My math teacher did this! 0.999...=1? Reddit r/theydidthemath

Is 0.999...=1 i.e. 0.9(repeating) is really equal to 1?
We will use "an algebraic trick" to explain why 0.999...=1.
We first let x=0.999.. then multiply both sides by 10 so we get 10x=9.999...
Next, we will do 10x-x, which is 9.999...-0.999... and get 9x=9.
After dividing both sides by 9, we see that x=1.
But we set x=0.999... to begin with. So we can conclude that 0.999...=1.

Genuinely curious: If .9 repeating = 1, what does .8 repeating = ? Reddit r/askmath

This is from Reddit r/theydidthemath.

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#math #algebra #mathbasics

[bprpmathbasics]

Genuinely curious: If .9 repeating = 1, what does .8 repeating = ? Reddit r/askmath

[pretzelbomb]

Confusion around repeating decimals like this are a solid third of the reason fractions are useful.

[vladimirpain]

Best part is the infinite supply of markers in the background. This guy is ready to explain math no matter what.

[atlas]

My favourite explanation for why 0, 999… is equal to 1 is:
“Assuming it’s smaller than 1, what would we have to add for it to become 1? The answer has to be 0, 000… with infinite zeros which everyone will argue is just 0”

[milandavid]

I think what confuses a lot of people is that there is literally an infinite number of 9's after the zero without any sort of approximation or "handwaving".

[ingiford]

I also like the 1/3 + 1/3 + 1/3 = 1 but 1/3 is .3(repeating) and if you add 3 of them together you get .9(repeating)

[maximummathematics]

One way I like to think about it is to ask myself “If 0.9 repeating and 1 are different numbers, then I should be able to find a number between them”. Failure to construct a number between them proves they are the same. Great video!

[herrhartmann]

It becomes slightly more obvious if you consider that the 1 at the end of the first calculation is basically 9/9.
If 0.444... = 4/9, then it should make perfect sense that 0.999... = 9/9 = 1.

And as for the "we're not quite there yet" argument:
The apparent difference between 0.999... and 1 is only caused by our inability to write down an infinite number of nines on our page, screen or whiteboard.
If we could write out infinite nines, then the difference between our written number and 1 would become infinitely small. "Infinitely small" means the same thing as 0. And if the difference between two values is 0, then those values are equal.

[Anik_Sine]

Discovering = 1 and that -40°C = -40°F really broke my mind

[anderstroberg]

It's just an artifact of base 10. 1/3 is 0.333... in base 10, because base 10 can't exactly describe that, just like how, for example, base 2 can't describe 0.1 exactly.

But, if we go to base 3, 1/3 is simply 0.1. Multiply that with 3, and you'll get 1.

[generalrubbish]

Engineers: "You had me at 0.9"

[danishskiwarehouse]

The most impressive thing about this whole demonstration is how seamlessly he swaps between those pen colours

[LaughingOrange]

This comes down to "Infinity isn't intuitive". It is impossible to find the difference between 0.999... and 1, because as you go through each digit, the difference gets smaller and smaller, and approaches 0.

[bprpmathbasics]

We know sqrt(4)=2
Now what is sqrt(0.444...)=?

[ophiushus]

Thanks for coming up with this problem again!

When our maths teacher in high school tried to convince us about that identity, I just didn't believe it.
Had he gone via the decimal representations of 1/9 and its multiples, he might have succeeded.
I would have had severe problems refuting 9/9 = 1.

[bobbun]

I think the real reason people reject the claim made here is because they believe that every real number has a single decimal representation. That's not actually true, of course. Every terminating decimal (except zero, I believe) also has a non-terminating representation that ends in an infinite sequence of nines.

[ianfowler]

It is also good to note that 1/2+1/4+1/8+1/16 = 1 when written in binary (base 2) is just the repeating decimal: which is indeed = 1 in base 2. Not to be confused with 1/9 in base 10.

[LexPride]

I love showing my students that having a matching number of 9s in the denominator ALWAYS creates repeating numbers, e.g. 0218/9999 = 0, 021802180218... You can therefore write any of these repeated patterns as a fraction, and seeing 9/9 has rarely given them a reason to argument against it being the same as 1. Funnily enough, I feel as if it would create more of a resistance, if I showed the same fact to them with a variable being in play
Thanks for the videos!

[vothaison]

Math gets weird at infinity: Anything is possible.

[synpynir]

We're so used to the decimal system, that we think of things like "0.4", "1", and as the actual numbers, when they're only representations of the numbers. And the decimal system has this property (maybe even a flaw) that some numbers have more than one representation. The number 1 has another valid representation: 0.999... but that looks strange to us, because we expect those representations to be unique, so if the representation is different, the numbers should be different as well.

And that is why people are less bothered by 0.4444... There is no other representation of 4/9, so it's fine that this decimal repeating fraction is equal to 4/9. But exposes the flaw in our beloved decimal system.