MYTUTOR SUBJECT ANSWERS

852 views

How do I approximate an irrational square root without using a calculator?

Although this might look like a smart trick at the beginning, the truth is that as a physicist this kind of approximations turn out to be useful very often. This is quite likely to come up in the PAT as well!

It is important to get used to understanding symbolic expressions (no numbers!), so I will first derive the maths and then apply the resulting expression to a couple of illustrative examples.

Our aim is to obtain an approximate value for the inexact square root x of an arbitrary number n, i.e. to solve x = n1/2. In other words, we want to solve the equation f(x) = x2 = n. Suppose we know the exact root x0 of another number n0 which is very close to n, i.e. f(x0) = x02 = n0 (this is key!). We can do a Taylor series expansion of f(x) about x0 up to first order (first derivative f'(x)), which is simply:

f(x) = f(x0) + f'(x0)(x - x0).

The first derivative f'(x) = 2x, which evaluated at x0 is f'(x0) = 2x0. Substituting f(x) = n and f(x0) = n0 leads to:

n = n0 + 2x0(x - x0).

We are almost done! We know all the values of n, n0 and x0 and just need to solve for x. Rearranging:

x = x0 + (n - n0)/(2x0)

That's it! Remember that this expression is just an approximation for the actual square root of n, but a quite good one the closer n0 is to n.

Let's plug in some numbers. Suppose you were asked to approximate the square root of n = 66. Despite the solution being irrational, there is a close number n0 = 64 with a nice exact square root x0 = 8. Using the expression above we find:

x = 8 + (66 - 64)/(2*8) = 8 + 2/16 = 8 + 1/8 = 8 + 0.125 = 8.125

How good is this? It's just 0.01% off the actual value 8.12404...!

What if we chose n0 = 81 instead, the square root of which is x0 = 9? This would give x = 8.167 - worse, but still a reasonable result considering how far 81 is from 66. Never forget to quote both the positive and negative square roots in your final solution!

Would you be able to derive a similar expression for approximating a cubic root?

Sergio H. Uni Admissions Test .PAT. tutor, IB Spanish tutor, IB Maths...

2 years ago

Answered by Sergio, an Uni Admissions Test .PAT. tutor with MyTutor


Still stuck? Get one-to-one help from a personally interviewed subject specialist

18 SUBJECT SPECIALISTS

£25 /hr

Emily M.

Degree: Spanish and Philosophy (Bachelors) - Oxford, Brasenose College University

Subjects offered:.PAT., Spanish+ 8 more

.PAT.
Spanish
Philosophy and Ethics
French
English Literature
English Language
.MLAT (Modern Languages)
-Personal Statements-
-Oxbridge Preparation-

“Enthusiastic, patient Fluent French, Greek and Spanish speaker, happy to help with Oxford Admissions tests (PAT and MLAT), personal statements and oral exams.”

MyTutor guarantee

|  1 completed tutorial

£30 /hr

Mrinank S.

Degree: Engineering (Masters) - Cambridge University

Subjects offered:.PAT., Physics+ 3 more

.PAT.
Physics
Maths
Further Mathematics
-Oxbridge Preparation-

“Hey there! I'm a second year undergraduate student at the University of Cambridge, studying Engineering. I am really passionate about using science and maths to make the world around us a better place.  I am a patient and friendly tu...”

£25 /hr

Kamil F.

Degree: Mechanical Engineering (Masters) - Manchester University

Subjects offered:.PAT., Physics+ 2 more

.PAT.
Physics
Maths
Further Mathematics

“"It is the supreme art of the teacher to awaken joy in creative expression and knowledge." Albert Einstein”

About the author

Sergio H.

Currently unavailable: until 01/08/2015

Degree: Physics with Theoretical Physics (Bachelors) - Imperial College London University

Subjects offered:.PAT., Spanish+ 2 more

.PAT.
Spanish
Physics
Maths

“Third year Physics student at Imperial College. Looking to help you with the two fields I love, Physics and Maths, and with Spanish as a native speaker.”

MyTutor guarantee

You may also like...

Other Uni Admissions Test .PAT. questions

How are you to find 2007^2 − 2006^2 without a calculator?

How do I evaluate something like 3070^2-3069^2?

How many trailing zeros are there in 100! (100 factorial)?

If a bullet is shot into a suspended mass and does not pass through why can you not just use conservation of energy?

View Uni Admissions Test .PAT. tutors

We use cookies to improve your site experience. By continuing to use this website, we'll assume that you're OK with this. Dismiss

mtw:mercury1:status:ok