Wyatt N.

# what is the square root of 58

Robert J.

Wyatt,

You can use Newton's method.

f(x) = x^2 - 58

f'(x) = 2x

x(0) = 7

x(i+1) = x(i) - f(x(i))/f'(x(i))

x(1) = 7.642857

x(2) = 7.615821

...

x(10) = 7.615773106 <==Very close to v(58)

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11/19/12

Roman C.

tutor

Newton's method has a benefit of quadratic convergence (the new error is about the square of the old error with each step). Thus in the long run, the number of digits of accuracy doubles with each step. Such fast convergence only occurs if the root has multplicity 1, so that f(r)=0 but f'(r) ? 0, just like in this case.

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11/19/12

Roman C.

tutor

The comment system replaced my "not equal to" sign with a question mark.

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11/19/12

## 4 Answers By Expert Tutors

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5.0 (149)

I can provide your 'A-HA' moment

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4.6 (13)

Certified High School AP Calculus and Physics Teacher

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5 (2)

Kathryn D.

Roman, I haven't seen this method before... can you explain it in words because I am having a terrible time trying to follow the math with no idea how you are getting to the next step...

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11/18/12

Roman C.

tutor

This is based on the fact that (10a+b)2 - (10a)2 = 20ab+b2 = (20a+b)b. The right hand side of this equation is easier to compute which is why it is used.

For example, (20*7+6)*6 = 762-702, and (20*76+1)*1=7612-7602

Here, 76 is the largest integer fitting into v5800 and 761 is the largest integer fitting into v580000, etc.

The algorithm builds the answer up digit by digit so that it's square exhausts the number whose square root is taken. For every next digit, the largest one for which the answer so far doesn't exceed the true square-root is chosen.

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11/18/12

Roman C.

tutor

It is easy to verify the identity (10a+b)2 - (10a)2 = (20a+b)b, which is what is used starting at the second step onward. RHS is easier than LHS to calculate with.

For example (20*7+6)*6 = 762-702 and (20*76+1)*1 = 7612-7602 etc.

Notice also the 762 is the largest square not exceeding 5800 and 7612 is the largest square not exceeding 580000, etc.

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11/18/12

Michael B.

Now I know why people use slide rules!

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11/19/12

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