Author Topic: Rational, Irrational, and Integer. The correct definition.  (Read 6486 times)

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Offline Peter TaylorTopic starter

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Rational, Irrational, and Integer. The correct definition.
« on: December 21, 2022, 03:43:02 am »
This is the new and correct definition of a number.
1/ A number must be an integer.
2/ The term rational and irrational don't apply to a number by definition 1/.

An integer is any number that can be defined wholly. For instance, 7, and 1 / 7 can be defined wholly and are therefor integers.
An equation that cannot be defined wholly, such as root 2, where we don't know what number multiplied by itself equals 2, remains a question, and is not a number by definition 1/.
A number can't be rational or irrational. These terms have no meaning.

Thankyou.  :D
 

Offline TimFox

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #1 on: December 21, 2022, 03:48:08 am »
No.
 
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Offline gnuarm

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #2 on: December 21, 2022, 03:54:39 am »
Why are you making up your own math?  I'm not following what you are trying to do.
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Online SiliconWizard

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #3 on: December 21, 2022, 04:01:29 am »
Is this a philosophical issue you have with non-integer numbers (and well, why not, although that's a bit odd, but a deep philosophical approach about this could be interesting, maybe), or is it because you have a problem fully understanding them?

If that's the former, you may want to elaborate (although it's not guaranteed to get much traction.) Otherwise that's just your own definition with nothing to back it up except itself, which, you'll have to admit, is a bit circular.
 

Offline gnuarm

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #4 on: December 21, 2022, 04:17:44 am »
I know there are different levels of infinity.  There are an infinite number of integer numbers.  You can pick two integers so there are not integers between them.  Then, between these two integers, there are an infinite number of rational numbers. 

Here's where it gets tricky.  There are an infinite number of irrational numbers between any two rational numbers.  But can you select a pair of rational numbers, that have no other irrational numbers between them? 

Clearly, integers and rational numbers are different sizes of infinity.  But I'm not sure that is true for rational and irrational numbers. 

Don't even get me started about transcendental numbers.
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Offline Tomorokoshi

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #5 on: December 21, 2022, 04:33:00 am »
All numbers are imaginary anyway.
 
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Online ledtester

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #6 on: December 21, 2022, 04:37:02 am »
I know there are different levels of infinity.  There are an infinite number of integer numbers.  You can pick two integers so there are not integers between them.  Then, between these two integers, there are an infinite number of rational numbers. 

Here's where it gets tricky.  There are an infinite number of irrational numbers between any two rational numbers.  But can you select a pair of rational numbers, that have no other irrational numbers between them? 

Clearly, integers and rational numbers are different sizes of infinity.  But I'm not sure that is true for rational and irrational numbers. 

Don't even get me started about transcendental numbers.

The cardinality of the integers and rationals are the same. They are both "infinite" but the size of the infinity is the same. The cardinality of the irrational numbers is strictly larger according to how mathematicians define it.

Between two rational numbers there are an infinite number of irrational numbers.

Between two rational numbers there is always another rational number, e.g.: (a+b)/2.

 
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Offline IanB

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #7 on: December 21, 2022, 04:42:38 am »
This is a rehash of an old topic. Let's not go there again.

https://www.eevblog.com/forum/chat/integers-pi-and-number-lines/
 
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Online magic

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #8 on: December 21, 2022, 10:12:18 am »
Is this a philosophical issue you have with non-integer numbers (and well, why not, although that's a bit odd, but a deep philosophical approach about this could be interesting, maybe), or is it because you have a problem fully understanding them?
Hardly anyone who hasn't done at least a BS in maths understands what the irrationals are so it's the latter obviously :P

And yeah, Deja vu ::)
 

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Offline pcprogrammer

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #10 on: December 21, 2022, 11:46:23 am »
Well in the other thread Peter Taylor does give a hint of where he lives  8)

In Lala Land  :-DD

And that was already clear with his superb website that he made  :palm:

Offline TimFox

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #11 on: December 21, 2022, 02:43:54 pm »
Mathematics is something that happened to other people.
 
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Offline rsjsouza

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #12 on: December 21, 2022, 02:52:14 pm »
Why are you making up your own math?  I'm not following what you are trying to do.
It is to "decolonize math", as these vicious numbers are transphobes since they don't believe in trangenderism and categorize themselves in binary terms (rational/irrational). Burn the books!  :-DD
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Oh, the "whys" of the datasheets... The information is there not to be an axiomatic truth, but instead each speck of data must be slowly inhaled while carefully performing a deep search inside oneself to find the true metaphysical sense...
 
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Online Nominal Animal

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #13 on: December 21, 2022, 04:18:47 pm »
I've always found it comfortable how the classification of numbers could be used as a tool.  They are not arbitrary, and they are useful.

You start with integers, then rationals, and then irrationals.  You extend these into complex numbers, and you'll find integers and rationals and irrationals have interesting features.  For example, consider \$e^{-2 i \pi x}\$, where \$x \in \mathbb{R}\$, and what it means for \$x\$ to be integer, rational, or irrational.

You start with integers, and extend these into vectors, and you get regular rectangular lattices.  Integers correspond to lattice points, rationals to lines that cross an infinite number of lattice points, and irrationals to lines that intersect with a single lattice point.  Examine the properties of such lattices, extending to non-rectangular lattices (say, face-centric cubic, body-centric cubic, hexagonal, and so on), and you've arrived at crystallography.

From rationals, you can represent both the numerator and the denominator with a polynomial, and you arrive at rational polynomials, which have a lot of practical applications.

You start with integers, think about rationals and divisibility, and arrive at prime numbers, prime number decomposition, and all sorts of discrete math.

You combine the idea of irrational numbers, prime numbers, and rational polynomials, and you arrive at ideas like elliptic polynomials, which can lead you to elliptic cryptography.

Everything else like discrete logarithms, factorization, and so on, used everyday in cryptography and computers and networks, is extensible from the first principles like this, and easily understood at the conceptual level, even though the actual mathematical machinery may be hard to fully understand and utilize. The space of such concepts has an infinite number (hah!) of dimensions, with each conceptual approach their own dimension, and the elements along that dimension are the concepts in which it is useful to divide the number or value into.  For scalars, it is integers, rationals, and reals.  For complex numbers, you have unity (\$\lvert z \rvert = 1\$, i.e. \$z = e^{i x}\$, \$x \in \mathbb{R}\$), roots of unity, pure reals, pure imaginary numbers, et cetera.

Instead of a vast continuous infinite-dimensional space of concepts we'd be hopelessly lost in, these give us reasonable and trustworthy stepping stones we can walk along.  They may not lead to everywhere yet, but you can always build new ones to reach places not yet reached.

The only reason you'd want to redefine these concepts is if you insist on sitting at the singular starting point, with your eyes closed, hoping for an universe that was less complicated so you could fool yourself into believing that you understood it all, fully.  So foolish.
 
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Offline T3sl4co1l

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #14 on: December 21, 2022, 04:36:23 pm »
Why are you making up your own math?  I'm not following what you are trying to do.
It is to "decolonize math", as these vicious numbers are transphobes since they don't believe in trangenderism and categorize themselves in binary terms (rational/irrational). Burn the books!  :-DD

Got just a little chip on your shoulder there?

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Offline TimFox

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #15 on: December 21, 2022, 04:39:44 pm »
Is this a philosophical issue you have with non-integer numbers (and well, why not, although that's a bit odd, but a deep philosophical approach about this could be interesting, maybe), or is it because you have a problem fully understanding them?
Hardly anyone who hasn't done at least a BS in maths understands what the irrationals are so it's the latter obviously :P

And yeah, Deja vu ::)

I learned about irrational numbers in high-school mathematics at a US public school.
 
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Offline eugene

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #16 on: December 21, 2022, 05:57:51 pm »
This is the new and correct definition of a number.
1/ A number must be an integer.
2/ The term rational and irrational don't apply to a number by definition 1/.

An integer is any number that can be defined wholly. For instance, 7, and 1 / 7 can be defined wholly and are therefor integers.
An equation that cannot be defined wholly, such as root 2, where we don't know what number multiplied by itself equals 2, remains a question, and is not a number by definition 1/.
A number can't be rational or irrational. These terms have no meaning.

Thankyou.  :D

I doubt anyone would object to you inventing your own, private version of math. I also doubt that very few people would be interested, but I'm pretty sure nobody would object.

What lots of people object to is redefining terms that are already well defined in the old (and still correct) math. So, my advice, if you hope to get any reaction other than negative, is to stay away from existing terms like "integer", "rational", and "irrational" and coin your own NEW terms to go along with your NEW math. You don't always need to tear down something old to build something new.
90% of quoted statistics are fictional
 
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Online magic

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #17 on: December 21, 2022, 06:06:40 pm »
I learned about irrational numbers in high-school mathematics at a US public school.
Cool.

So I gather that the naturals are an extension of counting fingers beyond the actual number of fingers you might have, the integers are the naturals mirrored around zero, the rationals are fractions of the integers, and complex numbers are what happens when you imagine that sqrt(-1) is a thing.

So what the hell are the irrationals, then?
Something about continuous divisions and stuff?
But you have heard about those "atoms" and "quantum" stuff, right?
There is nothing real about real numbers.

The theory that they are an inside joke of heterosexual European men invented to confuse Americans doesn't even seem so far off :P
 

Offline TimFox

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #18 on: December 21, 2022, 06:08:34 pm »
"rationals" are ratios of integers
"irrationals" are not ratios of integers
 

Online magic

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #19 on: December 21, 2022, 06:10:20 pm »
"irrationals" are not ratios of integers
This definition is not a ratio of integers either ;)
 

Offline TimFox

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #20 on: December 21, 2022, 06:11:49 pm »
Take it up with the Encyclopedia Britannica
https://www.britannica.com/science/irrational-number
 

Online magic

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #21 on: December 21, 2022, 07:01:48 pm »
Britannica
Neat, let's see if we need to further restrict the Privileged Boys Club to continental Europe only :box:

Quote
irrational number, any real number that cannot be expressed as the quotient of two integers
OK, this definition is not an irrational number itself, but let's see the reals...

Quote
real number, in mathematics, a quantity that can be expressed as an infinite decimal expansion
So say I have a sequence of infinitely many zeros, followed by seven. Like, 0.(0)7.
What sort of real number is that? >:D
 

Online Nominal Animal

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #22 on: December 21, 2022, 07:06:19 pm »
In my opinion, complex numbers are between scalars and vectors.  (There are others there too, like quaternions.)

The \$i^2 = -1\$ thing is just a "mathematical" way of expressing the idea of rotation by 90°.  If you think of the one-dimensional number line concept, then such a rotation by itself doesn't make any sense, but if applied twice, you mirror the number line.  And that's exactly what happens.
The math boffins examined the concept for a bit, and realized that it also naturally aligned with the exponential function, so that \$e^{a + i b}\$ corresponds to a scaling (multiplication) by \$e^a\$, and rotation by \$e^{i b}\$.  Everything else stems from there.

So, while the "imaginary number" concept doesn't make much sense for real numbers (integers, rationals, irrationals), it does work as a stepping stone for extensions to more complex concepts.

In certain ways, complex numbers remind me of homogenous coordinates. Cartesian coordinates \$(x_1, \dots, x_N)\$ are represented by \$(X x_1, \dots, X x_N, X)\$ in homogenous coordinates, and somewhat similarly to complex numbers, homogenous coordinates can be used to express both translation and rotation (full transform) in a single operation.  (In fact, the vast majority of computer 3D graphics libraries use homogenous coordinates at least internally for transformations.)  A particularly interesting facet of homogenous coordinates is that they let one trivially distinguish between position vectors (\$(x_1, \dots, x_N, 1)\$) and direction vectors (\$(x_1, \dots, x_N, 0)\$).  Mathematically, there is no reason to distinguish between the two, but in 3D visualization, the ability to easily do so while applying the exact same transformations (with translation only affecting position vectors and not direction vectors) simplifies things quite a bit.

So what the hell are the irrationals, then?
Any number that you cannot express as the ratio of two integers (or an extension of the same concept for e.g. complex numbers).

A step further are transcendental numbers.  They are numbers you cannot obtain as a root of a finite-degree nonzero polynomial with rational coefficients, like \$\pi\$ and \$e\$.

But you have heard about those "atoms" and "quantum" stuff, right?
There is nothing real about real numbers.
The two 'real' mean completely different things: one is 'physically real', and the other is 'the set of numbers \$\mathbb{R}\$ that we call "reals"'.

The theory that they are an inside joke of heterosexual European men invented to confuse Americans doesn't even seem so far off :P
Except they weren't initially European and never even "mostly" European, men, or even heterosexual.  That's what makes "decolonizing math" so ridiculous, and extremely racist and bigoted concept.

In particular, Indians and Arabs are quite, quite offended by your suggestion, and for good reason.  Sthananga Sutra is about 2400 years old, and covers things like fractions and elementary number theory.  In the 800s Arabs made huge leaps forward in maths, especially people like Muhammad ibn Musa al-Khwarizmi.
« Last Edit: December 21, 2022, 07:08:55 pm by Nominal Animal »
 
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Online magic

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #23 on: December 21, 2022, 07:59:00 pm »
So what the hell are the irrationals, then?
Any number that you cannot express as the ratio of two integers (or an extension of the same concept for e.g. complex numbers).
Can the number of all real numbers be expressed as a ratio of two integers, or is it an irrational number? ;)

The two 'real' mean completely different things: one is 'physically real', and the other is 'the set of numbers \$\mathbb{R}\$ that we call "reals"'.
Yes, and hardly anyone knows what the latter actually are, that's my point which started this whole exchange.
All I hear is "irrationals are not rational this", "reals are not imaginary that".
So what they are, if they are not all those other things?

The theory that they are an inside joke of heterosexual European men invented to confuse Americans doesn't even seem so far off :P
Except they weren't initially European and never even "mostly" European, men, or even heterosexual.  That's what makes "decolonizing math" so ridiculous, and extremely racist and bigoted concept.
No, it's precisely true :D
People of Color invented Mathematics.
Heterosexual European Males came up with the formal concept of "Real Numbers".
Heterosexual European Males teach that their "Real Numbers" are no more, no less, but all numbers that are real.
Reality is a social construct now.
This is Colonization.
« Last Edit: December 21, 2022, 08:01:27 pm by magic »
 

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #24 on: December 21, 2022, 08:29:47 pm »
Good lord. :-DD
 


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