Math Journey Archive

Math Journey 4: Trips to Infinity

Uncle Bob

This math journey will involve several treks. The goal is to get a fuller perspective on the nature of infinity. What is it? Can we reach it? Are there more than one? To accomplish the goal we will make several trips using different routes such as numbers, arithmetic, sequences, geometry, and maybe even going around in circles.

Trip #1: To Infinity via Numbers

Pick a number. Any number. Now name a number that is larger than the number you picked. You are able to do this in every case – one reliable method is merely to add one (1) to the number. There is no largest number. If that is true, then what is infinity? Infinity historically has been regarded as both a metaphysical concept and a mathematical limit, but infinity is not a number. Let's see how close we can get to it.

The United States is trying to get to infinity by spending it. The news from politico.com on December 10, 2013, was that Representative Paul Ryan and Senator Patty Murray had unveiled a two-year budget agreement that they say will end years of bitter budget wars on Capitol Hill. The agreement would set the discretionary spending level for fiscal year 2014 at \$1.012 trillion. Written out, the amount is

\$1,012,000,000,000 and some loose millions.

The national debt gets us even closer to infinity – from TreasuryDirect.gov as of 12/19/2013 total public debt outstanding was

\$17,251,528,475,994 and 19 cents.

That's over \$17 trillion.

Let's get away from the red ink for a bit and think about protecting our money. Another (sad) December event was the breaking of the story that 40 million Target stores customers credit information was hacked. Cryptology, the science of making and breaking codes, is a fast-growing and much studied field these days, due in part to the growing use of electronics and etail in business. Codes that protect account information can be made safer but not unbreakable. The degree of safety is related to cost and is therefore a business decision.

One simple code is a type of puzzle that many people solve every day – the cryptogram. This puzzle encodes a message by swapping out each letter for a different one in the alphabet. The new lineup is called the replacement alphabet, and it changes from one message to the next. How many different alphabets are there? A basic formula in mathematics tells us there are

26x25x24x23x22x21x20x19x...x4x3x2x1

different orderings (permutations) of the 26 English letters. That's right, we multiply all the numbers from 1 through 26 together.

Before we look at this number, we have to tell you that it overcounts the usable substitution alphabets because some of the possible orderings have one or more letters remain in their original position. They can't be used because it would be like swapping an "I" for an "I." We need a count of new alphabets where all of the letters are in a new position. Mathematicians call these orderings "derangements." I know, it sounds crazy.

Here's a cryptogram for you to solve. My one clue is that the replacement alphabet can be called 'atbash.'

MLD R'EV HZRW NB ZYX'H. GVOO NV DSZG BLF GSRMP LU NV.

solution

If we had just two letters in our alphabet, AB, there would be only one derangement, namely BA. In the case of three letters, ABC say, there are six possible permutations, including ACB, with A remaining first and BAC, with C last, but only two derangements BCA and CAB. Derangements become much more numerous with a larger supply of letters – 9 derangements for a 4-letter alphabet and 44 derangements for 5 letters.

We could solve for the exact count of derangements of our full alphabet, but we really want to know about how large the number is, and so we will save time (and money?) by taking a short cut. Don't worry – we wouldn't do this with your credit information! Interestingly, it turns out that for larger alphabets the derangements are approximately 37% of the total possible permutations. So we return to the enormous product of one through 26 and take 37% of it. For the English alphabet there are roughly

148, 000, 000, 000, 000, 000, 000, 000, 000 derangements.

That's 148 trillion trillion derangements. Well, that's a lot of security, but as I mentioned, puzzle people like me solve cryptograms with relative ease. And we are no where near infinity, because I could type a 9 at the top of this page and fill the rest with zeroes to make a much larger number

OK. Let's stop messing around and go for a large, large number. A googol? That's a 1 followed by a hundred zeroes – not big enough. A googolplex? That's a one followed by a googol of zeroes. That is a vast number.

Infinity? No. Take any newspaper and let one single printed letter stand for a googolplex. Do all of the letters in the entire paper represent at least a fraction of infinity? No. You'd be richer getting a penny change from a dollar.

So we have failed to reach infinity in this our first trip; however, there are other routes available to us, and we will explore them in coming months.

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