Sometimes Mathematics doesn’t make sense. Sometimes it presents you something, a result that makes you question everything, whether every ounce of truth you’ve felt in this universe has been a lie. Well, that may be going a little far, but sometimes Mathematics is a mindfuck.
The notion of ‘limits’ are quite difficult to get your head around. Put simply, limits explain what happens to a function at infinity. Infinity is a very large concept, so understanding it intuitively can be quite difficult. Luckily though, this is where the rigour of mathematics really starts to shine. As I’ve briefly mentioned previously, Mathematics is a subject built on the idea of ‘proofs’. A side effect of this approach, is that even when dealing with something as complicated as infinity, we can trust the logic in the argument given. We can figure out a result first, then wrap our heads around it later. This is what happened to me when I was presented with the result that “0.999…=1”.
From our intuition, we think that every number that exists is represented by a unique set of decimals. So the very fact that the same number can be expressed by both 0.999… and 1 just seems wrong. However, the problem here lies in that tricky concept of infinity. It is true that if we restrict numbers to any finite amount of decimal places, then each number is represented by a unique set of digits, but when considering an infinite amount of decimal places, our thinking has to change.
Below is an equation that ‘proves’ the result, 0.999…=1. It is not the only proof (many others exist) but it is the argument that makes the most sense to me.
In the above equation we start with the number 0.999…, the first equality is essentially just a rewriting of 0.999… using some limit notation. It states that 0.999… is just the number 0.99….9 where the amount of 9’s keeps growing infinitely.The next step is a little bit trickier. In decimal notation for any given digit, the position of this digit is an indication of it’s size, for example in the number 237, the digit 2 actually refers to the number 200, ie. 237 = 200 + 30 + 7, or alternatively 237 = 2*100 + 3*10 + 7*1. This idea lets us write out the number 0.999… in the following way:
The funny ‘E’ in the first equation is just the sigma notation of the above equation. The next two steps use previously proven mathematical results in regards to limits, namely the geometric series and the algebra of limits. Trusting the maths, we have shown that 0.999…=1, but if it still doesn’t feel right I’ll present a worded argument based on how I wrapped my head around this result.
As 0.999… keeps going and going, getting closer and closer to 1. No matter how close you get, you can always get closer. So if we assume that 0.999… is equal to any number less than 1, we can always just consider more and more decimal places to find a number closer to 1 than the assumed number. This means we can’t take 0.999… to be equal to any number less than 1.
Conversely, if we assume that 0.999… is equal to a number greater than 1 then we would expect 0.999…=1 for some finite number of decimal places, as the number (which grows larger with each successive decimal place considered) should ‘pass through’ 1 to be equal to the number greater than 1. This isn’t the case so 0.999… cannot be equal to any number greater than 1.
This leaves us with only one option, that 0.999…=1.