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By Intelligitimate
#658040
Between any two real numbers, there are an infinite set of numbers. Between 2 and 3, there is 2.5, 2.55, 2.856, 2.67310467109470491, etc. However, between 1 and 0.999... there are no numbers whatsoever. There is not a number, even infinitely small, that comes between these two numbers. Hence, they are the same number.
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By Vivisekt
#658041
I've already said that on the first page, Intelligitimate.
By Cap
#658101
Numbers don't exist. We created them - they do what we define them to do.


Oh they do exist independent of man. Numbers do not just "do what we define them to do"... they appear to, but we can be wrong, theories are overturned all the time. Mathematics evolves, we realize that we were wrong. For all we know 2+2=4 could be wrong if someone proves it so. You are arrogant to assume that all current mathematics is "right", just as all previous mathematicians presumed about their ideas. 100 years from now someone could prove that all calculus is a pile of horse shit.

Doesn't the limit of something mean something which it approaches, but never actually reaches? By definition? Yeah yeah, we can calcuate it, and use it practically, and it's all good, but that doesn't mean it actually reached the limit, because that is impossible, no? What is a limit then?

The "distance" between the two numbers only approaches zero. It's asymptotic.

edit:

Schrödinger's Kitty, thanks for the recommendations.

The only one I've read is:
Zero: The Biography of a Dangerous Idea by Charles Seife
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By Andres
#658165
Mathematics evolves, we realize that we were wrong

Mathematics does not evolve, it grows. Unlike in physics, where a new theory displaces (and enlarges the area of applicability), new theories in mathematics do not displace old ones. The older ones are just as valid as before. Physics can be wrong, mathematics not (although that does not preclude a wrong proof).

For all we know 2+2=4 could be wrong if someone proves it so.

Again, no. 2+2 will always equal 4. The definitions are such that this comes out.
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By Vivisekt
#658186
DZ wrote:Oh they do exist independent of man. Numbers do not just "do what we define them to do"... they appear to, but we can be wrong, theories are overturned all the time. Mathematics evolves, we realize that we were wrong. For all we know 2+2=4 could be wrong if someone proves it so. You are arrogant to assume that all current mathematics is "right", just as all previous mathematicians presumed about their ideas. 100 years from now someone could prove that all calculus is a pile of horse shit.

What? I'm sorry for the confusion, DZ, but you a) don't seem to really know what you're talking about and consequently b) don't seem to understand what I said in my reply in the first place. Our modern Calculus is based entirely upon how we want to define and understand real numbers (dedekind cuts atm), which is, at the very fundamentals, an arbitrary thing - based upon the observations we feel like categorizing thusly. There are assumptions (logical, sound, but assumed nonetheless) at the core of mathematics and, like I said, there aren't actually numbers in nature. There is no math in nature. Numbers are just symbols that we use to artificially group and represent the transient and transcendent physical forces that we can observe; this allows the human recognition, classification, and prediction thereof.

We simply pattern the logic (re: theory) by which we manipulate these arbitrary creations (re: mathematics) after the interactions of these forces as best we can - keeping what seems to work, and marginalizing what does not. What we are right or wrong about where mathematics are concerned is simply how our creation (re: math and numbers) applies to the forces in nature - which, again, aren't numbers or mathematics. Calculus, therefore, functions and is navigable as we define it to be. There is no system floating in the aether - there are only physical properties, trends, actions, reactions, and a host of unknown. And even saying that may not be entirely accurate, because that is just how we quantify what we observe from our linear, physical perspective.

Math is to Nature as Language is to Nature. If I point to a shovel and say "this is a shovel", it isn't as if the word shovel or my understanding of "what a shovel is or does" is somehow intrinsic within the object. The letters come together to form the word "shovel" and then relate to that object only because the man-made structure of English specifies that it does. "S" corresponds to its sound because someone said that it does. Saying "Sh" together makes a specific blending, because someone said that it should. Math, as a system, is the same. The data is taken from the physical environment, but the system operates only as we have constructed it to operate. We make it mimic the physical world in a way that we can sort - it is not actually the physical world.

And I really wish people would stop using that stupid 2+2=4 example. You can make 2+2 equal ElephantAndCheese if you feel like it. It isn't some tangible piece of the universe, it is merely a component of the logic that we have fabricated and use to interpret and describe the physical universe that we can observe. It isn't actually out there somewhere - it is in our heads. You don't prove it wrong anymore than you prove a screwdriver wrong. It's a tool.


DZ wrote:What is a limit then?

Let us say that an open interval; (g,h); contains a number; x'0.
Let us also say that a F(x) is defined in (g,h).
A limit (L) thereof is the val, derived from the surrounding/nearby xvals, that F(x) would return when our x'0 equals that x.
F(x) can be made to go as close to L as one feels like putting it without actually encountering x'0 - there becomes no applicable mathematical difference between the two. And blah blah blah, I'm not going to go into detail about the epsilon/deltas.

This relates to our example in a specific way. Limits are used in the formal proof of the relationship, for reasons of uniformity, but that is not even necessary. Some people can't seem to understand that we aren't simply talking about some {.99...} infinite sum, pulled out of the air. What you see here is merely an imperfection in decimal notation - a representation - at which we arrive by dividing the whole 9 by itself. The infinite number {.99...} is derived entirely from an operation that already equals 1 (re: not sum infinity). Thus, as we try to move backwards from 1 into {.99...}, we discover that we always remain at 1 because there is infact no difference between the two. There can be no distance between infinity and a 1 that lies at the 'end' of it. It isn't an infinitely small distance, it is NO DISTANCE. It's impossible. They MUST BE representative of the same number, according to the logic of Calculus. This makes them the same number. That's all there is to it, and debating the point in this instance is purely philosophical (re: not mathematical).
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By arcis
#658199
The definitions are such that this comes out.


b x a = a x b

b times a = a times b
a + a + a = b + b

For example: a = 2, b = 3
2 + 2 + 2 = 3 + 3

Now set a = 0, b = 3

0 + 0 + 0 = ??

Three times nothing is nothing.
But what is and how to write "zero times 3"?
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By Vivisekt
#658207
arcis wrote:But what is [...] "zero times 3"?

The Number Zero multiplied by the Number Three...


arcis wrote:and how to write "zero times 3"?

0 * 3?


I'm not sure that I understand your question.
User avatar
By arcis
#658217
a)
a + a + a = b + b

b)
Set a = 2 and b = 3
2 + 2 + 2 = 3 + 3

c)
Now set a = 0 and b = 3
0 + 0 + 0 = ?

In b) you have on the right side of the equation only the numbers three and the + sign.

So what do you have to write instead of the ? in c) so that it matches the formula in a)
which is the "commutativ law" (I am not sure, if this is the correct expression in english for [a x b = b x a] or [a + b = b + a] )
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By Vivisekt
#658232
We call (a * b = b * a) the Commutative Property of Multiplication.


But, a statement such as this:
arcis wrote:a + a + a = b + b
Depends entirely upon what the values of these variables are. That's not part of the Commutative Property. Perhaps you're confusing it with the Distributive Property [ a(b + c) = a * b + a * c ], or something?



arcis wrote:Set a = 2 and b = 3
2 + 2 + 2 = 3 + 3

Now set a = 0 and b = 3
0 + 0 + 0 = ?

IF a = 2 AND b = 3 THEN
a + a + a = b + b is TRUE.

However. IF a = 0 and b = 3 THEN
a + a + a = b + b is FALSE.
a + a + a ≠ b + b
By Cap
#658242
What? I'm sorry for the confusion, DZ, but you a) don't seem to really know what you're talking about and


No, I don't, that's why I was going by intuition, asking for explanations, definition of a limit, etc.

consequently b) don't seem to understand what I said in my reply in the first place.


No, I didn't, because you didn't really expand upon it. I was still thinking in the mindset that numbers are somehow intrinsic in nature, hence my reply due to lack of understanding.

Thank you for elaborating so eloquently in your last post. I can now see what you meant more clearly. Indeed, the decimal system is a whore. :hmm:
Last edited by Cap on 09 Jun 2005 20:19, edited 1 time in total.
User avatar
By arcis
#658244
Next try:

a)
a x b = b x a

what can be expanded to

b)
a + a + a + a ..... + a = b + b + b + .... + b

"b" pieces of "a" are the same as "a" pieces of "b" (2 x 3 = 3 x 2)

Now the question is, how to write the right side of b), if a is zero?

But you are only allowed to use "b"`s and the "+" sign due to formula b)
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By Rust
#658252
Simple:

3 * 0 = "Sum '0' three times" = 0 + 0 + 0

0 * 3 = ""Sum '3' zero times" = 0

0 = 0


If you ask me to write, "3", zero times, then I cannot write it (the "3"), can I?

Since I cannot write anything, I write "nothing". '0' represents "nothing".
Last edited by Rust on 09 Jun 2005 20:41, edited 1 time in total.
User avatar
By arcis
#658272
4 x 3 = 3 x 4
3 + 3 + 3 + 3 = 4 + 4 + 4

2 x 3 = 3 x 2
3 + 3 = 2 + 2 + 2

but:

4 x 0 = 0 x 4
0 + 0 + 0 + 0 = 0 ?

Do you see the difference? I have troubles to imagine what happens if I should sum the 4 zero times.
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By Rust
#658277
Do you see the difference? I have troubles to imagine what happens if I should sum the 4 zero times


That is the point. You can't sum it, any times.

Again, if I ask you to write "3", zero times, then I'm asking you to not write it; to not write down anything!

'0' in this case represents the "nothing".

So if you ask me, how do I represent 3 x 0, then the answer is, with a '0'.

thus:

3 x 0 = 0 x 3
0 + 0 + 0 = 0
0 = 0

The property holds.
User avatar
By arcis
#658281
The property holds.


But it doesn`t match with this formula in the case of one number is zero.

a x b = b x a

a + a + a + a ..... + a = b + b + b + .... + b

On the left side only a`s and on the right side only b´s
By Cap
#658295
But if A is 0, then on the right side there are 0 Bs.
User avatar
By Rust
#658298
But it doesn`t match with this formula in the case of one number is zero.

a x b = b x a

a + a + a + a ..... + a = b + b + b + .... + b

On the left side only a`s and on the right side only b´s


No. What does the property say? To use your example:

2 x 3 = 3 x 2

2 + 2 + 2 = 3 + 3

In general terms, a x b = a + a + a ... 'b' times.

Lets put that into words:

"Sum the first term, 'a', the number of times given by the second term 'b'.

So now lets do 0 x 3 :

"Sum '0', three times" : 0 + 0 + 0

Lets do 3 x 0 :

"Sum '3', zero times": 0


--------------

The problem is that you're taking the property (a x b = b x a) and putting a "requirement" to it which is not true. Nowhere does the property, in and of itself, state or require that the result include the first term.

That's the equivalent of asking you to write down the word "hello" without using the letter 'l'. That you cannot does not mean there is something wrong the the word "hello", the letter "l" or our language; it means you're putting an unnecessary and impossible requirement.
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By arcis
#658305
0 Bs.


Sure.
But you can not write down the result due to the following formula.

a x b = b x a
a + a + a + a ..... + a = b + b + .... + b

Which was my original question.
User avatar
By Vivisekt
#658307
arcis wrote:But you can not write down the result due to the following formula.

a x b = b x a
a + a + a + a ..... + a = b + b + .... + b

That isn't 'a formula'.

(a * b = b * a) is the Communitive Property of Multiplication.

(a + b = b + a) is the Commutative Property of Addition.


1) They are not mashed together like that.
2) Formulas always retain the same number of variables. So, really, what you're forwarding as a 'second part' of the CPM doesn't even make sense in any sort of formulaic way.

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