The power of 11111

Today, the eleventh of January of the year 2011, is written as 11111, i.e. a string of five ones together.

The power of 11111

This is called a repunit i.e. a repeated sequence of ones. There are repunits which are also primes like 11. After 11, a string of nineteen ones i.e. 111...(repeated 19 times and written as R19) is another prime, other examples of repunit primes are R23, R317 and R1031.

No larger repunit prime than R1031 is perhaps known! The number one is itself unique in many ways; i.e. apart from it being used to denote the top or number one position in any situation or subject. One and zero are of course unique binary digits, the universal basis for all digital systems and computations.

Anything multiplied by one, remains the same, but anything multiplied by zero also vanishes. Anything (i.e. any number), raised to the power of zero, becomes one, but anything (however large or small) raised to the power of one does not change and is the same number! This has led to the amusing adage “Show me a useless exponent and I will show you one!”. In the adventures of Alice in Wonderland (or rather in its sequel Through the looking glass) one of the riddles posed to Alice (by the Red Queen) is “What is one and one and one...?” Alice replies, “I have lost count!” Of course you can build up all the numbers using only zero and one which is actually the basis for the binary system.

The number eleven again has interesting properties. The powers of eleven give the binomial co-efficients in the binomial expansion, as is well known in school algebra, i.e. 11, 121, 1331 etc. If you want to expand (a+b) to the power of n, ex: n=2, coefficients are 1, 2, and 1 for a squared, product ab and b squared. A convenient way to remember this is the so-called Pascal’s triangle, which successively gives the power of eleven. The first few powers of eleven are also palindromic (i.e. read the same both ways), ex: 11, 121, 1331 etc.

Eleven dimensions also play a unique role in unified theories of basic interactions in physics. The universe is supposed to have started with eleven dimensions, four expanding and the remaining shrinking and becoming compact to minuscule dimensions which could be probed in the Large Hadron Collider! The eleventh of November this year will also have the sequence 111111.

Source: http://www.deccanherald.com/content/127937/snippets.html