Approximation Mathematics and Precision Mathematics

[Xenophon PCDGS]

As can the study of Latin and Ancient Greek

How did you guess that I was in the Classics stream at St Bede's College, Manchester from 1956 onwards.....

 

[bb21]

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)

If you take my department as a sample, no.

(... which is a shame as they all look at me when I talk about fares, etc, as some sort of alien just landed from Mars.)

 

[SS4]

Other approximations I often get away with are:
sin(x)=x
tan(x)=x

Which is why radians is much better than degrees . Mind you what's wrong with cos(x)=1-x^2/2 for appropriately small x?

 

[Searle]

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

you mean you didn't like my maths at the challenge?

 

[ASharpe]

Which is why radians is much better than degrees . Mind you what's wrong with cos(x)=1-x^2/2 for appropriately small x?

Too complicated, I'd sooner reach for a calculator.

Too accurate too, this thread is for approximations

 

[Searle]

Which is why radians is much better than degrees . Mind you what's wrong with cos(x)=1-x^2/2 for appropriately small x?

Too complicated, I'd sooner reach for a calculator.

Too accurate too, this thread is for approximations

Unless you fancy approximating cos(x) = 1, it's the smallest Taylor Approximation you'll get for it

 

[ASharpe]

Unless you fancy approximating cos(x) = 1, it's the smallest Taylor Approximation you'll get for it

I did Maclaurin series today with the Undergrads. One of them was amazed at how accurate sin(x)=x was given how few steps it takes to get there. It's also an easy way to convince people that radians are the best.

 

[Strat-tastic]

Forgive me for missing something; it's been a long time since A level maths. But how does sin x ~= x? They're nothing like each other for most values of x :-?

 

[SS4]

Forgive me for missing something; it's been a long time since A level maths. But how does sin x ~= x? They're nothing like each other for most values of x :-?

It's only the case for small values of x. It's even called the Small Angle Approximation. If you're working with small angles it comes in very handy. I remember using it in physics when working with subtended angles of faraway planets. It's quite misleading to suggest it's for all x.

I did Maclaurin series today with the Undergrads. One of them was amazed at how accurate sin(x)=x was given how few steps it takes to get there. It's also an easy way to convince people that radians are the best.

Calculus is also a good one

d/dx sin(x) = cos(x) is much easier than d/dx sin(180x/pi) = 180/pi cos(180x/pi)

 

[Strat-tastic]

It's only the case for small values of x. It's even called the Small Angle Approximation. If you're working with small angles it comes in very handy. I remember using it in physics when working with subtended angles of faraway planets. It's quite misleading to suggest it's for all x.

Thank you

 

[Butts]

And here was me thinking that Train Spotters were weird, judging by this thread they have nothing on Mathematicians

 

[Xenophon PCDGS]

And here was me thinking that Train Spotters were weird, judging by this thread they have nothing on Mathematicians

Oh dear, as a non-smoking Hyde supporter with a penchant for mathematics, currently on the RailUK website, how does this leave me in your eyes..

 

[Searle]

Unless you fancy approximating cos(x) = 1, it's the smallest Taylor Approximation you'll get for it

I did Maclaurin series today with the Undergrads. One of them was amazed at how accurate sin(x)=x was given how few steps it takes to get there. It's also an easy way to convince people that radians are the best.

There are people at university who still use degrees over radians in their work? o.O

 

[Butts]

Oh dear, as a non-smoking Hyde supporter with a penchant for mathematics, currently on the RailUK website, how does this leave me in your eyes..

Sorry Paul, I'll have to plead the "fifth amendment" here

 

[Gathursty]

I think anything beyond 5 decimal places or significant figures is overkill for day to day mathematics. A handful of my students still get a bit confused between significant figures and decimal places so I just say the column your first number is in (ie: 10987) is your first significant figure, so to round 10987 to 3 significant figures, you're rounding to the 9 column, aka the hundreds.

I did a bit of calculations when bashing, namely the percentage of stations covered in a weekend, stations bashed per hour and average mileage covered between stations. I tended to average driving 10 miles between stations, taking into account getting from Wigan to further flung regions and back to begin bashing, which i thought was pretty good but it's hard to get a definite answer for that pesky Travelling Salesman problem!

 

[krus_aragon]

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.

In fact, GCSE Mathematics (WJEC) papers state on the cover to take pi as 3.14 (if you have no calculator with you...)

 

[Searle]

I think anything beyond 5 decimal places or significant figures is overkill for day to day mathematics.

I'll use 3 decimal places usually, although for Numerical Analysis (iterating results instead of finding exact amounts), we're told to use a standard 10 decimal places on the computer package, as you need added precision. Then again, I wouldn't say it's day to day mathematics :P

 

[bb21]

I'll use 3 decimal places usually, although for Numerical Analysis (iterating results instead of finding exact amounts), we're told to use a standard 10 decimal places on the computer package, as you need added precision. Then again, I wouldn't say it's day to day mathematics :P

Significant figures!! You do not round 0.000000000025 to ten decimal places under any circumstance for scientific calculation, for example.

 

[Xenophon PCDGS]

I'll use 3 decimal places usually, although for Numerical Analysis (iterating results instead of finding exact amounts), we're told to use a standard 10 decimal places on the computer package, as you need added precision. Then again, I wouldn't say it's day to day mathematics :P

I agree that three, rather than two, decimal places should be used in generalised day-to-day matters, as this does give a more precise answer.

 

[Searle]

Significant figures!! You do not round 0.000000000025 to ten decimal places under any circumstance for scientific calculation, for example.

When we're talking about approximations of values tending to the real value, for the purposes of the course, we would take 10 d.p. and claim that they are equal (if that is the error term, for example).

 

[ainsworth74]

Significant figures!! You do not round 0.000000000025 to ten decimal places under any circumstance for scientific calculation, for example.

Indeed not! Because as we well know 0.000000000025 is better known as 0...

 

[bb21]

When we're talking about approximations of values tending to the real value, for the purposes of the course, we would take 10 d.p. and claim that they are equal (if that is the error term, for example).

Just testing. You should have made it clearer.