Approximation Mathematics and Precision Mathematics

[Searle]

Mod note - Split from this thread.

Unlike some, I can cope with a rounding error :roll:

I'm a mathematician, I don't deal with this "rounding" business. Exact figures only please

 

[ASharpe]

I'm a mathematician, I don't deal with this "rounding" business. Exact figures only please

Off-topic but, a few days ago I was doing a rough calculation for someone (spherical cows in a vacuum etc.) and used 3.0 as pi. It didn't go down well at all, I'll stick with 22/7 in future.

 

[bb21]

3 is too much of an approximation that only works in the most rough cases. 3.14 is usually sufficient for the vast majority of cases.

Searle is a pure mathematician whereas I work in applied maths, where absolute accuracy is rarely the objective, as it is usually not possible. In my area an accuracy of 25% is considered a great achievement.

 

[Wolfie]

I'm a mathematician, I don't deal with this "rounding" business. Exact figures only please

To how many significant figures?

Guess what my degree was.....

 

[causton]

Meanwhile I don't have any particular interest in mathematics yet one song has made me remember the first few digits of pi so well... not too sure I can do the rap out loud though!

Hard & Phirm - Pi
https://www.youtube.com/watch?v=BDMBtQjS1bQ

 

[ASharpe]

I'm a physicist (although I do a bit surface/materials science), precision isn't always expected of me provided I give a sensible estimate of the errors too.

The only time I do proper maths is when I'm helping with the undergrad Engineering maths modules.

Other approximations I often get away with are:
sin(x)=x
tan(x)=x
g=10m s^-2
Speed limit = 1' per nanosecond
temperature=300 K

 

[Starmill]

This required a new thread? Jesus I would not enjoy being left alone in a room with bb21 and Searle eh!!!!

Have to hand it to you too Brian such a catchy title

 

[Butts]

This required a new thread? Jesus I would not enjoy being left alone in a room with bb21 and Searle eh!!!!

Have to hand it to you too Brian such a catchy title

How many Mathematicians does it take to produce an interesting thread ...:roll:

 

[ASharpe]

How many Mathematicians does it take to produce an interesting thread ...:roll:

1 - well, that order of magnitude anyway.

 

[Xenophon PCDGS]

How many Mathematicians does it take to produce an interesting thread ...:roll:

I feel somewhat aggrieved not to have been invited to join in the debate as many forum members are aware that I obtained my First in Mathematics at Manchester University in 1966, some 48 years ago....:roll:

Perhaps at the age of 69, I might now be seen as being in my dotage...

 

[AM9]

I feel somewhat aggrieved not to have been invited to join in the debate as many forum members are aware that I obtained my First in Mathematics at Manchester University in 1966, some 48 years ago....:roll:

Perhaps at the age of 69, I might now be seen as being in my dotage...

Ah but your qualifications were in imperial numbers, we're all using SI ones now.

 

[Strat-tastic]

 

[bb21]

Ah, the squaring a circle problem. This is similar to the Diagonal Paradox.

The fundamental flaw is that the perimeter of the circle is not the same as a sum of infinitesimals at right angles with each other. Taking this process to the extreme you will obtain the limit of the area, but not the limit of the perimeter. After each step the perimeter remains the same, provided that the new edges are parallel to the sides of the original square, hence impossible to take the limit mathematically (as you cannot approach the supposed "limit" artificially closely). This is the best explanation I have managed to come up, for first year students. Maybe someone else has a better one?

Has this fallacy ever been disproven? I am not aware of one myself.

Edit:

The Diagonal Paradox may be a starting point for the proof I reckon, but of course for a straight line it is easy.

 

[Emyr]

Nothing like a good Fermi Problem to soothe fears of inadequate accuracy.

 

[Oswyntail]

Ah, the squaring a circle problem. This is similar to the Diagonal Paradox....

Not to mention Achilles and his tortoise

 

[ralphchadkirk]

I have no idea what you are all talking about.

 

[D6975]

Ah but your qualifications were in imperial numbers, we're all using SI ones now.

And now that windspeeds are higher in kmph than they used to be in mph, we're having to build stronger bridges, buildings etc...

 

[bangor-toad]

Has this fallacy ever been disproven? I am not aware of one myself.

Hi there,
I've not come across a formal mathematical proof but there is a simple(!) way to look at the paradox.

Looking at the circle diagram, when you draw the square box around the circle, you have "4n" deviations of d from the circle. (One for each corner)
For each time you put in place a square deviation on the perimeter the number of little deviations increases by 2^(n+1) whilst d decreases. The variation d is proportional to the square root of 1/((n+1)^2)

The total variation is the sum of the number of deviations multiplied by the deviation of each one.
As n increases to infinity you can see the deviation never disappears to zero by looking at the Taylor expansion of the deviation: 1-n+n^2-n^3+n^4...

Multiply that all together and there's always a deviation between the approximation using this method and the "normal" method of calculation of the perimeter of a circle.

OK, that's enough brain exercise for now. Time for lunch.
Mr Toad

 

[D6975]

The reason it gives an incorrect answer is because the thinking is fundamentally flawed.
It's the sum of the hypotenuses of all the little triangles that tends to the circumference of the circle, not the sum of the other two sides.

 

[ASharpe]

Nothing like a good Fermi Problem to soothe fears of inadequate accuracy.

 

[DarloRich]

what on earth are you people talking about?????????????

 

[Yew]

what on earth are you people talking about?????????????

Mathematics, fortunately we havent hit number theory yet..

 

[amateur]

..... Or group theory

 

[DarloRich]

physics and maths nerds thinking you are so clever with your numbers and theories and stuff

 

[amateur]

Or set theory!

 

[Abpj17]

is there a correlation between people interested in trains and those that studied maths?

(bonus point for anyone to come up with a testable hypothesis)

 

[Xenophon PCDGS]

Using the Atiyah-Singer Index Theorem as a base for forum discussion, how many different types of that have recently been put forward for refinement of logic of that specific theorem?

 

[amateur]

is there a correlation between people interested in trains and those that studied maths?

(bonus point for anyone to come up with a testable hypothesis)

And those who are on the autism spectrum

 

[Xenophon PCDGS]

And those who are on the autism spectrum

Mathematics can be of great help to students when taught correctly as it fosters discipline of thought processes.
--- old post above --- --- new post below ---

is there a correlation between people interested in trains and those that studied maths?

One could ask the same with regards to people with a liking for chess and those who have studied mathematics.

 

[Oswyntail]

Mathematics can be of great help to students when taught correctly as it fosters discipline of thought processes......

As can the study of Latin and Ancient Greek