rice in one cubic centimeter, and then to estimate the radius of a sphere that could encompass a volume of so many grains. (My estimate was a radius of approximately 8 kilometers.)