If we’re measuring a length of pi centimeters, why does it look finite but the number of digits is going on forever? It looks like that it’s moving small amount of time every time. Same with 0.5555555… or any number with infinite decimal places. I can’t wrap my head around it.
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Imagine you have a pizza. Can you divide it in half? Sure. Can you divide it in thirds? Sure.
Ok, now imagine you have ten pizzas. Without cutting any of them, can you divide them in half? Sure, 5 and 5. Ok, now divide it in to thirds. 3, 3, 3…and 1. Hmmm.
Ok, what if I let you divide that left over pizza? Great, you say now its easy again, I can just cut it in thirds! Waaaaaait just a minute I say. You can divide that left over pizza, but it has to be in pieces of 10. So now you have 3 pizzas and 1 pizza in 10 slices. Now can you divide it in thirds? Ok, 3 pizzas + 3 slices of pizza 3 times, but you are still left with one extra slice of pizza. Ok, now divide THAT slice of pizza into 10 even smaller slices. Wash. Rinse. Repeat.
Thats how the decimal system works, each time you move down a decimal point you are dividing a smaller quantity by 10 and then figuring out how many you need. Any left over is divided into another 10 pieces, and you keep trying to get closer and closer and closer.
Since we have chosen 10 as the base (thanks to our 10 fingers and 10 toes most likely) we’ll run into lots of situations where whatever we are measuring can’t be represented cleanly. We could have chosen a different base, say 12, in which case its easy to represent 1/3 (and 1/4 and 1/6) but we can no longer represent 1/5 cleanly. No matter what base we choose there will always be some numbers which we can represent as finite and some which will be infinite.
But keep in mind that is just how we are representing those numbers, it doesn’t mean the actual quantity they represent is infinite, as each decimal we add is a smaller and smaller fraction of the overall length, it eventually reaches a point where the other digits are meaningless.
**Fun fact:** Using just the first 40 digits of pi you can accurately calculate the circumference of the ENTIRE VISIBLE UNIVERSE to a margin of error about the size of a hydrogen atom. In the real world will probably never need to worry about even 40 digits of precision because the tools we use to measure are going to be less accurate than whatever we are trying to measure.
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