October 17th, 2014, 03:54 AM  #21 
Senior Member Joined: Jul 2014 From: भारत Posts: 1,178 Thanks: 230  There was someone who introduced money in India and he did bring the concept of 25 paise. But is it still used today ?

October 17th, 2014, 04:45 AM  #22 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,675 Thanks: 2655 Math Focus: Mainly analysis and algebra 
This concept of 'a completed infinity' is very unclear to me. As CRG is fond of pointing out, there are relatively few areas of mathematics in which it is correct to use the word 'infinity'. Either we use the concept to mean 'unbounded' or the area has a moe refined few and 'infinity' iss too general a term to have meaning in the. context. You talk about the real line going from $\infty$ to $+\infty$, but neither of them are points on the line. (They are on the extended real line, but that's different). In mainstream calculus $\pm\infty$ are notational devices depicting the unbounded growth of a quantity. Set theory does have infinite sets, but the word infinity is meaningless in that context. You must talk about more refined concepts to make any sense. Any argument that infinite sets don't exist seems flawed, because clearly the natural numbers exist, and all of them have shared properties. The same goes for the rationals and the algebraic numbers. So we can't claim that these sets don't exist because we can define what these numbers are, and given any number we can tell what type it is. 
October 17th, 2014, 04:55 AM  #23  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,675 Thanks: 2655 Math Focus: Mainly analysis and algebra  Quote:
This isn't a problem, there are plenty of areas of mathematics you can study without needing a concept of the infinite, but you can't make everybody else discard it just because you don't follow the reasoning. The main problem is that your argument is not mathematical, it is philosophical. And therefore your thinking is more akin to religion than to mathematics. The Catholic church spent many centuries trying to exclude the infinite (and zero) from mathematics because of a perceived incompatibility with their beliefs. But they failed because logical thought makes the concept necessary in many branches of scientific endeavour. Once the concept is found to be necessary, it can be said to exist and can therefore be studied as Cantor did, rigorously. You therefore must argue against the mathematical rigour of proven results. This is by definition rather difficult to do because reason is not on your side. It has already been deployed against you.  
October 17th, 2014, 05:38 AM  #24  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Quote:
I've only ever seen this in writings by nonmathematicians, personally. Quote:
As taught today calculus (following Cauchy) does not use infinite or infinitesimal numbers anywhere, so you don't need to worry about it. It does use the symbol $\infty$ but this does not refer to a number (within calculus), it's a pure formalism. If you want to actually use infinite numbers you need to go beyond basic calculus. Quote:
At the risk of repeating myself: basic calculus does not use infinite or infinitesimal numbers at all. The first exposure to infinite numbers in that field would probably be complex analysis with the Riemann sphere. (But it's arguable whether that's infinite in any meaningful sense anyway  it's just the top point of a sphere, right?) Quote:
If you'd like to be stuck in the 18th century you could adopt his view on the matter. If you want to be a finitist in the modern world  and that's possible!  you'll need better reasons. Quote:
Quote:
Again, you need to clarify your reasons here. Are you saying you can derive a contradiction from ZF?  
October 17th, 2014, 06:18 AM  #25 
Senior Member Joined: Apr 2014 From: Glasgow Posts: 2,157 Thanks: 732 Math Focus: Physics, mathematical modelling, numerical and computational solutions 
I don't understand why so many people argue about the infinite, infinity, endlessness, limits, etc... It's been studied to death already in the literature, so read it and learn from it. Limits apply to functions, so $\displaystyle \lim(0.\dot{3})$ makes no sense to me. $\displaystyle 0.\dot{3}$ is just a constant. The whole point of a limit is to investigate what value a function converges to as the dependent parameters vary to some limit. You might not like the idea that a limit can be infinity, but this is not a problem in mathematics. For example, you can define a function, in the domain of positive x, such as $\displaystyle f(x) = \frac{1}{x}$ and then say $\displaystyle \lim_{x\rightarrow 0} f(x) = \infty$ or $\displaystyle \lim_{x\rightarrow \infty} f(x) = 0$. As your number gets closer and closer to 0, f(x) gets larger and larger without bound, or as your number gets larger and larger, f(x) converges to zero. Mathematics deals with any attempted evaluation of the function at zero by saying that the function is "undefined" at $\displaystyle x=0$. A proper definition of the the above function is in fact $\displaystyle f(x) = \frac{1}{x}, x \in \mathbb{R}^+, x \neq 0$. After all, division of a finite number must occur with another finite number (at least in the most common number systems, like the naturals or the reals). I don't see what's so problematic about understanding this and what is unsatisfactory about it. The statements are clear, the concept is clear and anything worked out in a real physical situation is compatible with this. Is infinity required to describe reality? Possibly not. Is infinity (and other related concepts such as cardinality) required to describe abstract mathematics? Yes, depending on what field you're studying. Also, the set of real numbers contains irrational numbers, but this doesn't mean those numbers aren't exact, they just cannot be represented exactly using a decimal number system and are instead specified exactly using algebraic letters or surds. The decimal number system can represent rational numbers using the recurring notation, for example $\displaystyle 0.\dot{3} = \frac{1}{3}$ $\displaystyle 0.\dot{0}7692\dot{3} = \frac{1}{13}$ There is a quirk where a number can have two decimal representations, which needs taking into account in very few circumstances (such as Cantor's proof): $\displaystyle 0.\dot{9} = 1$ but again, this is just a quirk with the decimal number system; it is a number system where some numbers have more than one representation. The above case for decimal numbers is true based on the definitions of recurring numbers. Because it is true as a matter of definition, it cannot be disproved, just changed or redefined depending on applications. 
October 17th, 2014, 06:22 AM  #26  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Quote:
A similar statement in set theory: $$ \forall\nu\in\omega,\nu<\infty $$ (where the symbols "$<\infty$" mean "is finite").  
October 17th, 2014, 06:27 AM  #27  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Quote:
Does this bother you? Does that mean you reject the rational numbers, too?  
October 17th, 2014, 07:25 AM  #28 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,675 Thanks: 2655 Math Focus: Mainly analysis and algebra  It's important to draw a distinction between a number system, such as the Real Numbers, the Rational Numbers, etc.. and a numeral system which is merely a representation of numbers that exist within a number system. Numeral systems include the decimal system. I realise that you were trying to do just that. You are correct in saying that limits apply to functions, but we can consider the decimal system to be a function that maps a sequence of numbers $\cdots a_3 a_2 a_1 a_0 . b_1 b_2 b_3 \cdots$ to the function $$f(x)=\sum_{i=0}^\infty a_i {10}^i + \sum_{j=1}^\infty \frac{b_j}{10^j}$$ 
October 17th, 2014, 07:28 AM  #29 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Note that the sequence a_i must be finite (i.e., there is some N such that a_n = 0 for all n > N).

October 17th, 2014, 07:36 AM  #30 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,675 Thanks: 2655 Math Focus: Mainly analysis and algebra 
Yes. I was considering your bracketed case.


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