I know there are proofs on this but I’m not great at math, can someone dumb it down for me?
In: Mathematics
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Let .99999… be called X
What is 10X? It must be 9.99999…
What is 10X – X? It must be 9.0.
10X – X also equals 9X.
So if 9X = 9.0, then X = 1.0 as well.
2 simple solutions I like:
1/3 = 0.33333333…….
2/3 = 0.66666666666……..
3/3 = 0.9999999999………
But 3/3 also equals 1
The other is:
x=0.99999999……….
10x=9.9999999999
Subtract x from each side and you get:
9x=9
x=1
The very simple answer is there’s no “room” between 0.999… and 1 for another number.
The three dots (…) is called an ellipsis and it means “keep doing that forever” in maths. And for 0.999… it means “keep writing nines forever”.
But let’s see if we can find any room between 0.999… and 1. The easiest way would be to subtract 0.999… from 1. What does that equal? You’ll have to “carry the 1” and turn 1.0 in to 0.¹0 repeatedly.
1.000 – 0.999… = 0.¹000 – 0.999… = 0.100 – 0.099… = 0.0¹00 – 0.099… = 0.010 – 0.009…, etc, which might end up with 0.000…1 .
But what would “0.000…1” mean? “Zero point zeroes forever and then one” doesn’t make much sense.
**Alternate approach from the other direction**
What does 0.999…. mean?
* 0.9 = 9/10, is nine tenths (leaves one tenth of “room”)
* 0.99 = 99/100 is ninety hundredths (leaves one hundredth of “room”)
* 0.999 = 999/1000 is nine hundred and ninety nine thousandths (leaves one thousandths of “room”)
* 0.9999999 = 9,999,999/10,000,000 is nine million, nine hundred and ninety-nine thousand, nine hundred and ninety-nine out of ten million (leaving one ten-millionth of “room”)
Here’s another way to look at it. If 0.99999… does *not* equal 1, there must be some number greater than 0.99999… and less than 1, because the real numbers are defined as infinitely divisible – there are infinite numbers between any two numbers you can name.
What is that number between 0.99999… and 1? How can you express that number in decimal if every place in 0.99999… has a 9, and there’s no digit greater than 1?
A simpler explanation:
Let’s start with a simpler number, 0.9. How much different is 0.9 to 1? They differ precisely by 0.1.
Now, let’s add more trailing digits, 0.99. How much different is 0.99 to 1? They differ by 0.01.
You see, each trailing digit we add, the difference drops an order of magnitude. So, for 0.99999, the difference would be 10^(-5).
Now, let’s say 0.99999 goes on infinitely. What would be the difference between 0.9999 recurring and 1? It would be, according to our observation, be 10^(-∞), which would be 0.00000…. That number would be infinitesimal.
And hence, when two numbers have zero difference, those two numbers are the same.
On a more ‘philosophical’ level, it’s important to remember that there’s a difference between numbers, and our *representations* of numbers.
There’s only one “one” in the abstract sense, but we can represent it in numerous different ways. We can write “one”, or use the roman numeral I, or we can use digits. But even using digits, we don’t have a unique representation. We can write “1”, but we can also write “1.0”, or “1.000”.
It’s also the case that our representations don’t perfectly cover every case. It is literally impossible to write out 1/3 using only decimal digits; we need to “cheat” but putting a line over some of the threes to indicate an infinitely repeating decimal. But critically, this isn’t something intrinsic to “1/3”; rather, it’s a consequence of our base 10 number system. If we used a base 12 number system, with the digits [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, a, b], then we could write 1/3 as 0.4. No repeating decimal!
So all the math proofs here are 100% correct, but if there’s still a part of your brain that needs to say “but look how it’s written, it’s clearly a different number!”, remember that the digits are a different thing from the concept, and that the way we write things means that sometimes things look odd. Just like 2/3 is 0.66666…, and manages to be 2/3 despite it seeming like adding ever more 6s will never quite be enough to get there, 0.99999… manages to be 1. Because you’re not just “adding more 9s” – there’s an *infinite number of 9s.* And infinities are hard to write, and hard to conceptualize.
If 0.999…. wasn’t 1, then 1 – 0.999… isn’t 0. Then you can divide by it. Then, 1 / (1 – 0.999…) is a number that is bigger than all of the natural numbers (natural numbers are the counting numbers: 1, 2, 3, …). However, every real number must be smaller than some natural number (round the number up). This is a contradiction.
3 and 10 are relatively prime. It is impossible to perfectly represent 1/3 as a base 10 decimal.
As I understand it, there is a number system called the surreal numbers where 0.99999… and 1 are different numbers.
Possibly simpler explanation since the OP is struggling with assertions that look like tautologies.
IF .999… = 1 then 1-.999…=0, yes? Not asserting that .999… = 1, just saying what would be true IF IT WERE.
So what’s 1-.999…? 0.000…
It is NOT 0.000…1
The really really key point, OP, is that string of 0s doesn’t end. There is no “1” anywhere in it because you never get to the end. It doesn’t have an end. Infinity isn’t just “a really huge string of numbers with an end somewhere we can’t imagine”, it literally has no end. It can’t end with a 1 because it can’t end at all.
And 0.000… is just an inconvenient way of writing “0”.
So if 1-.999… = 0, then 1 = .999…
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