Algebra, Analysis, Geometry

Pi

A number that needs no introduction, π is an MVP when it comes to Maths. Many people have one on their shirts, even I’m thinking about getting a tattoo with it. Why? Cause people like π, it’s their connection to mathematics, beyond the mundane arithmetic of everyday life and let’s face it, it sounds sweet! For many, it’s their first introduction to infinity.

Pi, or π, is defined as the ration of the circumference of a circle to its diameter:

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This often leads to confusion among people because as I told you in an previous post, π is ‘irrational’, which means that it cannot be expressed as a fraction. The thing we need to remember is that a fraction has integer elements. But with π either the circumference or the diameter will be irrational. This is interesting and strange: it means that if you can write the value of the diameter, you will never be able to write the exact value of the circumference as a decimal, and vice versa.  It’s kinda hard to measure a circle, right? We don’t really have a circular ruler, do we?

The idea of π as a constant has been around for millennia. The Egyptians estimated it at 25/8 (or 3.125), while the Mesopotamians gave it a value of 3.162.

tabel pi

Archimedes was the first to examine π in depth. By drawing polygons both inside and outside the circle, and calculating their perimeters, he was able to estimate for π between 223/71 and 22/7, which is where the common aproximation of π as 22/7 comes from. Since Archimedes’ time the accuracy of π has been increasing, especially with the IT development for which we are thankful for billions of digits.

archimedes

The symbol of π was introduced by William Jones in 1706 in his book Synopsis Palmariorum Mathesios. However, π can also be represented as an infinite series of numbers. The Indian mathematician and astronomer, Madhava, produced the following series:

serie1

This can be used to estimate π, but it’s slow. The very famous Swiss mathematician Leonhard Euler used the series:

serie2

While another intersting series was given by John Wallis which he published in 1656. It starts off with:

serie3

Without drowning too deep into the mathematics, these series show some of the many properties of π; and perhaps this is the reason for its enduring appeal.

How’s π affecting your everyday life? Well think about the speedometer or how to calculate the volume of every tin can. 😉

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Algebra, Geometry

Types of numbers, part III

Hey guys! This is the last post about numbers, I promise! In this last one we’ll talk about some numbers that have geometrical tendencies. They are pretty spectacular!

When you say “five squared equals twenty-five”, did you ever thought why we call it ‘squared’? Well, the Greeks were big on geometry and applied it to numbers as well. Twenty-five is a square number because you can arrange twenty-five dots to form a five-by-five square. In fact, twenty-five is the fifth square number, or n=5. We grow familiar to this family of square numbers from early age and we play with it very often in our problems, only because they’re fun to analyse. They have one of the most important properties in algebra, they are always positive!

Now let me introduce to you a less well-known set: the triangular numbers (1, 3, 6, 10, 15, 21,…). They earned their name by forming triangles of dots.

Knowing these two kinds of numbers, we can easily see that some numbers are both square and triangular. I’ll show some of them below!

square triangular numbers

36

 

Square and triangular numbers are only two of many geometric (or figurate) number sets, and the table below shows the first few along with their formulas – just replace n with any number and you will find the corresponding geometric number: Geometric numbers even exist in three dimensions; for example, there are ‘tetrahedral’ numbers that are the sums of triangular numbers and form a pyramid with a triangular base.

geometric numbers

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Algebra, Analysis, General, Geometry

The Language of Mathematics

“Mathematics knows no races or geographic boundaries; for mathematics the cultural world is one country.” David Hilbert (1862-1943)

I started studying Maths in Romanian, and I have to admit, I was a very dull kid. Why I say that? Because I didn’t have the mind of a genious, I didn’t have a vision of the world or etc. In time, I’ve come to the conclusion that if you want to know something, you have to know to ask the right questions. And every science man knows that asking question makes the whole algorithm! So I started asking myself what if I want to talk to people about Maths, my parents, friends, colleagues, would I know how to explain it? And what if I wanted to talk to a foreign person, how would I talk to him, would he understand me? Is Math the same for everybody?

Well of course it is, it’s a universal language! The simbols are the same for each country! I bought a book written in English, which was about Pythagoras’ theorem when I was 13 years old, and I just learned about it at school. I understood some words and even some sentences, but it didn’t matter because all I needed was explained to me with symbols I knew how to read from Maths class. During college I had the opportunity to share my knowledge with exchange students, and professors from other countries for whom I held presentations in Analysis and Geometry. I must admit that my English isn’t so fluent as I wished for, but they understood me completely as I did too when they were in my place. That’s the beauty of it!

Amazingly, Maths may be universal in the truest sense of the word. And it is fr this reason that the Search for Extraterrestrial Intelligence uses binary representations of π and prime numbers to broadcast our presence to anyone who might be listening. Why would that be? Maybe because they might talk another language and not understand the word hello, but a circle is a circle for them too, and they would know what π is, and the binary concept would be obvious (on/off, day/night).

So what do you think now? Isn’t it beautiful?  Maths is dynamic and its ever-present nature is its most powerful quality.

pi.bw

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