I know the Arithmetic proof and everything but how to explain this practically to a kid who just started understanding the numbers?
In: 3202
If they’re different numbers, there must be a number in between them. In between 0.999 and 1 there is a 0.9994, or 0.9997, and so on. But in between 0.999… and 1 you can’t ever find a number in between them. If those two numbers are really different then you should be able to find a number in between them
.9999… = 1 isn’t really a math trick, it’s just a side effect of converting fractions into decimal formatting. 1/3 = .3333…. They are the same exact quantity, just written in different ways.
You have no issues with 1/3 + 1/3 + 1/3 = 3/3 = 1 right?
Well 1/3 written as a decimal is .3333… and three of those makes .9999… The quantities you are working with have not changed, you’re just writing them out differently.
It’s like how hola and ciao both mean hello, it’s just a different way of writing the same thing.
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1 – 0.999…. = 0.0000… an *infinite* string of zeros. The nines never stop, so the zeros never stop either, and the last little 1 on the end never gets to exist.
I thought the simplest way to say it is
if you can’t find a number between two numbers then they are the same number, and 0.000…0001 isn’t a real number because you can’t have infinite zeros but also end at a 1
It’s the limitation of the decimal number system.
3 * 1/3 = 1
3 * 0.333… = 0.999… = 1
Not a proof but logical reasoning that maybe helpful to understand the statement.
Q: What is the largest number?
A: It does not exist.
Q: What is the smallest positive real number?
A: Does not exist.
Q: What is the largest real number less than 1?
A: Does not exist.
Conclusion: 0.999… cannot be less than 1 otherwise it will be the largest real number less than 1 which does not exist.
How do we know something simple like 1≠3? Well, one way to definitively show it is by bringing up the number 2, which is between them, so they obviously cannot be equal.
This is actually something we can do for all numbers on the number line. If two numbers are different, they will be different points on the line, with something in between them. Alternatively, if two numbers are equal, you can’t find a point between them, since they’re the same point.
For example, 4≠5, as we can find a number (like 4.5) between them. 6.1≠6.2 as we can find numbers between those (like 6.11).
On the other hand, we know something like 1=1.0, even though they are written differently, because there is nothing between them.
Now, what is there between 0.9999…. and 1?
The sum of an infinite geometric series is a/(1-r), where ‘a’ is the first term, and r is the ratio between successive terms. 0.9999…. can be made into an infinite geometric series by separating out the digits. 0.9999… = 0.9+0.09+0.009+…
In this case, a=0.9, r=0.1, the formula becomes 0.9/(1-0.1)==1.
You can use this formula for other repeating digit numbers.
If they are just starting to understand numbers, maybe don’t bother with this part yet?
If you have to explain it….
If you have 1, and you subtract nothing from it (we’ll, a thought and prayer more than nothing, but not a whiff beyond that)… you still have one.
So, if I give you an Apple, then I grab it back, rub it off, and give it to you again…. You have 1 Apple still.
The easiest way for me to understand was this: if they are NOT the same number, then there must be something in between them, but what could come between 0.99999…. and 1?
For me, it’s like actual values are over there —>
<— and how we write them is over here
They’re two separate things, and what we’re ending up with is a situation where “how we write them” doesn’t quite work with the actual value. 0.999… is the same as 1. It’s an artifact of decimal notation.
While we’re at it, it’s kind of broken that the vast, vast majority of values cannot be expressed as a decimal. Or a fraction. That sort of notation works for a miniscule fraction (ha) of all the numbers there are. Like approaching 0% of all numbers. That’s why we use symbols like e and pi, because we literally can’t write them.
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