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January 23rd, 2019, 10:25 AM   #71
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Quote:
Originally Posted by AplanisTophet View Post
It should be obvious you and Maschke are implementing physical limitations that I am not.
But you are. You've repeatedly claimed that you can measure an arbitrary real number of inches. That is factually false. No statement in the English language could be more false. If you don't understand why, the choice would be to try to explain it to you, or to give up on you entirely. Or perhaps suggest that you take a basic course in physical science at the high school level.

But ok, I'll explain it. Suppose you claim you can measure 1/pi inches. Can you please suggest the nature of a the physical apparatus that could make such a perfectly exact measurement? You'd win the Nobel prize in physics for devising such an apparatus.

All physical measurement is approximate. Are you under a different impression?

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Originally Posted by AplanisTophet View Post
These are theoretical line segments of length $x$ inches where $x \in (0,1)$
Aplanis, this shows an appalling lack of understanding of the nature of physical science.

Last edited by Maschke; January 23rd, 2019 at 10:35 AM.
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January 23rd, 2019, 10:32 AM   #72
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Mr. sintan, is your middle name cos?

If so, then .33333333...... of your name is missing.....
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January 23rd, 2019, 10:59 AM   #73
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Quote:
Originally Posted by Maschke View Post
But you are. You've repeatedly claimed that you can measure an arbitrary real number of inches. That is factually false. No statement in the English language could be more false. If you don't understand why, the choice would be to try to explain it to you, or to give up on you entirely. Or perhaps suggest that you take a basic course in physical science at the high school level.

But ok, I'll explain it. Suppose you claim you can measure 1/pi inches. Can you please suggest the nature of a the physical apparatus that could make such a perfectly exact measurement? You'd win the Nobel prize in physics for devising such an apparatus.

All physical measurement is approximate. Are you under a different impression?



Aplanis, this shows an appalling lack of understanding of the nature of physical science.
Find where I said we must be able to physically measure something. I said the opposite, that my symbols were theoretical, from the start. Given that, what the hell Maschke?

Answer my question on how we come up with uncountably many symbols then. That was your claim originally, not mine.
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January 23rd, 2019, 11:11 AM   #74
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Quote:
Originally Posted by AplanisTophet View Post
Find where I said we must be able to physically measure something. I said the opposite, that my symbols were theoretical, from the start. Given that, what the hell Maschke?
You said a line segment of pi inches exists. That is utter nonsense and shows deep misunderstanding on your part regarding the nature of physical science.

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Originally Posted by AplanisTophet View Post
Answer my question on how we come up with uncountably many symbols then. That was your claim originally, not mine.
It was yours. I defy you to quote me saying anything remotely like that. You've completely taken leave of your senses here.
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January 23rd, 2019, 11:17 AM   #75
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Quote:
Originally Posted by Maschke View Post
You said a line segment of pi inches exists. That is utter nonsense and shows deep misunderstanding on your part regarding the nature of physical science.
I answered your question before you even asked it.

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Originally Posted by AplanisTophet View Post
Just in case you don't have a magnifying glass that powerful, for each real number x (of which there are uncountably many), let 1/3 = x/3x.
Quote:
Originally Posted by AplanisTophet View Post
If you are restricting your notations to formulas that can be produced using a formal language [having at most a countably infinite alphabet] that takes into account only finite sentences, then sure, there are only countably many.
It was a joke.

But, you still have to explain this now given your amusing reversal:

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Originally Posted by Maschke View Post
No reason we can't have uncountably many formal symbols. Just let each real number be a symbol. I don't know if "symbol" is formally defined anywhere but in logic you can make any set into an alphabet of symbols. Typically the alphabet is countable but in some applications it's uncountable.
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Originally Posted by Maschke View Post
FWIW I found a specific reference regarding uncountable alphabets. Here's the link ...

https://books.google.com/books?id=eq...page&q&f=false

And here's the relevant paragraph.
So how is it we have "uncountable alphabets" if you wish now to only abide by physical laws?

Last edited by AplanisTophet; January 23rd, 2019 at 11:26 AM.
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January 23rd, 2019, 11:36 AM   #76
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Geezzz....methinks time to close this thread
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January 23rd, 2019, 12:00 PM   #77
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Quote:
Originally Posted by v8archie View Post
As an addendum to the above, you can find the fraction that represents a recurring decimal quite easily:

Given
$$x = 2.46358358358 \ldots = 2.46\overline{358}$$
We multiply by the a power of ten that gets one of the recurring sets of digits to the right of the decimal point: in this case, $100$.
\begin{align}x &= 2.46\overline{358} \\ 100x &= 246.\overline{358} \end{align}
And then by a power of 10 that gets the recurring part to the left of the decimal point, in this case 1000.
\begin{align}100x &= 246.\overline{358} &(1) \\ 100000x &= 246358.\overline{358} &(2)\end{align}
We still have the recurring part to the right of the point because that's what recurring means - the digits repeat for ever.

Now we subtract the first line $(1)$ from the second $(2)$, causing the decimal parts (to the right of the point) to cancel each other:
\begin{align} 99900x &= 246112 \\ x &= \frac{246112}{99900} \\ \text{factorize} \qquad &= \frac{2^5 \times 7691}{2^2 \times 3^3 \times 5^2 \times 37} \\ \text{and cancel} \qquad &= \frac{2^3 \times 7691}{3^3 \times 5^2 \times 37} \\ &= \frac{61528}{24975}\end{align}
Sincere thanks @v8archie and @JeffM1

I follow most of what you've explained but I am a real novice and some of the terms and symbols you probably take for granted are new to me.

To help develop my understanding I will certainly look to pick up a calculus text and would be keen to delve deeper into logic and paradoxes as I find these areas particularly fascinating.

If I could please pick your collective brains one more time to recommend some good reads in these areas I would be grateful.
This thread has been hugely educational for me so thanks to all who have contributed.

Yours, The Troll
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January 23rd, 2019, 12:39 PM   #78
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Quote:
Originally Posted by sintan View Post
Sincere thanks @v8archie and @JeffM1

I follow most of what you've explained but I am a real novice and some of the terms and symbols you probably take for granted are new to me.

To help develop my understanding I will certainly look to pick up a calculus text and would be keen to delve deeper into logic and paradoxes as I find these areas particularly fascinating.

If I could please pick your collective brains one more time to recommend some good reads in these areas I would be grateful.
This thread has been hugely educational for me so thanks to all who have contributed.

Yours, The Troll
It depends on your mathematical knowledge so far and your experience with proofs and rigor.

Knowing about decimal expansions is very intimately connected with knowing precisely what a real number is. Thus it seems inevitable that you will have to learn in the future what a real number is exactly and how mathematicians construct the real numbers. You will have to learn how mathematicians deal with infinity.

The following books seem relevant to you:

1) How to prove it - Velleman
Details about what proofs are in mathematics, how to prove things yourselves, set theory, infinity, etc. Not very interesting topics compared to what follows, but necessary to be able to follow the rest.

2) The real numbers and real analysis - Bloch
One of the most detailed books on what numbers are. Constructs, N, Z, Q, R from scratch. Decimal expansions are defined and it is proven every number has one. In order to truly understand the issues you raised in this thread, this book is a must.

If you're interested in learning this just to satisfy your curiosity, please send me a private message, I might be able to help you further significantly.
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January 23rd, 2019, 01:19 PM   #79
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As a further addendum to the above, something which might interest sintan is that we now have all the necessary pieces to prove that $$0.\overline9 = 1$$ a fact which causes much controversy over the internet.

\begin{align}0.\overline9 = 0.999\ldots &= 0 + \frac9{10} + \frac9{100} + \frac9{1000} + \cdots \\ &= \frac9{10}\left(1 + \frac1{10} + \left(\frac1{10}\right)^2 + \left(\frac1{10}\right)^3 + \cdots \right) \\ &= \frac9{10} \left(\frac1{1 - \frac1{10}}\right) \\ &= \frac9{10} \left(\frac1{\frac9{10}}\right) \\ &= \frac9{10} \cdot \frac{10}9 \\ &= 1\end{align}

What is happening here is that there are two sequences, each represented by decimals, which both have the same limit. Indeed, every terminating decimal has two representations in this manner, one terminating, the other having the final digit decremented by 1 (more or less) and with an infinite tail of 9s appended. e.g. $$0.25 = 0.24999\ldots$$
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January 23rd, 2019, 03:09 PM   #80
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Every terminating decimal except 0.
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