Today I’m going to show a variety of proofs that 9.9 Recuring is equal to 10 and not just approximately. One of the things I love the most about these proofs is it shows how incompetent we humans are with the term infinity.
For example, Hilbert’s hotel – a thought experiment about a hotel with infinite rooms; one could get their head around this fairly easily at first but are perplexed when somebody asks “could you fit an infinite amount of passengers inside such a hotel”.
Another example could be the concept of smaller infinities, to begin, think about the ordinary integer infinity (from one Ad Infinitum) and the decimal infinity (0.0000….,0.999… and the numbers in between)
However, decimals are between integers so you could have a larger infinity when counting both the integer infinity and the decimal infinity in between. Strange, isn’t it?
Now let’s move on to The proofs!
To begin let’s talk about one proof used rather frequently,
We start off with 10 and divide it by three – (try it for yourself on a calculator!) you get 3.333 recuring now if you times that by three you get 9.999 recuring. But if 3.33… is 1/3 of 10 then timesing it by 3 should give us 3/3 therefore 9.99… is equivalent to 10
Sadly though, this proof does feel somewhat unsatisfying since if you don’t believe 9.99… is equal to ten you wouldn’t believe 3.33 is 1/3 of ten
Not finished yet