Re: Smarter Scots


Custos Custodum wrote:
On 9 May 2006 04:10:05 -0700, "Chicmac"
<charles.mcgregor@xxxxxxxxxxxx> wrote:


Chicmac wrote:
Chicmac wrote:
Custos Custodum wrote:
On Wed, 3 May 2006 22:11:36 +1200, "Mad Prof" <mad@xxxxxxx> wrote:


"uNkulunkulu" <izulu@xxxxxxxxxx> wrote in message
news:k0Z5g.63407$wl.19314@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx

"Mad Prof" <mad@xxxxxxx> wrote in message
news:e39cv8$j2h$1@xxxxxxxxxxxxxxxx
Are Scots any smarter than other nations? We often laugh at the Yanks
who's
knowledge of geography is sadly lacking but if you went on the streets
of
a
major Scottish town and asked a person at random to point to say 'South
Dakota' or Luxemberg on a map would they be able to do it? I doubt it.
Ask
them to solve a simple algebra problem

sin(x)+x^2=0

As this equation ha a trigonometric function (sin(x)) it has to be a
trigonometric problem and not algebraic. At first glance it is obvious to
those with even a smattering of high school maths that the answer can not
be
zero unless x=0


Fraid not - that is the trivial solution - there is also another solution.
You can see this by drawing the graph of sin(x) and x^2 and looking at the
intersections.(one of which is your solution at x=0) But other than a
graphical solution how to find x?

I don't think it can be solved algebraically - you have to resort to
numerical methods such as successive approximation or truncated
series. Alternatively you can learn to use a programmable calculator
and get the result x = -0.8767 to 4 decimal places.
Actually it can very simply, since the question does not specify
radians, (which you have understandably assumed).

So if I choose to use degrees.

Then it can immediately be seen that the solution in that instance must
be <<1.

This allows us to make the small angle approximation, where sin x = x
where x is in radians and x<<1 radian.

there are 57.3 degrees in a radian, therefore if we are working in
degrees with a very small angle then the approximation means that


x/57.3 = x^2

so x=1/57.3 = 0.000345 approx.

Chic

Oops, I forgot the neg. sign.

x/57.3 =-x^2

x= -0.000345

Jings I must have been pissed when I did this.

Having used some mathematical lateral thinking in switching from
radians to degrees in order to obviate the need for a calculator, i.e
trying to be clever, I then go on to demonstrate my intrinsic need of a
calculator in order to do simple arithmetic.

The answer should have been 1/57.3 = 0.01745 degrees.

Also I was talking pish with x as a negative. Not allowed so the
transposition needs to be
x^2 = -sin x

Classic case of concentrating on the clever bit and ignoring the
simple.

Indeed. I think you're barking up the wrong tree (or should that be
log?) with this approach, Chic. Sin(x) = x is a useful approximation
to make when performing calculations using known values, but it is
fraught with dangers when solving equations involving unknowns.
Since |sin(x)| <= 1 for all real x (complex x hurts my brain too
much), regardless of units chosen, we can also state that, for this
problem, |x| <= 1. It doesn't necessarily follow that x << 1. If you
deliberately choose x very small to allow your approximation, you get
x + x^2 = 0. Factoring gives x(1+x) = 0, so x = 0 or -1. Now x = 0 is
the trivial solution. The other solution, x = -1, contains an
indeterminate error because of the error inherent in the original
approximation. Remember that sin(x) = x isn't just an approximation;
it's a limit. i.e. sin(x) --> x as x --> 0. Ignoring arithmetical
errors, the answer you got was basically the x = 0 solution (because
you assumed it to be very nearly so at the outset), offset by the
error introduced by your approximation.

[ObScot: Just in case anyone is wondering what this has to do with
Scottish culture, I give you:
The Maclaurin Series for sin(x):
sin(x) = x - (x^3)/3! + (x^5)/5! - (x^7)/7! + ... ]


x = - 0.01745

Shove it into your equation:-

sin(x)+x^2=0

and you get

- 0.0003045 + 0.0003045 = 0

And nope it is not just an error due to approximation.

Gotta go out so can't go into any more detail.

.

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