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Integers, Pi, and Number Lines.
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ledtester:
Perhaps this is the angle the OP is trying to express:


--- Quote ---Natural numbers were created by God, everything else is the work of men — Kronecker (1823–1891).

--- End quote ---

Indeed, you could even create the natural numbers from the basic axioms of set theory. Once you have the natural numbers you can create the integers, then the rationals, the reals, complex numbers, etc.

From:

https://www.cantorsparadise.com/kronecker-god-and-the-integers-28269735a638
Someone:

--- Quote from: tooki on July 04, 2022, 01:44:32 pm ---Honey, you can’t just go around making up your own rules and definitions about mathematics.  :palm:
--- End quote ---
Seem to work for people in other fields on this forum ;) ... "no no, what I define that word to mean is XXXX"
Nominal Animal:

--- Quote from: Someone on July 05, 2022, 12:18:00 am ---
--- Quote from: tooki on July 04, 2022, 01:44:32 pm ---Honey, you can’t just go around making up your own rules and definitions about mathematics.  :palm:
--- End quote ---
Seem to work for people in other fields on this forum ;) ... "no no, what I define that word to mean is XXXX"

--- End quote ---
Yes, human communication is complex.

When you want to discuss mathematics or physics, you need to agree upon the terms used to refer to things.  That means there are two layers of definitions: there is the English (or human language) definition, and then there is the definition of the physical or mathematical theorem, quantity, object, or rule.  It is the latter that you cannot make up as you go; and the former you are allowed to redefine as you wish, as long as you use specific, detailed, unambiguous human language to do so.

For example, the metal called "Tungsten" in English, is "volframi" in Finnish.  "Addition" in English is "yhteenlasku" in Finnish, and so on.  No problem there.

Trouble happens, when terms are redefined without expressing the definition in precise language.  Graphics artists, for example, can have huge problem grasping a correct intuitive understanding about physical measurables like gravity and volume and especially weight versus mass, because they traditionally redefine these terms with their own definitions that are close enough to cause serious headaches, and have a hard time switching to a new set of definitions.  (And vice versa, I guess!)

This is also why you see myself mentioning things like "quaternions are multiplied using the Hamilton product", because in the context of quaternions, multiplication is a very specific operation with very specific rules, even though it is just "multiplication".  It is always useful to define the human-language terms one uses for mathematical and/or physical references, unless a single commonly accepted definition exists.

The same goes with acronyms: the first layer is the words the acronym expands to –– SPI can refer to Serial Peripheral Interface or to Service Provider Interface in programming, for example –; and the second layer is the thing those words refer to.  For example, if we agree that MOSI refers to Master Out, Slave In, we still need to be in agreement that Master is the one who provides the clock (SCL) line, and Slave is the other one.  Again, the former can be redefined as needed, but the latter must not, or confusion results.
ledtester:

--- Quote from: magic on July 04, 2022, 10:04:15 pm ---Two pages and still no one has pointed out OP's fundamental error:

Almost all real numbers cannot be described as a solution to any equation, and yet mathematicians will insist that they are there >:D

--- End quote ---

for the record, I did point this out early on:

https://www.eevblog.com/forum/chat/integers-pi-and-number-lines/msg4270255/#msg4270255


IanB:

--- Quote from: Peter Taylor 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.

--- End quote ---

Since I don't think anyone has really pointed this out yet, I will do so:

You talk about irrational numbers, but you overlook the rational numbers. If you remove all the irrational numbers from the number line, there is still an infinite number of rational numbers, that are not integers, left over.

For example, 1/7 is a rational number that is not an integer. You may protest that is is not exact, and cannot be evaluated. But on the contrary, if we evaluate 1/7 in base 7 arithmetic we get 1/10 (base 7) = 0.1 (base 7) exactly. So 1/7 is clearly exact, and can be evaluated, and is not an integer. The same is true of all the other (non-integer) rational numbers.

Furthermore, if you take any two non-integer, but rational, points on the number line, however close together, you can always find an exact, rational, number that lies between them. So you can zoom into the number line as much as you like, and still find numbers filling the space. Any time you think there might be a gap, you can find infinitely many exact numbers occurring inside that gap.
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