First positive Real number = 1/∞ ?

[T0M]

Can anybody with the correct education agree or disagree ? Is that a fact in Maths or not ?

Let A be the first positive Real number, then:

A = 1 / ∞

T0M

[C1ay]

Originally Posted by T0M

Can anybody with the correct education agree or disagree ? Is that a fact in Maths or not ?

Let A be the first positive Real number, then:

A = 1 / ∞

T0M

See infinity from MathWorld. 1 / ∞ is undefined but the limit of 1/x as x approaches infinity is 0.

[T0M]

Originally Posted by C1ay

See infinity from MathWorld. 1 / ∞ is undefined but the limit of 1/x as x approaches infinity is 0.

Thank you. So they're wrong!
Saying 1 / ∞ = 0 is rounding!

T0M

[kamil]

Originally Posted by T0M

Thank you. So they're wrong!
Saying 1 / ∞ = 0 is rounding!

T0M

It just shows that having a function 1/x=y. and if this function was to intercept the x axis (x=0) then y would have to be infinite.

x=0.5 y=2

x=0.25 y=4

x=0.125 y=8

x=0.00001 y=100000

As the value of x proportionaly decreases the value y proportionaly increases.
But in order to keep on halfing 0.125 til u get to zero u would have to devide it an infinite amount of times.......ahhaaa.... infinte. Therefore:

1/0=infinite

This is hard to imagine just like its hard to imagine an infinite amount of apples. Thats why we use the limit concept.

[kamil]

The point in the number line next to zero? This doesnt make any sense. Because there is always a distance between two points. But in a case where there isnt a distance between two points (such as this one I assume) then the distance between them would be zero. If we add on a distance of 0.4 to teh point 1.5 we will get the point 1.9. But if we get the point zero, and add on a distance of zero, we will get zero.
0.000....000001 is acctually equal to zero, just like 9.9999.....=1.

PS: How do you type in the infinite sign?

[C1ay]

Originally Posted by T0M

Thank you. So they're wrong!
Saying 1 / ∞ = 0 is rounding!

T0M

They are not rounding. Have you studied limits in first year calculus yet?

[T0M]

Originally Posted by C1ay

They are not rounding. Have you studied limits in first year calculus yet?

Well I gotta thank you again for forcing me to research on "limits" (which you probably guessed I didn't know since I kept avoiding to mention them ).
So, which I understood: "limits" is a method by which one can simplify when needing to avoid infinite in an equation by pushing "infinite" to the "limit". Just like rounding! (oh wait a moment! It IS rounding!).
I find that cool, for it simplifies, thus it's logic. But only as a method. I found in my research that "limits" as a theory, is categorized in a branch of Mathematics named "Generalized abstract nonsense". Nonsense. And I agree!

P.S.: Have you noticed the 1 in your nickname, as the 0 in mine? We just need a -1!

Originally Posted by kamil

The point in the number line next to zero? This doesnt make any sense. Because there is always a distance between two points. But in a case where there isnt a distance between two points (such as this one I assume) then the distance between them would be zero. If we add on a distance of 0.4 to teh point 1.5 we will get the point 1.9. But if we get the point zero, and add on a distance of zero, we will get zero.
0.000....000001 is acctually equal to zero, just like 9.9999.....=1.

PS: How do you type in the infinite sign?

The problem is that you are trying to give ∞ a value. Therefore assuming a value to that first Real number, which we know it hasn't (other than 1/∞).
So, let's call A the "first" of the Real numbers (as we call ∞ the "last") then:

A = 1 / ∞

And of course:

∞ = 1 / A

This means, in your words (and talking about maths):
There's always a distance between two points.
There's always another point in the distance.

P.S.: I just copy ∞ from Windows' Character Map. I'll let you know as soon as I know how to type it.

[CraigD]

Originally Posted by T0M

… Let A be the first positive Real number …

Before proceeding to consider the rest of this proposition, let’s examine the terms used, and see if the statement is meaningful.

The term “first” has a pretty precise mathematical meaning. In a finite or infinite countable set, it is the element mapped to the Natural number 1. Note that, since there are many possible schemes (algorithms) for mapping the Natural numbers (1, 2, 3 …) to a given countable set, the precise meaning of the term “fists” is dependent on the scheme. Let’s not worry about this potential ambiguity for now, though, but look instead at some popular mappings of the set of Natural numbers to some well-known number sets.

The Integers (… -2, -1, 0, 1, 2, 3 …) can be mapped to the Naturals with this scheme for generating an integer (Z) from 2 Naturals (W1 and W2):
Z = W1 - W2.
It’s reasonable, then, to assert that the first Integer is generated by 2 instances of the first Natural, that is:
0 = 1-1,
and that the first positive Integer is:
1 = 2-1.

A Rational number (Q) can be generated from an Integer (Z) and a Natural (W) by:
Q = Z / W,
so by similar reasoning, the first Rational is
0 = 0 / 1,
and the first positive Rational is:
1 = 1 / 1.

This technique can be applied to many number sets, giving many reasonable “first” numbers in each set.

We run into trouble, however, when trying to apply it to generate a good candidate for the first Real number. There’s no simple scheme for generating Reals from Naturals, or numbers that can be generated from Naturals, an accepted fact for which many proofs exists, one of the most famous being 19th century Math superstar Cantor’s ”diagonal argument”. For this reason, the set of Real numbers is called “uncountable”.

Since there’s no scheme to map the Reals to the Naturals, the phrase “the first positive Real number” seems to describe a thing too poorly defined to be mathematically useful. Introducing the infinity symbol (∞) doesn’t get rid of this underlying trouble, but seems to lead back to the confusion being discussed in Infinite = 1.

[CraigD]

Originally Posted by T0M

…

…P.S.: I just copy ∞ from Windows' Character Map. I'll let you know as soon as I know how to type it.

A pragmatic, off topic note: You can type ∞ by typing Alt+8734 using your numeric keypad.

To type a character from the Windows Character Map:
1) make a note of its Unicode (for ∞, U+221E)
2) convert the hexadecimal part of the code to base 10 (221E16 = 873410)
3) Press and hold the Alt key
4) Using the numeric pad keypad part of your keyboard, type the 4-digit decimal number you got in step 2 (if the number is less than 1000, bad with 0s, eg: A = 0065)
5) Release the Alt key.

This works well (in Windows apps that support Unicode) if you routinely use a few unusual characters, and are able to memorize their codes.

When posting in html, you can specify special characters by their ISO character code inside the html escaping characters. ∞ would give you ∞, if Hypography allowed html use, which, to the best of my knowledge, it doesn’t. You can find all the ISO characters at www.w3.org, though only a subset of them work in most browsers

[CraigD]

Originally Posted by T0M

…

…P.S.: I just copy ∞ from Windows' Character Map. I'll let you know as soon as I know how to type it.

A pragmatic, off topic note: You can type ∞ by typing Alt+8734 using your numeric keypad.

To type a character from the Windows Character Map:
1) make a note of its Unicode (for ∞, U+221E)
2) convert the hexadecimal part of the code to base 10 (221E16 = 873410)
3) Press and hold the Alt key
4) Using the numeric pad keypad part of your keyboard, type the 4-digit decimal number you got in step 2 (if the number is less than 1000, bad with 0s, eg: A = 0065)
5) Release the Alt key.

This works well (in Windows apps that support Unicode) if you routinely use a few unusual characters, and are able to memorize their codes.

When posting in html, you can specify special characters by their ISO character code inside the html escaping characters. ∞ would give you ∞, if Hypography allowed html use, which, to the best of my knowledge, it doesn’t. You can find all the ISO characters at www.w3.org, though only a subset of them work in most browsers