I know it sounds stupid but hear me out. I’ll try to explain my dilemma as best I can.
19+9 is 28
If I put it in a basic method it looks like this:
19
+¹9
28
You carry the one.
3 thirds is 0.9 reoccurring, meaning that it goes on forever right? So because it goes on forever, if you carry and infinite amount of “1”‘s. Yes the “1” gets 10 times smaller each time, but if you have an infinite amount, eventually they should all add up to a whole 1 right? And then again? And then again again? Yes it will take longer and longer each time, but because it’s infinite it will eventually happen.
But 3 thirds is equivalent to 1 right? And 1+1 is 2. Sooooo… is 1+1 equal to infinity or 2 depending on the context? Or is 3 thirds different to 1?
In: Mathematics
The bit that nobody seems to have answered yet is that an infinite number of ones after the decimal point doesn’t add up to infinity, because they shrink too fast.
Another interesting example of this is: if you add 1/2 + 1/4 + 1/8 etc what answer do you get? The problem bivouac suggestion is that you have an infinite number of fractions, so it must equal infinity.
But consider that if you start with a half and add 1/4 you get to 3/4, which is halfway between 1/2 and 1
Now you add 1/8 to get 7/8 which is halfway between 3/4 and 1
Each fraction you add gets you halfway closer to 1, so the answer is that all those infinite fractions add up to 1.
Is that at all helpful?
3 thirds is equal to one, but 1/3 isn’t equal to 0.333 (repeating).
Three doesn’t divide equally into one, so we just go with the closest equivalent which is “zero point three repeating”. If you were to take three copies of the decimal representation of 1/3 (aka 0.333…) and add them together, you will get 0.999…
This is infinitesimally* close to 1, but isn’t actually 1.
* An infinitesimal is like the opposite of an infinity. Instead of being unfathomably large, it is unfathomably small. The mathematical definition would be something like “a quantity that is closer to zero than any standard real number, but is not zero.”
Three thirds is 0.999… = 1.
Three thirds plus three thirds is 1.999… = 2.
None of these values equal infinity.
0.9 repeating has an infinite number of digits, but it doesn’t have an infinite value. That’s where you’re confusing yourself.
We know what it’s value is (it’s 1.)
Any number with an infinite number of digits still has a value. We don’t know all the digits, but we know the value. Like, I can’t tell you all the digits of pi, but I can tell you the value of pi is between 3.14 and 3.15.
This is my absolute favourite part of maths hands down.
I think the problem you’re seeing is that 0.999 recurring is equal to 1:
0.999… = x
10x = 9.999…
10x – 1x = 9.999… – 0.999… = 9
9x = 9
x = 1
It’s a weird proof but what it comes down to is that 0.999… becomes one.
.99999999999… *is* 1
What’s the difference between the two numbers? 0.00000… it’s all zeros, so there’s no difference.
This is just a visual artifact of a base-10 system struggling to accurately represent 1/3 in decimal form.
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