05 Dec 14
Originally posted by twhiteheadSo now you have a more free interpretation of the "=", right?
Only on condition that the '=' does not mean quite the same as it does in normal equations. In this case the '=' is defined slightly differently.
Now meaning... like... almost equal?
Do Americans have other mathematics compared to the rest of the world...?
05 Dec 14
Originally posted by twhiteheadWhat about equations like d²y/dx² + ky = 0, does '=' have its usual meaning there?
I am being serious. In this instance the final 'sum' of the sequence is defined as being the limit of the partial sums of the sequence. It is not the case that an infinite number of terms are actually added to give exactly 1.
05 Dec 14
Originally posted by FabianFnasNo, I am saying it has a specific definition in that instance that is different from the one usually used. This is quite normal in mathematics. They symbols for addition and multiplication don't always mean the same thing either.
So now you have a more free interpretation of the "=", right?
Now meaning... like... almost equal?
No. Meaning 'the sequence converges to'.
Do Americans have other mathematics compared to the rest of the world...?
I am not american. Are you?
http://en.wikipedia.org/wiki/Series_(mathematics)
In mathematics, given an infinite sequence of numbers { an }, a series is informally the result of adding all those terms together: a1 + a2 + a3 + · · ·.
Note the use of the word 'informally'.
Formally, no such claim should ever be made.
05 Dec 14
Originally posted by twhiteheadI see what you're getting at. I think that it is not the equal sign that's any different from normal, but what is meant by 'sum'. The situation is similar with derivatives and for that matter integrals, where the object is defined by either a limit or some kind or a delta-epsilon argument. But it's the object that is defined that way, the equals is just equal.
No, I am saying it has a specific definition in that instance that is different from the one usually used. This is quite normal in mathematics. They symbols for addition and multiplication don't always mean the same thing either.
[b]Now meaning... like... almost equal?
No. Meaning 'the sequence converges to'.
Do Americans have other mathemat ...[text shortened]... . [/quote]
Note the use of the word 'informally'.
Formally, no such claim should ever be made.
06 Dec 14
Originally posted by DeepThoughtYes, I agree, that's a better way of putting it.
I see what you're getting at. I think that it is not the equal sign that's any different from normal, but what is meant by 'sum'. The situation is similar with derivatives and for that matter integrals, where the object is defined by either a limit or some kind or a delta-epsilon argument. But it's the object that is defined that way, the equals is just equal.
06 Dec 14
Originally posted by twhiteheadFabianFnasDo Americans have other mathematics compared to the rest of the world...?
Do Americans have other mathematics compared to the rest of the world...?
I am not american. Are you?
http://en.wikipedia.org/wiki/Series_(mathematics)
In mathematics, given an infinite sequence of numbers { an }, a series is informally the result of adding all those terms together: a1 + a2 + a3 + · · ·.
Note the use of the word 'informally'.
Formally, no such claim should ever be made.[/b]
twhitehead: I am not american. Are you?
So your part of the world uses other definitions, and are vague about definitions.
In your link http://en.wikipedia.org/wiki/Series_(mathematics) I cannot find any alternative definition of the "=" symbol.
In accordance of yor alternative definition of the "="-symbol, you don't believe that 0.111... isn't exactly one ninth, only almost, right?
06 Dec 14
The symbol (and it's just a symbol) a_1 + a_2 + a_3 + · · · is taken by mathematicians to denote an infinite series, which in turn is taken to be the limit of the partial sums:
a_1 + a_2 + a_3 + · · · := lim(a_1 + · · · + a_n),
where the symbol := means "is defined to be equal to." Of course, the limit of a sequence of partial sums does not always exist as a real or complex number.
07 Dec 14
Originally posted by SoothfastSo you define the symbol ":=", fine with me.
The symbol (and it's just a symbol) a_1 + a_2 + a_3 + · · · is taken by mathematicians to denote an infinite series, which in turn is taken to be the limit of the partial sums:
a_1 + a_2 + a_3 + · · · := lim(a_1 + · · · + a_n),
where the symbol := means "is defined to be equal to." Of course, the limit of a sequence of partial sums does not always exist as a real or complex number.
But the symbol "=" has another meaning, well defined, not sloppy.
07 Dec 14
Originally posted by FabianFnasNo, := means the same as =, to indicate "the value of the quantity on the left is the same as the value of the quantity on the right," only it makes clear that the equality is set by definition. After all, the symbol
So you define the symbol ":=", fine with me.
But the symbol "=" has another meaning, well defined, not sloppy.
a_1 + a_2 + a_3 + · · ·,
as with most any symbol in mathematics, has no meaning until it is given one by a definition. Thus we say that a_1 + a_2 + a_3 + · · · equals the value of the limit of the sequence of partial sums of the sequence {a_n} by definition. In practice := is seldom used, because context makes clear when = is given by definition. So we have:
s = a_1 + a_2 + a_3 + · · ·
if and only if
n
Σ a_k → s as n → ∞
k=1
08 Dec 14
Originally posted by SoothfastOf course in physics we'll happily take a series, each of whose terms is a formally divergent integral, and claim the whole thing converges.
No, := means the same as =, to indicate "the value of the quantity on the left is the same as the value of the quantity on the right," only it makes clear that the equality is set by definition. After all, the symbol
a_1 + a_2 + a_3 + · · ·,
as with most any symbol in mathematics, has no meaning until it is given one by a definition. Thus w ...[text shortened]... have:
s = a_1 + a_2 + a_3 + · · ·
if and only if
n
Σ a_k → s as n → ∞
k=1
08 Dec 14
Originally posted by DeepThoughtI'm sure that's one of the reasons why I decided not to pursue physics beyond the undergraduate level. I'm an order freak, and the seemingly chaotic and cavalier treatment of the Queen of the Sciences at the roguish hands of physicists drove me crazy.
Of course in physics we'll happily take a series, each of whose terms is a formally divergent integral, and claim the whole thing converges.
08 Dec 14
Originally posted by SoothfastThe classic example of that kind of game going well is the Dirac delta function, which mathematicians initially dismissed as invalid. Happily Dirac wasn't to be put off and as he continued to get good results with it they were forced to take it seriously and the field of distributions was born.
I'm sure that's one of the reasons why I decided not to pursue physics beyond the undergraduate level. I'm an order freak, and the seemingly chaotic and cavalier treatment of the Queen of the Sciences at the roguish hands of physicists drove me crazy.
In a sense the Millennium prize for showing that Yang-Mills theories exist is a game of showing that the functional integrals we get correlation functions out of are well defined. Theoretical physics is not applied maths - applied maths is (or should) involve taking well defined mathematical theorems and applying them to the real world in a consistent way. Theoretical physics involves taking the real world and seeing what theories we can make up to describe it. When things go well new fields of mathematics open up.