Identity

Mathematics is a complex and beautiful science. It is a logical poetry that can be to some just as majestic as the writings of Shakespeare, Cummings, Emerson, or Blake. Several years ago I was introduced to this Identity riddle of sorts.

In mathematics, each value, whether it is an integer or a decimal or a fraction or anything really, has its own value that is unique to it. The number "5" is the number "5." There isn't a lot of argument in this. Surely you can depict the number 5 in several different ways, but there is only one value for that number. 5 is not equal to 7, nor is it equal to 0.673, nor is it equal to the Sin (60 Degrees).

By now, I hope you get the idea that numbers have their own identity. But the proof I am about to show you may alter your view of mathematics and perhaps even reality in general.

The sequence is quite simple really. I'm going to show you several variations of the same fraction and then I am going to suggest that if patterns hold, and in this case they most certainly do, then there is a discrepancy as the value approaches the number 1.

Enough of this abstract jibber-jabber. Let me work through the problem with you now. This problem involves the fractions and decimal values when a number is divided by 9.

When a number is divided by the value of 9, a really interesting pattern seems to emerge in the decimal value that results. I'm going to give you a couple examples and I'm going to see if you are able to decipher the pattern.

For instance, when the number one is divided by 9, the result in decimal form is .111 repeating. This means that until infinite the number 1 will reoccur. Visually it is depicted as follows:
Great. So, just so that everyone is up to speed, there are essentially two things that we learned from this one fraction. First, that the decimal reoccurs until infinite, and that the decimal number is the same number that is being divided by. In this case that would equal 1.

But, lets look at a couple other examples of numbers divided by 9 to see if there is an emerging pattern that we may be able to formulate.

Consider the fractions and decimal results from the number 4, 7, and 8:

As you can see, there is most definitely a pattern that is beginning to emerge when values are divided by the number 9. In all of these cases, the result is a recurring decimal until infinite and the number that is recurring is the same value as that being divided by. Excellent.

Well, with this in mind, that the number that is being divided by creates a decimal that is recurring until infinite while also being the same number as the value being divided by in the decimal, a minor problem arises when one considers the case of the number 9 divided by itself.

Any number divided by itself is 1. This is the result of the multiplicative identity which proves that all numbers that are divided by itself render the same number, 1.

However, this is slightly disconcerting given our current pattern involving numbers that are divided by 9. If we notice that 1 divided by 9 is 1 "repeating," and 4 divided by 9 is 4 "repeating," and 7 divided by 9 is 7 "repeating," then surely the same must hold for 9 when divided by 9 is 9 "repeating." According to the multiplicative identity this just isn't the case.

This strikes me as a little perplexing, and the desired effect of this blog post would be to elicit similar feelings in you. There is a definite conundrum here. There are two numbers that can be proven separate ways that equal the same thing, but are just not the same. In my opinion, this defies most of what mathematics is, a proof of why something is as it is. And though in math you can prove things several ways and arrive at the same result, the result here is different.

I hope you enjoy pondering this complexity as I have.