A New Vignette or two: Mathematics for Teachers’ Pleasure

Bill BartonThe University of Auckland

… seeks to speak to teachers about the mathematics that they deal with on a daily basis.

… is about contemporary mathematics:its themes,its problems,its excitements,and its applications.

“I hope you draw from mathematics a living stimulus for your teaching.”

“This book is designed solely as a mental spur, not as a detailed handbook.”

The international Klein Project is a joint project of ICMI and IMU.

It seeks to reach ALL upper secondary teachers—not just those who are already enthusiastic mathematicians, but it must also entice those who can rediscover their love for mathematics.

International MathematicalUnion (IMU)

The Klein Project is neutral with respect to the school curriculum: its structure, content, assessment, teaching modes, philosophy.

Klein materials are not intended as classroom resources—they are material for teachers.

(However we know that some teachers do use these materials in the classroom).

A Klein Vignette is a short piece about contemporary mathematics.

Vignettes are written with the intention that teachers will:…want to READ them;…want to KEEP reading them;…want to read MORE about the topic;…want to read ANOTHER one.

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Calculators & Power Series

Actually this is not (yet) a Vignette because it is not contemporary mathematics, but the application is contemporary, and may well be of interest to Yr 12 or Yr 13 students.

It can be found on the Klein Project WEBSITE (not the blog). Google “Klein Project” and it is the second item, then click on “Klein Vignettes”.

How does a calculator know all the values of sin(x) or the exponential or logarithmic functions? Surely it does not store all the values to many decimal places?

The answer lies in the field of power series, that is, series of the form:

Provided that |x| < 1 then this series usually converges quickly (depends on an).

Hence, if we can find a power series that will approximate sin(x) and other functions sufficiently accurately, then we have a way to evaluate those functions

Let us try it with sin(x). Let us assume that a power series can be found, so we have:

Can we find the coefficients ?

First, put x = 0.Then we have sin(0) = a0 so a0 = 0

We now have:

sin(x) = a1x + a2x2 + a3x3 + …

Differentiate:

cos(x) = a1 + 2a2x + 3a3x2 + …

And again, put x = 0.Then we have: cos(0) = a1 so a1 = 1

If we keep differentiating and putting x = 0, then we can find all subsequent terms:

But this is an infinite series—our calculator surely does not evaluate an infinitude of terms ? It turns our that this series converges quickly for all values of x. Indeed, even evaluating the first two or three terms gives us very good approximations.

Let us name the following partial sums:

How good are they as approximations?

y = sin(x)y = S5

y = S3

At x = 1, the error inS3 is about 0.008, and S5 is less than 0.001

We can improve these approximations considerably by using Chebyshev Polynomials … but I will leave that for you to read in the Vignette.

If we have a little more time, however, let us look at how we find power series expansions for another function or two.

A very simple power series is when every coefficient is equal to 1. This gives us:

which, you may remember,

We can create power series for other functions by substituting, for example, -x2 for x, or by differentiating both sides.

But look what happens if we integrate both sides…

Start with the simple power series but (for reasons that will come clear) write t instead of x:

Now integrate from 0 to x and multiply by -1:

So then:

Simon Newcombe (1835-1909)Frank Benford (1883-1948)

Invariant under change of scaleInvariant under change of base

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