Author Topic: Integers, Pi, and Number Lines.  (Read 3088 times)

0 Members and 1 Guest are viewing this topic.

Offline Peter Taylor

  • Regular Contributor
  • *
  • Posts: 61
  • Country: au
    • Pete's Electronic Workshop
Integers, Pi, and Number Lines.
« on: June 29, 2022, 11:01:29 pm »
In classic education, we are taught that a number line contains an infinite number of irrational numbers between each integer. I will show that it should contain only integers.

dy / dx is never evaluated because dx is infinitely small and zero.

When working with Fourier Transforms for example, the terms are introduced into a formulae as a pair to help solve it, and then disappear at the end without ever being evaluated.

The imaginary number "square root -1" is never evaluated, but introduced into a formulae at the start to help solve it, and disappears at the end.

Pi also is not evaluated, but remains the ratio of the circumference of a circle to its radius.

"Square root 2" likewise is not evaluated, but remains a question: "What value multiplied by itself equals 2 ?".

These are termed irrational "numbers", having no exact value.

But these are not numbers, or values, but questions, or formulae.

If we remove all these non-numbers, or questions, or ratios, or any unevaluated formulae from our number line, we are left only with integer numbers.

So, a number line should contain only integer numbers.

QED.
« Last Edit: June 29, 2022, 11:21:35 pm by Peter Taylor »
 

Offline aeberbach

  • Regular Contributor
  • *
  • Posts: 69
Re: Integers, Pi, and Number Lines.
« Reply #1 on: June 29, 2022, 11:46:27 pm »
Similarly you can take an opaque tube and put a ball bearing in it, and seal the ends. Then because you can no longer know the exact position of the bearing, it is no longer there! (Ignore that rattling noise.) ;)
 
The following users thanked this post: tooki

Offline John B

  • Frequent Contributor
  • **
  • Posts: 552
  • Country: au
Re: Integers, Pi, and Number Lines.
« Reply #2 on: June 29, 2022, 11:49:24 pm »
These are termed irrational "numbers", having no exact value.

But they do have exact values, or positions on the number line (complex numbers excluded), just they can't be expressed as the ratio of 2 integers.
 
The following users thanked this post: mathsquid, tooki

Offline RJHayward

  • Super Contributor
  • ***
  • Posts: 1151
  • Country: us
Re: Integers, Pi, and Number Lines.
« Reply #3 on: June 30, 2022, 01:55:50 am »
Actually, 'Infinitely small' and 'zero' are not the same.
In CALCULUS mode, you need a really good school instructor.  You see; Calculus being a methodology, a new method of approach to arriving at usable numeric result.  It's NOT instinctual, and so it seems impractical.  For example, the mind-excercise that says things like; "dx goes to zero, as you repeat forever..."
But that is only one fragment.  With cleverness, it can be seen, that as that 'dx' portion moves a certain way, that another 'fragment', 'dy' also, in turn, is moving some characteristic way. 
   It's almost as nebulous a concept as saying, for example, one WIZARD shrinks (you) to half size, and then another WIZARD makes you 4X bigger, with end result; you are twice original size.  The little, localized exchanges make no real sense, but combined the two actions DO make a nice, clean result.
  A lackluster, dopey but competent MATH teacher might not be able to teach this essence.
  Now you've got me wondering about Issac Newton and his thoughts on inventing The Calculus...
 
The following users thanked this post: tooki

Online Someone

  • Super Contributor
  • ***
  • Posts: 3603
  • Country: au
    • send complaints here
Re: Integers, Pi, and Number Lines.
« Reply #4 on: June 30, 2022, 02:02:39 am »
Some nonsense to reply to the nonsense:

"Square root 2" likewise is not evaluated, but remains a question: "What value multiplied by itself equals 2 ?".

These are termed irrational "numbers", having no exact value.

But these are not numbers, or values, but questions, or formulae.
Only because people tend to think in units of integers, other problems are thought of in other units:
https://www.explainxkcd.com/wiki/index.php/1292:_Pi_vs._Tau
https://en.wikipedia.org/wiki/Radian_per_second
https://en.wikipedia.org/wiki/Normalized_frequency_(signal_processing)
Their ranges overlap on the continum of scalar values with integers, but are equally "nice/round" as integers. Engineering/physics is all about the units you choose to define the problem.
 

Offline SiliconWizard

  • Super Contributor
  • ***
  • Posts: 10164
  • Country: fr
Re: Integers, Pi, and Number Lines.
« Reply #5 on: June 30, 2022, 03:15:51 am »
In classic education, we are taught that a number line contains an infinite number of irrational numbers between each integer. I will show that it should contain only integers.

Why did you leave out the rational numbers though? They must be feeling lonely now.

But these are not numbers, or values, but questions, or formulae.

Irrational numbers can only be expressed indirectly indeed. Why would that mean they are not numbers? What's your definition of a number? Do you include infinity in your imaginary "number line"? What is infinity anyway?

And even rational numbers can only be expressed indirectly, if all you have are integers. I guess you don't see them as numbers either?

What are really integers anyway?

 ;D

Isn't it time for a beer?
 
The following users thanked this post: mathsquid, ebastler

Online ledtester

  • Super Contributor
  • ***
  • Posts: 2503
  • Country: us
Re: Integers, Pi, and Number Lines.
« Reply #6 on: June 30, 2022, 03:38:19 am »
...
"Square root 2" likewise is not evaluated, but remains a question: "What value multiplied by itself equals 2 ?".
...

Not every real number is describable. Even after you remove all describable numbers you are still left with an uncountably infinite number of numbers.

Here's a relevant blog post:

http://www.goodmath.org/blog/2014/05/26/you-cant-even-describe-most-numbers/

and here's one by someone who quit studying EE and instead turned to math when he came upon this revelation:

https://blog.ram.rachum.com/post/54747783932/indescribable-numbers-the-theorem-that-made-me
« Last Edit: June 30, 2022, 03:51:16 am by ledtester »
 
The following users thanked this post: newbrain

Offline hamster_nz

  • Super Contributor
  • ***
  • Posts: 2743
  • Country: nz
Re: Integers, Pi, and Number Lines.
« Reply #7 on: June 30, 2022, 03:47:50 am »
But if you view Pi as being a number line wrapped around a unit circle, then you can quite easily have Pi as "half way around"... and 2 Pi will take you back to where you started from.

Or are circles just "questions, or formulae" and need to be removed?
Gaze not into the abyss, lest you become recognized as an abyss domain expert, and they expect you keep gazing into the damn thing.
 
The following users thanked this post: Someone

Online Someone

  • Super Contributor
  • ***
  • Posts: 3603
  • Country: au
    • send complaints here
Re: Integers, Pi, and Number Lines.
« Reply #8 on: June 30, 2022, 07:34:31 am »
Or are circles just "questions, or formulae" and need to be removed?
Nah nah nah that's old 2d thinking, you need to start with spheres!
 

Online newbrain

  • Super Contributor
  • ***
  • Posts: 1439
  • Country: se
Re: Integers, Pi, and Number Lines.
« Reply #9 on: June 30, 2022, 11:34:18 am »
Not every real number is describable.
In addition to the links from ledtester, here's a simple video from Numberphile with Matt Parker:
Nandemo wa shiranai wa yo, shitteru koto dake.
 

Offline snarkysparky

  • Frequent Contributor
  • **
  • Posts: 364
  • Country: us
Re: Integers, Pi, and Number Lines.
« Reply #10 on: June 30, 2022, 12:06:41 pm »
pi has an exact value. No one will ever know what it is though.
 

Offline TimFox

  • Super Contributor
  • ***
  • Posts: 5047
  • Country: us
  • Retired, now restoring antique test equipment
Re: Integers, Pi, and Number Lines.
« Reply #11 on: June 30, 2022, 01:52:16 pm »
Remember:  "irrational" in this context means that it is not a ratio of integers, it does not imply insane or other English meanings of "irrational".
 

Offline xrunner

  • Super Contributor
  • ***
  • Posts: 6516
  • Country: us
  • hp>Agilent>Keysight>???
Re: Integers, Pi, and Number Lines.
« Reply #12 on: June 30, 2022, 02:10:58 pm »
Remember:  "irrational" in this context means that it is not a ratio of integers, it does not imply insane or other English meanings of "irrational".

And imaginary numbers are not "imaginary".  :)
[hp] Hewlett . Packard
 

Offline kosine

  • Regular Contributor
  • *
  • Posts: 153
  • Country: gb
Re: Integers, Pi, and Number Lines.
« Reply #13 on: June 30, 2022, 03:40:39 pm »
The ancient Greeks referred to "irrational" numbers as "alogos" (although other terms were used). A good translation would be "inexpressible" in terms of whole numbers. Many such quantities can be constructed geometrically, however, and were then called "magnitudes" to distinguish them from "arithmos" (numbers).

Gauss thought imaginary numbers should have been termed "lateral", with positive numbers being called "direct" and negative numbers "inverse".
 

Online Nominal Animal

  • Super Contributor
  • ***
  • Posts: 4114
  • Country: fi
    • My home page and email address
Re: Integers, Pi, and Number Lines.
« Reply #14 on: June 30, 2022, 06:55:54 pm »
Even though $$\varphi = \frac{1 + \sqrt{5}}{2}, \quad \psi = \frac{1 - \sqrt{5}}{2}$$ are irrational numbers, I can calculate the number \$F_n\$,
$$F_n = \frac{\varphi^n - \psi^n}{\varphi - \psi} = \frac{\varphi^n - \psi^n}{\sqrt{5}}$$
for any positive integer \$n\$ without ever calculating any kind of an approximation for \$\varphi\$ or \$\psi\$.

I bet you can, too, because the sequence \$F_0=0, F_1=1, F_2=1, F_3=2, F_4=5, \dots, F_n = F_{n-1} + F_{n-2}\$ is the sequence of Fibonacci numbers, OEIS A000045.

Thus, while you cannot specify the value of irrational numbers exactly, you can specify their behaviour and use, which makes them exceedingly useful as numbers.  Many other irrational numbers (like \$\pi\$, \$e\$ for example) have similar use cases.



Let's say we have a continuous function \$f(x)\$.

The slope of this function is \$f^\prime(x)\$:
$$f^\prime(x) = \frac{d ~ f(x)}{d ~ x} = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$
The right side does not mean that \$\frac{d f(x)}{d x} = \frac{f(x + h) - f(x)}{h}\$; you can't just ignore the limit operator \$\lim\$ like that!

What that right side actually means (being the definition of a derivative), is that the value of the expression at \$x\$ is determined by the behaviour of the ratio between the change in \$f\$ and the change in \$x\$, as the change in \$x\$ tends towards zero.

That is, we never, ever try to divide anything by zero there.  Instead, we can think of the process as looking at how the ratio changes as \$h\$ gets closer to zero, and extrapolate the value the ratio would be if we could calculate it at \$h = 0\$.

The power of calculus is that instead of having to do that rigmarole for every value of \$x\$ we're interested in – especially considering how careful one would need to be and how much work it would be to obtain the correct ratio! –, calculus tells us directly how we can obtain the slope as a function, \$f^\prime(x)\$, from the mathematical expression of \$f(x)\$ itself, with very little work.

For example, if we have $$f(x) = 3 \sin(x) + 2 \cos(3 x^2 - x + 2)$$calculus tells us that $$f^\prime(x) = \frac{d f(x)}{d x} = 3 \cos(x) + (2 - 12 x)\sin\left(3 x^2 - x + 2\right)$$

In short, because the limit operator tells us the behaviour of a function at a value where the function itself is not defined, there is absolutely no "division by zero".  And it is a powerful operator, because even in cases where we have something like \$\lim_{x \to c} \frac{f(x)}{g(x)}\$ with \$f(c) = g(c) = 0\$ (or \$\pm\infty\$) and both \$f(x)\$ and \$g(x)\$ continuous (or differentiable) around \$x = c\$ but perhaps not at exactly \$x = c\$, it can still yield the correct answer (via l'Hôpital's rule).



Simplifying all this hard work done by mathematicians over the centuries by claiming my number line is different is, well, pretty arrogant.  Does your number line work as well as the one we use right now does?
 
The following users thanked this post: jpanhalt, RJHayward

Offline mathsquid

  • Regular Contributor
  • *
  • Posts: 239
  • Country: us
  • I like math.
Re: Integers, Pi, and Number Lines.
« Reply #15 on: June 30, 2022, 08:40:22 pm »
None of this makes any sense.
 

Online tggzzz

  • Super Contributor
  • ***
  • Posts: 16127
  • Country: gb
    • Having fun doing more, with less
Re: Integers, Pi, and Number Lines.
« Reply #16 on: June 30, 2022, 08:58:40 pm »
None of this makes any sense.
K

All will become clear if you look at the other topic starter emitted by the OP.

TL:DR your statement is correct.
There are lies, damned lies, statistics - and ADC/DAC specs.
Glider pilot's aphorism: "there is no substitute for span". Retort: "There is a substitute: skill+imagination. But you can buy span".
Having fun doing more, with less
 
The following users thanked this post: mathsquid, langwadt, tooki

Offline TimFox

  • Super Contributor
  • ***
  • Posts: 5047
  • Country: us
  • Retired, now restoring antique test equipment
Re: Integers, Pi, and Number Lines.
« Reply #17 on: June 30, 2022, 09:00:48 pm »
19th century mathematics is something that happened to other people.
 

Offline eugene

  • Frequent Contributor
  • **
  • Posts: 428
  • Country: us
Re: Integers, Pi, and Number Lines.
« Reply #18 on: June 30, 2022, 09:09:17 pm »
In classic education, we are taught that a number line contains an infinite number of irrational numbers between each integer. I will show that it should contain only integers.

When I was taught about the number line in grade school we also learned about different number systems. Specifically real numbers, integers, and so-called natural numbers. Do recall that part of your classical education?

Quote
dy / dx is never evaluated because dx is infinitely small and zero.

Neither dy or dx are ever zero, but they do approach zero. In classical education the concept is introduced in the form of a limit.

Quote
When working with Fourier Transforms for example, the terms are introduced into a formulae as a pair to help solve it, and then disappear at the end without ever being evaluated.


I don't know what you mean by that. Nothing disappears without being evaluated. I assure you that Mr. Fourier was not a fool.

Quote
The imaginary number "square root -1" is never evaluated, but introduced into a formulae at the start to help solve it, and disappears at the end.

Again, I don't understand what you mean. If you're working with mathematical statements and surprised that terms have disappeared, then either you lack a basic understanding of the math or you're doing it wrong.

Quote
Pi also is not evaluated, but remains the ratio of the circumference of a circle to its radius.

I evaluate pi regularly.

Quote
"Square root 2" likewise is not evaluated, but remains a question: "What value multiplied by itself equals 2 ?".

Likewise, I evaluate sqrt(2) regularly. The fact that I can't write it down with a finite number of decimal digits is nothing more than a minor inconvenience.

Quote
These are termed irrational "numbers", having no exact value.

Yes they do! Who told you they don't have exact values? Don't trust that person to teach you math. They must be incurably stupid!

Quote
But these are not numbers, or values, but questions, or formulae.

No, they're numbers, just like 1 or 10.25. Again, don't trust the person that told you they aren't.

Quote
If we remove all these non-numbers, or questions, or ratios, or any unevaluated formulae from our number line, we are left only with integer numbers.

If we remove those numbers from the set of real numbers then we'll have a different set of numbers. You can ponder that set all you want, but I don't see how it will be useful to me. I need all the reals to get much of my work done.

Quote
So, a number line should contain only integer numbers.

Now we're back to the elementary school concepts of real numbers, integers, and natural numbers. Nobody is stopping you from working with only the set of integers. If you want to talk about it with someone you might find a third grade math teacher. That's where I learned this stuff.

EDIT: the campaign to stop people from driving while drunk is for the safety of others. Posting while high doesn't present a safety problem, but it does make you look like a fool.

So don't post while high.
« Last Edit: June 30, 2022, 09:17:12 pm by eugene »
90% of quoted statistics are fictional
 
The following users thanked this post: tooki

Offline SiliconWizard

  • Super Contributor
  • ***
  • Posts: 10164
  • Country: fr
Re: Integers, Pi, and Number Lines.
« Reply #19 on: June 30, 2022, 09:10:05 pm »
When you see that: https://equitablemath.org/
everything becomes possible.
 :-DD
 
The following users thanked this post: jpanhalt

Offline jpanhalt

  • Super Contributor
  • ***
  • Posts: 2154
  • Country: us
Re: Integers, Pi, and Number Lines.
« Reply #20 on: June 30, 2022, 09:19:39 pm »
@SiliconWizard

"DEI" is sweeping the known universe.   Thanks for the link.

I don't remember where I saw this first, but it certainly seems consistent with that scenario:
« Last Edit: June 30, 2022, 09:23:12 pm by jpanhalt »
 
The following users thanked this post: dunkemhigh, John B, SiliconWizard

Online Nominal Animal

  • Super Contributor
  • ***
  • Posts: 4114
  • Country: fi
    • My home page and email address
Re: Integers, Pi, and Number Lines.
« Reply #21 on: June 30, 2022, 10:01:15 pm »
None of this makes any sense.
:wtf: You! You.. ALT-numeric bigot, you!

Oops, sorry, my new-speak translator was accidentally left on.  What I was actually trying to say was, I think these threads are further proof of how little humans make sense in general...  Even in universities, the weight has shifted to the first syllable, uni.  All Shall Think And Speak As One, with as little information content as possible, so as to not offend anyones sensibilities.
 
The following users thanked this post: Circlotron

Offline RJHayward

  • Super Contributor
  • ***
  • Posts: 1151
  • Country: us
Re: Integers, Pi, and Number Lines.
« Reply #22 on: June 30, 2022, 10:15:17 pm »
Math Squid;
   It's sometimes hard to make sense, but Nominal Animal has best grasp of how 2 or more 'weird' things can still be resolved, if you can somehow postpone doing exact calc on the individuals, and, instead focus on the relationship between the two:
   Maybe try 3x Pi / 4x Pi. Quick glance, and just CANCEL the weirdos, resulting in concrete answer, without having to solve EVERY individual term.
 

Offline RJHayward

  • Super Contributor
  • ***
  • Posts: 1151
  • Country: us
Re: Integers, Pi, and Number Lines.
« Reply #23 on: June 30, 2022, 10:20:57 pm »
Maybe could install a 'CONFUSED' icon, along with 'thanks' and 'reporte de la Moderatique'!

   'Je suis Reporte de dis-raespect la familla'
(He takin bout your MAMA).
 
The following users thanked this post: tooki

Online tooki

  • Super Contributor
  • ***
  • Posts: 8626
  • Country: ch
Re: Integers, Pi, and Number Lines.
« Reply #24 on: July 01, 2022, 07:35:01 pm »
In classic education, we are taught that a number line contains an infinite number of irrational numbers between each integer. I will show that it should contain only integers.

dy / dx is never evaluated because dx is infinitely small and zero.

When working with Fourier Transforms for example, the terms are introduced into a formulae as a pair to help solve it, and then disappear at the end without ever being evaluated.

The imaginary number "square root -1" is never evaluated, but introduced into a formulae at the start to help solve it, and disappears at the end.

Pi also is not evaluated, but remains the ratio of the circumference of a circle to its radius.

"Square root 2" likewise is not evaluated, but remains a question: "What value multiplied by itself equals 2 ?".

These are termed irrational "numbers", having no exact value.

But these are not numbers, or values, but questions, or formulae.

If we remove all these non-numbers, or questions, or ratios, or any unevaluated formulae from our number line, we are left only with integer numbers.

So, a number line should contain only integer numbers.

QED.
Having your other thread about this closed by a mod wasn’t enough, eh?  :palm:
 


Share me

Digg  Facebook  SlashDot  Delicious  Technorati  Twitter  Google  Yahoo
Smf