# Switzerland

### Mathematicians

Leonhard Euler was born in Basel Switzerland in 1707. Amazingly enough he took no mathematics in school and it was his father taught who began teaching him math at home. His interest in math led him to read math textbooks and hire a tutor to teach him mathematics. At the early age of 14 Euler began general studies at the University of Basel. Another famous mathematician Johann Bernoulli soon recognized Euler's aptitude and convinced him to focus his studies on math.

As a mathematician Euler found himself immersed in math which held implications for many other areas. He worked on government projects concerning ship building, fire engines, cartography(map making), magnetism and other machines. Mathematically he built on the early work of other mathematicians in areas of calculus, number theory, and differential equations. He also devised many new theorems and proofs. Euler held positions in Universities all over Europe including Berlin(Germany) and St. Petersburg (Russia). Meanwhile he married and had 13 kids!

### Problem

We are most familiar with some of Euler's mathematical notation f(x) for a function e for the base of natural logs and the symbol for pi. In the first activity we will be looking at functions for converting currencies. Here are some world currencies and their equivalents.:

Conversion Table

 CHF Switzerland Francs INR India Rupees Function 5 CHF = 181.625 INR I(x)= INR Indian Rupees EGP Egypt Pounds 3 INR = 0.403270 EGP P(x)= CLP Chile Pesos EGP Egypt Pounds 155 CLP = 1.50072 EGP P2(x)= ILS Israel New Shekels RUR Russia Rubles 10 ILS = 64.4591 RUR R(x)= ILS Israel New Shekels CLP Chile Pesos 1 ILS = 142.122 CLP C(x)=

Now based on the information given in the above table we can calculate a number of currency conversions. We will represent them using function notation as Euler would want. Let I(x) be the function that converts Swiss Francs to Indian Rupees and P(x) be the function that converts Indian Rupees to Egyptian Pounds etc. Complete the table by finding the functions.

We know that when performing multiple functions consecutively we can use a composition of functions. For example f(g(x)) is a composition of f(x) and g(x) where we first evaluate g(x) for some value of x and then take the result and substitute it for x in the function f(x).

What is the composition of functions for converting Swiss Francs to Egyptian Pounds?

Construct a new function called G(x). What is the above composition of functions represented as a single function?

We also know that many functions have an inverse. If a function f(x) goes from a unit x to a unit f(x), then the inverse of the function goes from a unit f(x) to a unit x. We can construct the inverse of a function by swapping the x the f(x) and then solving for f(x) which we now call . What is the inverse of the function C(x)?

The inverse of C(x) converts from what currency to what currency?

Finally, using the table determine the function, M(x) for converting between Swiss Franks and Russian Rubles and enter it in the box below to unlock your next colour.

M(x)= *x

### Geometry on a Sphere

Geometry as we know it on a sphere can be very tricky. Imagine you are somewhere on earth. Imagine you walk NORTH 5 kms, EAST 5 kms then SOUTH 5 kms. If you have ended up back where you started – where are you?

Try solving the International Color Challenge!