In our last column we dwelt on the fact that individual numbers have their quirks, and looked closely at just two numbers: 2 and 3. We continue this in today’s column.
The powers of 2, obtained by multiplying 2 by itself a certain number of times, are 1, 2, 4, 8, 16, 32, 64, … and the powers of 3 are 1, 3, 9, 27, 81, 243, 729, … (1 is part of both lists; that is because the zeroth power of any positive number is taken to be equal to 1).
Here are some striking features of the two power sequences. Every natural number can be written as a sum of unequal powers of 2. For example, 3 = 2 + 1, 5 = 4 + 1 and 50 = 32 + 16 + 2. Moreover, there is just one way of writing each such expression. This is the basis of the binary system. Using this one can find the answers to many puzzles involving weights and a pan balance. For example, using a set of weights measuring 1, 2, 4, 8, 16, 32 and 64 kg, every integral weight from 1 kg till 127 kg can be measured.
If we try this using the powers of 3 we find soon enough that it cannot be done; we cannot write 2 as a sum of unequal powers of 3; nor 5 or 6. Strangely, not even one number between 1100 and 2100 can be written this way; the largest number before 2012 that can be so written is 1093, and the smallest number after 2012 that can be so written is 2187 (numerologists can check whether some strange events happened in the year 1093, and can make a few choice predictions for 2187!).Hence, using 1, 3, 9, 27 and 81 as weight measures seems a poor choice. But if we allow the placing of weights in both the pans, then every integral weight becomes measurable. We can write 2 = 3-1 (place 3 kg in one pan and 1 kg in the other one), 5 = 9-3-1, 6 = 9-3, and so on. Every natural number can be written as a sum and difference of unequal powers of 3.
These features are special: if we require that every natural number should be expressible as a sum of unequal powers of some fixed k, then k = 2 is the only possibility; and if the requirement is that every natural number should be expressible as a sum and difference of unequal powers of some fixed k, then k = 3 is the only possibility! The odd thing is that both these expressions are unique. For example, the only way of writing 10 as a sum of unequal powers of 2 is 10 = 8 + 2, and the only way of writing 10 as a sum and difference of unequal powers of 3 is 10 = 9-1; try it!
Shailesh Shirali and Sneha Titus are the editors of At Right Angles.
At Right Angles will run a fortnightly math column in this space. At Right Angles is a resource for school mathematics, published by the Azim Premji Foundation and the Community Mathematics centre, Rishi Valley. It provides a national level platform where interested individuals can access resources, read about mathematical matters, contribute their own writing and interact with one another. It will be published in print and online at http://www.azimpremjiuniversity.edu.in/content/publications. For a print copy of the magazine please send your postal address to AtRightAngles@apu.edu.in
Editor’s note: The graphic that appeared with this column on September 18 was sourced from http://www.shyamsundergupta.com/triangle.gif