Mathspp: mathematics and programming

Pt En I have folded a piece of paper in half hundreds of times in my life. And probably so did you. Folding a piece of paper in half is fairly easy: just bend the piece of paper until the corners meet, and then crease. That is it. And with this method one can also fold a piece of paper in $4$, in $8$, etc. We just have to successively divide the sections of the paper in half. But what if we wanted to fold a piece of paper into thirds, as in the picture above? Some people are good at doing that, but they don't really measure anything: they just do it approximately by looking at the paper and folding where it seems about right. I guess it goes without saying, but mathematicians don't like things to be "about right", they want them right... and even though I wasn't a mathematician, when I was a child I thought that maybe there was a way for me to successively fold different parts of the paper in half, until one of the creases would be the crease at...

Pt En Being able to do basic arithmetic calculations in your head is a great skill. Not because it is sexy but because it is useful in your daily life: it can help you check the change you are given when shopping, it can help you know if you will have enough money to pay for your groceries, it can help you estimate how much things cost after the discounts, etc... This often reduces to being able to sum and subtract decently; sometimes you need to make a couple of small multiplications, but that is it. More likely than not, you don't need to compute averages every day. But sometimes you just want the scoring average of your team for the past few games, or the average price per person of a given meal, or the average time you spent stuck in traffic this past week... And averages may appear nastier than simply adding or subtracting, because averages also require you to perform a division: in fact, you have to add all the numbers you want and then divide the total by ho...

Pt En An obvious way of creating rational approximations for irrational numbers is by truncating its decimal expansion. For example, $3$, $3.1$ and $3.14$ are all rational approximations of $\pi $; as fractions, those approximations would be written $3$, $\frac{31}{10}$ and $\frac{314}{100} $. Notice how $\frac{314}{100}$ has $100$ as the denominator and yet only produces an approximation correct up to two decimal places. Claim: by using continued fractions one can obtain better rational approximations for irrational numbers. Method: if $x $ is an irrational number, instead of truncating its decimal expansion, we can truncate its continued fraction. Taking $\pi $ as an example, we have $$\pi = 3 + \frac1{7 + \frac1{15 + \cdots}} $$ and by taking $$\pi \approx 3 + \frac17 = \frac{22}{7} $$ we get the approximation $\pi \approx 3.14285\cdots$: it is correct up to two decimal places just as $\frac{314}{100} $, but $7$ is a much smaller denominator than $100$. (And al...

Pt En What if I told you that right now there is a place on Earth where there is no wind blowing to the sides? None at all! How can I know that? All we need is what is usually called the Hairy Ball Theorem . In less rigorous contexts, one can phrase the Hairy Ball Theorem as such: Theorem (Hairy Ball I): if you have a hairy ball, regardless of the way you comb its hair there will always be a spot where the hair points right up. In this particular image, the hair is pointing up both on the top and on the bottom. More formally, the Hairy Ball Theorem can be formulated like so: Theorem (Hairy Ball II): every continuous vector field over $S^2$ has at least a point where the tangential component is $0$. From this theorem it is actually quite easy to establish our interesting fact! If we think of the wind at the Earth's surface as a continuous vector field, the Hairy Ball Theorem says that there must be a point where the wind isn't blowing to the sid...

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