## Posts tagged ‘calculus’

### Rational approximations of √2

Introducing undergraduates to rational approximations of √2 can be an opportunity to insidiously tell them about many parts of mathematics they certainly don’t want to hear about. In a less pessimistic way, I would say this is a nice way to illustrate the use of several theories in abstract mathematics.

First, you may want to tell them it is not a rational number, which could be easy, unless they never heard about factoring integers.

Then, you could use classical sequences from high school classes: it is easy to check that iterating converges to ±√2, and setting they will even be able to give an explicit formula with a geometric sequence. There is of course a well-known algorithm which speeds up considerably the computation : *Newton’s method*. This can be illustrated geometrically by drawing the graph of a function having √2 as a root (for example ):

- take a rough approximation, such as x=1
- imagine the function is affine (replacing the graph by its tangent)
- use the approximation of the function to calculate an approximation of the solution
- use this newly found rough solution to iterate from step 1

This method requires iterating , and converges considerably faster. Both methods yield sequences of rational numbers converging to √2, so should be considered as methods to obtain fractional numbers giving good approximations of √2. (more…)

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