Mathematical obscuration?
Moderator: msmod
Mathematical obscuration?
let's consider the simplest example. We have two number 0.5 and 0.5.
1) if we add them we get just 1;
2) but if we multiply them we get 0.25
Could someone explain why adding two positive numbers results in a bigger number them multiplying these numbers? Why it is so?
Logically say that multiplying numbers that are bigger than 0 and less than 1 gives smaller outputs that adding them. Is it logical? In all other cases it is not the case, for example, 2*5=10 but 2+5=7 or 350*10=3500 but 350+10=360 and so on.
P.S. In my opinion this question has a practical application in a plenty of the real world problems! In particularly, in calculating distances and other parameters.
1) if we add them we get just 1;
2) but if we multiply them we get 0.25
Could someone explain why adding two positive numbers results in a bigger number them multiplying these numbers? Why it is so?
Logically say that multiplying numbers that are bigger than 0 and less than 1 gives smaller outputs that adding them. Is it logical? In all other cases it is not the case, for example, 2*5=10 but 2+5=7 or 350*10=3500 but 350+10=360 and so on.
P.S. In my opinion this question has a practical application in a plenty of the real world problems! In particularly, in calculating distances and other parameters.
Re: Mathematical obscuration?
First, it isn't always so. 3 + 3 < 3*3.Could someone explain why adding two positive numbers results in a bigger number them multiplying these numbers? Why it is so?
Let a = (1+x), b = (1+y)
ab = 1 + x + y + xy = (a + b) + (xy – 1).
So your statement is true only if xy < 1.
If we think of the numbers as representing physical measurements such as distances, the statement is meaningless:
3 m + 3 m = 6 m, 3 m * 3 m = 9 m^2.
But we can't say 9 m^2 > 6 m, because they have different units. (Further, the values depend on the units: compare 0.006 km and 0.000009 km^2.)
Joe
Re: Mathematical obscuration?
Thank you, Joe. You're right. Now I see it when it concerns unit. However, let's forget about units for a moment and consider the logic of the mathematical operations. It seems that the multiplication operations with numbers in range: 0<numbers<1 have another logic. So 0.3 + 0.3 = 0.6 but 0.3 * 0.3 = 0.09 (less than the result of the numbers addition)! It contradicts the general logic: multiplication of the same two numbers should result in a bigger number than that of adding these numbers!
Re: Mathematical obscuration?
That is true for the natural numbers greater than one, because it's a special case of the algebraic explanation quoted above.multiplication of the same two numbers should result in a bigger number than that of adding these numbers!
But what about for rational numbers, i.e. numbers of the form n/m where n and m are integers
Consider the numbers 1/n and 1/m. When we multiply them, we get 1/nm but when we add them we get (n+m)/nm.
Numbers do what they do, they don't care what you think they should do! If you have a theorem, you have to prove it, and that requires stating your premises. If you mean "for the natural numbers greater than one" or for some other special case, say so, and then do the algebra quoted above.
Joe
Re: Mathematical obscuration?
By the way, one more obscuration Gabrial's Horn whose Volume is finite but Surface Area is infinite:
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Re: Mathematical obscuration?
Another bizarre one is the "1 + 2 + 3+ 4 + ... = 1/12" result (which I think is used in physics)!
Re: Mathematical obscuration?
That is indeed bizarre. I'll bite, what strange use of "=" are you using?
Joe
Joe
Re: Mathematical obscuration?
I'm not quite sure, I think it is used to denote regularisation of the sum? I've seen this formula in several places. The Wikipedia article is here: http://en.wikipedia.org/w/index.php?tit ... edirect=no
From what I understand (which is not a great deal), it's based on using analytic continuation of the Riemann zeta function to arrive at the value of 1/12 for ζ(−1). And ζ(s) is defined by: sum from n = 0 to ∞ of 1/n^s, for complex s with Re(s) > 1.
Why does this lead people to write 1 + 2 + 3 + 4 + ... = 1/12 ? Because ζ(−1) isn't defined by the same rule as for Re(s) > 1, is it? If it were, the LHS WOULD surely be written as 1 + 2 + 3 + 4 + ... .
Cont.
From what I understand (which is not a great deal), it's based on using analytic continuation of the Riemann zeta function to arrive at the value of 1/12 for ζ(−1). And ζ(s) is defined by: sum from n = 0 to ∞ of 1/n^s, for complex s with Re(s) > 1.
Why does this lead people to write 1 + 2 + 3 + 4 + ... = 1/12 ? Because ζ(−1) isn't defined by the same rule as for Re(s) > 1, is it? If it were, the LHS WOULD surely be written as 1 + 2 + 3 + 4 + ... .
Cont.
Re: Mathematical obscuration?
Cont.
Maybe whenever that bizarre equation is written, the equals sign is used in the understanding that it is some sort of manipulation of something. (Obviously the sum diverges in terms of using limits of partial sums. It turns out not to be able to be summed using Cesaro summation, or Abel summation. Also, an equals sign is used in the series 1 – 1 + 1 – 1 + ... = 1/2, which is found using Cesaro summation. So would the equals sign in 1 + 2 + 3 + 4 + ... = 1/12 be something similar (not Cesaro summation, but some other method)? Ramanujan summation of 1 + 2 + 3 + 4 + ... also gives 1/12 as the answer according to Wikipedia.)
Wikipedia also says this result is used in areas of physics: http://en.wikipedia.org/w/index.php?tit ... no#Physics . It says it is used in string theory, and also for calculating Casimir force in 1D in quantum electrodynamics.
Cont. in one more post
Maybe whenever that bizarre equation is written, the equals sign is used in the understanding that it is some sort of manipulation of something. (Obviously the sum diverges in terms of using limits of partial sums. It turns out not to be able to be summed using Cesaro summation, or Abel summation. Also, an equals sign is used in the series 1 – 1 + 1 – 1 + ... = 1/2, which is found using Cesaro summation. So would the equals sign in 1 + 2 + 3 + 4 + ... = 1/12 be something similar (not Cesaro summation, but some other method)? Ramanujan summation of 1 + 2 + 3 + 4 + ... also gives 1/12 as the answer according to Wikipedia.)
Wikipedia also says this result is used in areas of physics: http://en.wikipedia.org/w/index.php?tit ... no#Physics . It says it is used in string theory, and also for calculating Casimir force in 1D in quantum electrodynamics.
Cont. in one more post
Re: Mathematical obscuration?
I'm quite curious, how do we know using such results for infinite series is valid, or that results derived from these will be true physically?
Re: Mathematical obscuration?
The graph on the article you cite:
http://en.wikipedia.org/w/index.php?tit ... no#Physics
is clear enough.
The RHS is not the sum of the LHS, rather it is the f(0) of a smooth function that approximates
f(n) = 1+2+3+…+n
So it's not (obviously) equality in the usual sense. Thanks for bringing it to my attention.
Joe
http://en.wikipedia.org/w/index.php?tit ... no#Physics
is clear enough.
The RHS is not the sum of the LHS, rather it is the f(0) of a smooth function that approximates
f(n) = 1+2+3+…+n
So it's not (obviously) equality in the usual sense. Thanks for bringing it to my attention.
Joe
Re: Mathematical obscuration?
Jacobian and the Conception of Limit.
Limit of Riemann sum when delta(x) goes to 0 is Integral. In other words, delta(x) is infinitely small. If take Jacobian for changing variables we get some coefficient used to calculate a new area. So if we have x,y coordinates and then change them with t,h in some conditions we may get another figure having other area. To calculate area of a new figure we use Jacobian. If I understand right Jacobian is also applied to infinitesimal area formed by delta(x), delta(y) or delta(t), delta(h). So if Jacobian = 2 then area of each infinitesimal rectangle formed by delta(t) and delta(h) should be 2*delta(x)*delta(y). How may the latter match the conception of Limit when all delta go to 0 (infinitely small values)? It induces the conclusion that Limit is relative to a size of such a new figure
Thank you.
Limit of Riemann sum when delta(x) goes to 0 is Integral. In other words, delta(x) is infinitely small. If take Jacobian for changing variables we get some coefficient used to calculate a new area. So if we have x,y coordinates and then change them with t,h in some conditions we may get another figure having other area. To calculate area of a new figure we use Jacobian. If I understand right Jacobian is also applied to infinitesimal area formed by delta(x), delta(y) or delta(t), delta(h). So if Jacobian = 2 then area of each infinitesimal rectangle formed by delta(t) and delta(h) should be 2*delta(x)*delta(y). How may the latter match the conception of Limit when all delta go to 0 (infinitely small values)? It induces the conclusion that Limit is relative to a size of such a new figure
Thank you.
Re: Mathematical obscuration?
...continuation of my previous post. If following Jacobian's conception then the precision of calculating areas of such new bigger figures by using Jacobian will be decreasing. Just imagine if delta(t)*delta(h) = 1000000000*delta(x)*delta(y)! It seems that Jacobian violates the conception of Limit when delta(something) goes to 0, doesn't it?!
Thanks a lot.
Have a nice day
Thanks a lot.
Have a nice day
Re: Mathematical obscuration?
I don't understand the problem here. Even if the coefficient of a differential is very large, one can still make the product very small by taking a very, very small differential.