Actually 0.99... is the same as 1. They both represent the same number
105It's so dumb and it makes perfect sense at the same time. There is an infinitely small difference between the two numbers so it's the same number.
31There was also a veritasium video about this.
It was interesting.
7His videos always are
1
If .99....9=1, then 0.999...8=0.999...9, 0.99...7=0.999...8, and so forth to where 0=1?
0The tricky part is that there is no 0.999...9 because there is no last digit 9. It just keeps going forever.
If you are interested in the proof of why 0.999999999... = 1:
0.9999999... / 10 = 0.09999999... You can divide the number by 10 by adding a 0 to the first decimal place.
0.9999999... - 0.09999999... = 0.9 because the digit 9 in the second, third, fourth, ... decimal places cancel each other out.
Let's pretend there is a finite way to write 0.9999999..., but we do not know what it is yet. Let's call it x. According to the above calculations x - x/10 = 0.9 must be true. That means 0.9x = 0.9. dividing both sides by 0.9, the answer is x = 1.
The reason you can't abuse this to prove 0=1 as you suggested, is because this proof relies on an infinite number of 9 digits cancelling each other out. The number you mentioned is 0.9999...8. That could be a number with lots of lots of decimal places, but there has to be a last digit 8 eventually, so by definition it is not an infinite amount of 9 digits before. A number with infinite digits and then another digit in the end can not exist, because infinity does not end.
49Your way of thinking makes sense but you're interpreting it wrong.
If you can round up and say "0,9_ = 1" , then why can't you round down and repeat until "0 = 1"? The thing is, there's no rounding up, the 0,0...1 that you're adding is infinitely small (inexistent).
It looks a lot less unintuitive if you use fractions:
1/3 = 0.3_
0.3_ * 3 = 0.9_
0.9_ = 3/3 = 1
6No, because that would imply that infinity has an end. 0.999… = 1 because there are an infinite number of 9s. There isn’t a last 9, and therefore the decimal is equal to 1. Because there are an infinite number of 9s, you can’t put an 8 or 7 at the end, because there is literally no end. The principle of 0.999… = 1 cannot extend to the point point where 0 = 1 because that’s not infinity works.
4There is no .99...8.
The ... implies continuing to infinity, but even if it didn't, the "8" would be the end, so not an infinitely repeating decimal.
3If you really wants to understand the concept , you need to learn about limits
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This goes back to an old riddle written by Lewis Carroll of all people (yes, Alice in Wonderland Lewis Carroll.)
A stick I found,
That weighed two pound.
I sawed it up one day.
In pieces eight,
Of equal weight.
How much did each piece weigh?
(Everyone says 1/4 pound, which is wrong.)In Shylock's bargain for the flesh was found,
No mention of the blood that flowed around.
So when the stick was sawed in eight,
The sawdust lost diminished from the weight.21That's just pretentious. Oh your magic stick was exactly two pounds? The only right answer is "a little bit less than 1/4 pound"? Your stick weighted about 2 pounds, the pieces weigh about 1/4 pound. Get your wonderland shit out of here Lewis.
29He had another good one too... imma have to look it up because I don't have it memorized...
John gave his brother James a box:
About it there were many locks.
James woke and said it gave him pain;
So gave it back to John again.
The box was not with lid supplied
Yet caused two lids to open wide:
And all these locks had never a key
What kind of box, then, could it be?As curly headed Jemmy was sleeping in bed,
His brother John gave him a blow on the head.
James opened his eyelids, and spying his brother,
Doubled his fists, and gave him another.
This kind of a box then is not so rare
The lids are the eyelids, the locks are the hair.
And any schoolboy can tell you to his cost
The key to the tangles is constantly lost.7Love the reasoning. Reminds me of how "1 + 1 equals 3 for suffuciently large valies of 1" is actually true when talking about physical objects, since there's always some rounding involved.
1
written by Lewis Carroll of all people
I mean, he was a mathematician and a poet. Is it really that surprising he wrote a poem about math?
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i've seen a few people leave more algebraic/technical explanations so i thought i would try to give a more handwavy explanation. there are three things we need:
- the sum of two numbers doesn't depend on how those numbers are written. (for example, 1/2 + 1/2 = 0.5 + 0.5.)
- 1/3 = 0.33...
- 1/3 + 1/3 + 1/3 = 1.
combining these three things, we get 0.99... = 0.33... + 0.33... + 0.33... = 1/3 + 1/3 + 1/3 = 1.
it's worth mentioning the above argument could be refined into an actual proof, but it would require messing around with a formal construction of the real numbers. so it does actually explain "why" 0.99... = 1.
13One of the pieces is actually 0.33333....4
8If you cut perfectly, which is impossible because you won't count or split atoms (and there is a smallest possible indivisible size). Each slice is a repeating decimal 0.333... or in other words infinitely many 3s. (i don't know math well that's just what i remember from somewhere)
4Technically no
0.3333.... repeats infinitely. The 0.333...4 is not an infinitely repeating number. And since 0.333... is, there's no room to add that 4 anywhere
Which is why adding them up you get 0.999..... which is exactly and completely equal to 1
4
“just ask me the question” is great all by itself.
4If you take into account quantum fluctuations each piece will have a uniquely different mass at any given moment of time.
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