0.99999999… is equal to 1
Why is 0.99999999… equal to 1? This is one of those questions which always puzzled me to begin with. How on Earth can one number be the same as an entirely different number? How can 0.9999999… going on towards infinity be the same thing as 1? The answer to this lies in fractions. The number 1 can be divided into third, which can be either written as a fraction or as a decimal of the type; 0.33333333… (off into infinity). If we add 0.333333333… to itself, we get 0.6666666666… add another and we get 0.9999999999… But since 3 thirds are equal to 1, so 0.99999999… must be equal to 1.
Taking this understanding of the number one, we can construct other decimals that might give us a greater understanding of infinity. Lets look at the hypothetical number, which I call the Infinite Nines;
…99999999999999.999999999999999…
When I first cam up with this number I was looking to define the largest number possible. I know what you are thinking. There is no such thing as a largest number, because it is always possible to add one more. But bare with me…
Clearly, this is a very large number with nines stretching out ahead and behind into infinity. But as we already know 0.99999999… is equal to one. So, at some infinitesimal value on this infinite number line, one of the 9’s have to switch to a zero. This, in turn, puts a one in the next register causing that to make a zero and so one and so forth right the way up the number line like a series of dominoes, each one setting the other off. And then we come to the number 1.0000000…, but as we can see our sum does not end there as we have an infinite series of nines — stretching up to the highest possible value — which also have to be knocked by the approaching wave of dominoes. What we are left with in the end is something like the following.
…0000000000000000.000000000000000…
If our last number was infinitely big, then this number is only infinitesimally larger than it by a factor of about; 0.0000…0001. And yet no matter how far or how long we travel up the number line, we will never reach the number 1 at the other end. We know it must be there, somewhere, but we will never discover it and so how do we truly know it is there? To put it another way: What is the difference between this number and nothing? The answer is nothing.
I have said that the infinity of zeros shown here is slightly larger than the block of infinite nines, but we know that this isn’t true, because 0.999999… is equal to 1. So both this numbers are actually equal to each other. So, are the infinite nines (or zeros) the biggest number we can imagine? The best test for that is again always to add 1 and see what the result would be. But in this instance since this infinity is also indistinguishable from zero, adding one to it will only result in 1, which is nowhere near close to an infinite number in the largest sense.
If zero equals infinity, then 1 plus infinity equals 1. But this entails that all the numbers that we can write, which aren’t supposed to be infinite, are themselves based off of infinity. In this sense, infinity can be thought to be the ground state of all numbers. This in itself bares similarity to the concept of the Dirac Sea, which requires an infinite sea of negative particles in order to create a neutral state or even a single positive particle. In the same way zero could be thought of as the sum of all the negative and positive numbers in existence; making it akin to a ground state infinity.
