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? Can it be contained? Is there more than one infinity? To accomplish the goal we will make several trips employing different modes 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.

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The United States is trying to get to infinity by spending it: thebalance.com reports that President Biden’s budget for FY 2022 totals $6.011 trillion, eclipsing all other previous budgets. Written in full the number is
$6,011,000,000,000.

The national debt gets us even closer to infinity. From US Deb Clock, as of May 23, 2022 total public debt outstanding was more than $30,467,000,000,000. That's over $30 trillion.

Let's get away from the red ink for a bit and think about protecting our money. These days we commonly hear stories of thousands if not millions of people having private information stolen – this in spite of the security measures taken. Cryptology, the science of making and breaking codes, is a fast-growing and much studied field these days, and it is a necessity in e-commerce and other online transactions. 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

 

26 x 25 x 24 x 23 x 22 x 21 x 20 x 19 x…x 4 x3 x 2 x 1

 

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 over counts 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.

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 have not come 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 three more trips.

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Here's a cryptogram for you to solve. My one clue is that the replacement method has been referred to as “atbash.” Look it up.



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