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#176 2013-02-10 02:05:37

Agnishom
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From: The Complex Plane
Registered: 2011-01-29
Posts: 18,872
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Re: A few questions

bobbym wrote:

Hi;

You can also get quite close using:

gif.latex?\large%20\bold{%205^{44}=%20\frac{10^{44}}{\left%20(%202^{10}%20\right%20)^4%20\cdot%202^4}}

Hi bobbym,
What should I be doing after getting this form?

Hi stefy,
How? If I could have approximated the logarithm of 5^44, then there would have been no problem at all. Can you do it?

Last edited by Agnishom (2013-02-10 02:07:04)


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#177 2013-02-10 02:18:08

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 90,615

Re: A few questions

10^44 is a 45 digit number.

2^10 ≈ 1000 = 10^3

(10^3)^4 = 10^12

10^44 / (10^12 * 2^4) = 10^32 / 2^4

Now 2^4 = 1.6 *10

10^32 / ( 1.6 * 10) = 10^31 / 1.6 ≈ 10^30.

10^30 is a 31 digit number. But this is only an approximation.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#178 2013-02-10 02:23:52

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 18,872
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Re: A few questions

Yes, yes, I get it

Well, what had stefy been talkiing about?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#179 2013-02-10 02:28:35

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 90,615

Re: A few questions

You can use Taylor's series to get log(5), but it will require a little bit of calculus.

If you want to see how...

Last edited by bobbym (2013-02-10 03:15:30)


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#180 2013-02-11 01:24:08

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 18,872
Website

Re: A few questions

Indeed I want to see,

My problem is that I am not acquainted enough with calculus


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#181 2013-02-11 01:26:39

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 90,615

Re: A few questions

You are a little young for your school system to have taught you it. They prefer a gradual introduction to it.

I am using log(x) to denote the common logarithm and ln(x) as the natural log.

You start with the Mclaurin series

gif.latex?\small%20\inline%20\dpi{120}%20\bg_green%20\fn_cs%20\ln\left%20(\frac{1+x}{1-x}%20\right%20)=2%20x+\frac{2%20x^3}{3}+\frac{2%20x^5}{5}+\frac{2%20x^7}{7}+...+

If we substitute x = 2 / 3 in the we get:

gif.latex?\small%20\inline%20\dpi{120}%20\bg_green%20\fn_cs%20\ln\left%20(\frac{1+\frac{2}{3}}{1-\frac{2}{3}}%20\right%20)=%20\ln(5)\approx%20\frac{2\%202}{3}+\frac{2}{3}%20\left(\frac{2}{3}\right)^3+\frac{2}{5}%20\left(\frac{2}{3}\right)^5+\frac{2}{7}%20\left(\frac{2}{3}\right)^7%20=%201.600261284211901

So ln(5) ≈ 1.600261284211901

But we needed log(5) not ln(5)! How do we get it?

We use this relationship

gif.latex?\small%20\inline%20\dpi{120}%20\bg_green%20\fn_cs%20\log(x)=\frac{\ln(x))}{\ln(10)%20}

gif.latex?\small%20\inline%20\dpi{120}%20\bg_green%20\fn_cs%20\log(5)=\frac{\ln(5))}{\ln(10)%20}

gif.latex?\small%20\inline%20\dpi{120}%20\bg_green%20\fn_cs%20\log(5)=\frac{\ln(5))}{\ln(2)%20+%20\ln(5)}

gif.latex?\small%20\inline%20\dpi{120}%20\bg_green%20\fn_cs%20\log(5)%20\approx%20\frac{1.60026}{0.693\,%20+1.60026}%20\approx%200.698970004

which is quite close to the true value.

Last edited by bobbym (2013-02-11 02:37:12)


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#182 2013-02-11 01:54:28

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 18,872
Website

Re: A few questions

Let me try it out


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#183 2013-02-11 02:03:23

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,657

Re: A few questions

Hi bobbym

gif.latex?\frac{\ln{\left%20(1+\frac{2}{3}%20\right%20)}}{1-\frac{2}{3}}\neq%20\ln{5}


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#184 2013-02-11 02:07:24

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 90,615

Re: A few questions

Please hold on the latex maker is not holding up, I am aware of that and am editing the original post.

All done!

Last edited by bobbym (2013-02-11 02:27:57)


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Offline

#185 2013-03-14 04:32:52

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 18,872
Website

Re: A few questions

Please help me with the problem attached

View Image: Infinite Circles.PNG

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#186 2013-03-14 05:07:38

bob bundy
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Registered: 2010-06-20
Posts: 6,531

Re: A few questions

hi Agnishom,

I'm not getting this yet.  I've drawn a line L .  Now it says 'on a straight line l' so I assumed that the circles had their centres on the line, and that each circle touches the ones on either side of it.  That covers the 'externally tangential to circle(n-1) and circle(n+1) but apparently also tangential to L itself.  How does that happen?  And I haven't even got to the second sentence yet.

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#187 2013-03-14 08:05:57

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,657

Re: A few questions

I am getting 1/2.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#188 2013-03-14 10:42:50

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 90,615

Re: A few questions

Hi Agnishom;

bobbym wrote:

A picture is worth a thousand words.

For people like me who are geometrically challenged this problem is a nightmare. So far I have come up with 10^16 different possible drawings that to me fit those constraints.

Now I am busy looking up externally tangent, internally tangent, tangerine tangent... in the hopes of what externally tangent to the line means. Intersects at one point?

I see that bob has got his circles with their centers on the line, I have lifted my circles and it was hard to do, after all there are an infinite amount of them, to roll on the line. That takes care of externally tangent. Now this second bunch of circles, where do they go?

John McCarthy wrote:

As the Chinese say, 1001 words is worth more than a picture.

Maybe I shouldn't need the diagram...

Last edited by bobbym (2013-03-14 19:08:18)


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Offline

#189 2013-03-14 13:59:39

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 18,872
Website

Re: A few questions

Bob, if you want I can give a diagram


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#190 2013-03-14 14:12:12

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 18,872
Website

Re: A few questions

This is the only diagram I can imagine.

A rough sketch though...

1001 Words ≈ 7 KB
1 picture > 100 KB

Obviously, a picture is worth more....

View Image: Infinite Circles.PNG

Last edited by Agnishom (2013-03-14 14:14:14)


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#191 2013-03-14 14:45:43

Nehushtan
Member
From: London
Registered: 2013-03-09
Posts: 615
Website

Re: A few questions

Agnishom wrote:

Please help me with the problem attached

This is what I got.


Let
be the centre of
and
the point of contact of
with
. Let
be the centre of
and
the point of contact of
with
.

Let

be the point on
such that
is a rectangle. Applying Pythagoras’s theorem to
gives
.

Let

be the point on
such that
is a rectangle. Applying Pythagoras’s theorem to
gives
.

Let

be the point on
such that
is a rectangle. Applying Pythagoras’s theorem to
gives
.

Then

Hence:

[align=center]

[/align]

Last edited by Nehushtan (2013-03-14 14:53:34)


160 books currently added on Goodreads

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#192 2013-03-14 18:19:41

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 90,615

Re: A few questions

Hi Agnishom;

Thanks for the drawing.

Last edited by bobbym (2013-03-14 21:19:25)


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Offline

#193 2013-03-14 19:18:18

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 18,872
Website

Re: A few questions

Thanks Nehushtan
I shall ask you if I have any doubts


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#194 2013-03-15 13:54:42

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 18,872
Website

Re: A few questions

Please upload a diagram, I really do not understand from "Applying Pythagoras Theorem..."


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#195 2013-03-16 04:09:30

Nehushtan
Member
From: London
Registered: 2013-03-09
Posts: 615
Website

Re: A few questions

Here you go.

View Image: Circles.png

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#196 2013-03-16 16:06:15

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 18,872
Website

Re: A few questions

Thank you smile


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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