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Re: Differentiation
Posted: Mon Sep 05, 2011 10:59 am
by Abdul Muntakim Rafi
You said what we did at first was...
\[x\frac{\mathrm{d} (x+x+x+x+..............+x)(x times)}{\mathrm{d} x}\]
But it is not what we did at first... At first what we wrote meant that
\[x\frac{\mathrm{d} x}{\mathrm{d} x}\]
But writing this was wrong... That's why we got $x$ as an answer...
Your equation shows that the derivative was $x^2$...
Re: Differentiation
Posted: Mon Sep 05, 2011 11:03 am
by Abdul Muntakim Rafi
The addition rule fails because of countability. When we say "x times", x needs to be in a countable set. Then you can have a bijection with a sequence from that countable set and you stop when you reach x in the sequence and call that being x times. But the kind of calculus we are doing involves uncountable sets, because when you say differentiate x^2 , you can not think x to be one point, you actually need to be able to pick x from an open interval around that point and that open interval is uncountable. So you can not say "x times". You can only define the phrase "times" with respect to elements of a countable set.
P.S. of course you can try to define new things, that's how mathematics develops. There are theories of calculus where you don't need the notion of limit, and some of them are very algebraic. I have not read any of those yet. There are many examples of calculus on discrete and countable sets as well, and many of them are very useful in economics, physics etc. But you will need to define them in a intelligent way first.
ভাইয়া, বাংলা অনুবাদ করেন। কিছু টার্ম সমস্যা করতেছে।
Re: Differentiation
Posted: Mon Sep 05, 2011 5:30 pm
by tanvirab
ঐগুলা বাদ দেও।
When we say "x times", x needs to be from a countable set.
এই এক লাইনেই সব আছে। "x times" বলতে কি বুঝায় এইটা নিয়া চিন্তা কর।
Re: Differentiation
Posted: Mon Sep 05, 2011 8:32 pm
by Abdul Muntakim Rafi
x times x is the same thing as finding out the sum of x, x's
Re: Differentiation
Posted: Mon Sep 05, 2011 8:37 pm
by Abdul Muntakim Rafi
মানে x times x হল x টা x এর যোগফল।
$a.b$ হল a সংখ্যক b এর যোগফল। অথবা b সংখ্যক a এর যোগফল।
$a.b=a+a+a+a+a+................+a$ এতে b সংখ্যক a আছে।
$a.b=b+b+b+b+......................+b$ এতে a সংখ্যক b আছে।
Re: Differentiation
Posted: Tue Sep 06, 2011 12:19 am
by tanvirab
OKay, think of it like this, what if I say, give me $\pi$ apples?
Re: Differentiation
Posted: Tue Sep 06, 2011 11:47 am
by Abdul Muntakim Rafi
First I will give you 3 apples... Then I will give you .1415........ part of another apple. But I will not be able to give you exactly $\pi$ apples... I will give you more or less than $\pi$ apples... Cause $\pi$ is an irrational number...
So what are you trying to say is that in order to express $x^2$ as sum of $x, x's$ ;x needs to be a rational number...
Is that it?
And Bhaiya, how will you define $a.b$ then?
Re: Differentiation
Posted: Tue Sep 06, 2011 12:15 pm
by tanvirab
That's the basic idea. When you say "$x$ times" you have to be able to count like 1 times, 2 times, 3 times .... and it will end at $x$ times. Clearly you cannot do that for all real number.
multiplication of real numbers can be defined using decimal representation, after you define multiplication of natural numbers using addition.