Applying Newton's Method without the need of a calculator

[miniradman]

Hello everyone

Although I do not have a specific homework question to ask, I do have a question which directly relates to a topic I do indeed have for homework.

One of our topics for the semester are approximating roots using Newton's method, and as I understand the value you one obtains after using this particular method is quite accurate. However, our course bans any use of a calculator whilst doing any of the topics including Newtons method. I've attempted to apply the method to cubic polynomials, and irrational square roots and I cannot seem to get past the first iteration. Once I start involving fractions which are unclean (eg. \frac{161}{11230} ) that I have to cube or root, I find my self taking up one or two whole pages of working out (converting decimals to fractions, squaring fractions, adding/subtracting fractions... all that good stuff) which is a luxury that I won't be getting in my exam.

I know this is a far fetched/slightly silly question, but... are there any easier ways to use Newtons methods without a calculator?

Because if we can use calculator, why take the approximation? When we can get an answer correct to so many decimal places?

My apologies if this is in the wrong forum.

 

[mfb]

I don't think you will have to calculate many numbers like the root of 11230 without a calculator with a precision better than a few percent (and that does not need so much time: the approximation 100 is already good up to 6%, and another iteration leads to 106, which is less than 0.03% away from the exact value).

are there any easier ways to use Newtons methods without a calculator?

Easier than what? There are many tricks to speed up calculations in general.

Because if we can use calculator, why take the approximation?

To understand how the calculator gets its approximation.

 

[miniradman]

I don't think you will have to calculate many numbers like the root of 11230 without a calculator with a precision better than a few percent (and that does not need so much time: the approximation 100 is already good up to 6%, and another iteration leads to 106, which is less than 0.03% away from the exact value).

Thank you for the reply mfb

One question is to approximate \sqrt{2} to 4 decimal places. Although I don't know of any reliable methods to compute these kind of calculations without the aid of a calculator.

Easier than what? There are many tricks to speed up calculations in general.

The method I normally use is to simply take the decimal value and convert it into a fraction and working with my fractions. However, that can be hard enough when multiplying to very large numbers together, let alone converting it back to a decimal number afterwards.

To understand how the calculator gets its approximation.

Surely... a calculator must use a different method

 

[vela]

One question is to approximate \sqrt{2} to 4 decimal places. Although I don't know of any reliable methods to compute these kind of calculations without the aid of a calculator.

The method I normally use is to simply take the decimal value and convert it into a fraction and working with my fractions. However, that can be hard enough when multiplying to very large numbers together, let alone converting it back to a decimal number afterwards.

You're going to have to trust that your instructor won't be a jerk and give you a problem with unwieldy numbers on the exam.

Applying Newton's method to find $\sqrt{2}$ is pretty easy to do by hand. If you think it's too much work, your arithmetic skills are very bad, in which case you seriously need to improve them, or you need a reality check as your expectations are out of whack.

Of course, there are efficient ways to do the calculations and inefficient ways. You get a feel for what works best by doing problems like these.

Surely... a calculator must use a different method

Like what? Remember, unlike humans, a calculator doesn't mind doing arithmetic out to 14 or 15 digits.

 

[mfb]

Surely... a calculator must use a different method

They usually do, but those methods are harder to understand and probably harder to calculate for a human.