471–480 of 647 posts
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Ask for a number between .9 repeated and 1
How about, ask for an integer between 1 and 2. Can't think of one? Guess they're the same number then.
There is no proof that will ever satisfy a person dead-set against this. Ever since I brought this home from school as a child, my whole family ribbed me mercilessly for it. If you tell a person that 3/6 = 1/2, they'll believe you - because they have been taught from an early age that fractions can have multiple "representations" for the same underlying amount. People mistakenly believe that decimal numbers don't hav…
> ...infinitely many 9s... How about we prove that an infinite number of 9s is impossible? Assume that we have a finite number of 9s. Add a 9. The result is not infinite. Add another 9. The result is still not infinite. We can repeat this process for an infinite amount of time and still not have an infinite number of nines. Any process that can not be completed in a finite amount of time can not complete and can not…
Isn't this somewhat ill-defined? You may have a finite number of 9s at any one point when adding more 9s, but there is no point "at infinity" where you can stop and look at how many 9s you've added because by definition there are still more 9s to add.
It's like trying to prove infinity is impossible:
> Assume that we have a finite number. Add 1 to that number. The result is not infinite. Add another 1. The result is still not infinite. We can repeat this process for an infinite amount of time and the number is still not infinity.
Sure, individual numbers "on the way" to infinity are not infinite, but that doesn't necessarily disprove the existence of infinity.
What if you have 0.9̅4? Can we say 0.9̅5 > 0.9̅4 > 0.9̅3? More on what happens if you allow this: https://mathwithbaddrawings.com/2013/08/13/the-kaufman-decim...
>What if you have 0.9̅4? Well, you fundamentally can't. If the 9s go on for forever then you never reach a point where you can add the 4. The definition of infinity precludes anything after infinity, because it never ends so you can never get there.
There is no proof that will ever satisfy a person dead-set against this. Ever since I brought this home from school as a child, my whole family ribbed me mercilessly for it. If you tell a person that 3/6 = 1/2, they'll believe you - because they have been taught from an early age that fractions can have multiple "representations" for the same underlying amount. People mistakenly believe that decimal numbers don't hav…
I would try to explain to them that numbers are a framework for us to understand both the observable universe and abstract ideas, depending on what we're using them for.
Like you said, it's hard for people to understand that numbers have multiple representations and to grasp the implications of those representations. I think that if you can communicate that different representations can have the same meaning, accepting those representations when they come across them may be easier.
Or, if they're experienced enough with math, I think going through Euler's identity in addition to the link could help.
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This is not an infinite decimal. The digit 1 is somewhere out there.
But this is not a compelling argument to somebody in this situation. While correct, it feels identical to saying "it just is".
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> If people don't accept the former, they can take out a pencil and paper to compute it themselves. After a few digits it will become obvious. How can they compute 1-0.999… when they clearly have no idea what 0.999… is?
They have to know that 0.999... means you never stop writing nines. Put 1.0 on top, 0.9 on the bottom. Start subtracting from left to right, and keep writing nines on the bottom as you go to the right. In no time you'll see that the answer is infinite zeros.
How do they know that that's a real number?
What is the largest number smaller than 1?
Assume x is the largest number smaller than 1.
(x + 1)/2 is a number larger than x but smaller than 1. Our assumption that x is the largest number smaller than 1 must be wrong. QED.
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.999... and 1 exist on a continuous line. If they are different numbers, name a number between them.
? Just because there is nothing between two numbers does not mean the two numbers are equivalent. What nonsense is this