Square root of 4.1- do you have to use calculus?

[Femme_physics]

Square root of 4.1-- do you have to use calculus?

In order to solve square root of 4.1, is there a simple arithmetic method or do you have to use calculus and use that "tangent line approximation formula"?

I wonder how do calculators do it... does anybody know?

 

[l'Hôpital]

I'm assuming you by "tangent line approximation method" you mean Newton's method. What exactly is wrong with it?

I suppose if you wanted a more "elementary method", you could the following:
<br /> x_{n+1} = \frac{1}{2} (x_n + \frac{4.1}{x_n} )<br />

That sequence will converge to your square root, and it's an analog of Newton's method.

 

[Femme_physics]

Oh, okay, I see the next topic in my book is Newton's method. Guess I should've read that first :shy:

 

[Femme_physics]

Okay, I read Newton's method. Both methods don't give the exact results (to all sig fig). Do calculators give the exact result? I wonder.

 

[G037H3]

Question: what book are you reading?

 

[Femme_physics]

Calculus textbook, of course! It's produced by the OpenU of my country (IL)

 

[phyzguy]

Okay, I read Newton's method. Both methods don't give the exact results (to all sig fig). Do calculators give the exact result? I wonder.

No, calculators don't give an exact result, in general. How could they, since they only display a finite number of digits? They use an approximation algorithm like the one described above.

 

[Borek]

In order to solve square root of 4.1, is there a simple arithmetic method

Define 'simple'.

http://www.homeschoolmath.net/teaching/square-root-algorithm.php

 

[HallsofIvy]

Okay, I read Newton's method. Both methods don't give the exact results (to all sig fig). Do calculators give the exact result? I wonder.

\sqrt{4.1}= \frac{\sqrt{41}}{\sqrt{10}}

Neither 41 nor 10 is a perfect square so their square roots are irrational. Further, they have no common factors so the ratio of \sqrt{41} to \sqrt{10} is irrational. It cannot be written as a finite decimal expansion nor as a ratio of integers (fraction). No, calculators do not give an exact result for a problem like that- the exact result cannot be written in any "place value" notation.

 

[slider142]

Note, however, that if the calculator displays n digits of the result, those n digits are the correct first n digits of the decimal expansion of the number. The approximative methods allow you to calculate the correct digits up to any precision you desire.

 

[Femme_physics]

Thanks Halls, guys. That pretty much clears it up for me :)