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:

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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

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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, General

Types of Numbers, part II

In the previous post we talked about how numbers have a varied social life, and they belong to different groups, like our chess clubs or gyms. In this post we’ll take a look at how we group numbers.

One of the most interesting groups of numbers is the prime number set, which is actually a subset of the natural numbers. A prime is a natural number with exactly two distinct natural-number divisors: one and itself. To make it more clear, a prime is a natural number that is evenly divisible only by one and itself. That is to say that if you divide a prime number by any other natural number you will get a fraction or decimal. We have a few conditions: a negative number cannot be a prime; and one itself is not a prime.

On the opposite part of the fence there are the composite numbers. A composite number is a natural number that has a positive divisor other than one and itself, which means that composite numbers are all the natural numbers that are not prime, except one. The number one is neither prime, nor a composite.

Also, I told you guys in the previous post that some numbers are perfect. Sounds nice, huh?  Well a ‘perfect’ number is one for which the sum of all its proper divisors (whole number divisors) is equal to the number. Let’s take 6 as an example; it’s divisors are: one, two and three; if we sum them up we have 6! These perfect numbers are quite rare and really cool. The next perfect number is 28, which has the following divisors: 1, 2, 4,7,14; summing them up we have 28! And so on.. The next perfect number would be 496, and after that it’s 8128.

I’ll come back tomorrow with more neat numbers! Cheers!

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Algebra, Analysis, General, Geometry

A Short History of Mathematics

Textbooks nowadays don’t show kids where Maths has come from, or when it was discovered. In my opinion Maths or any other part of the Science field was here long before humans can decipher it. It has been shown that crows can distinguish between sets of up to four elements, a fact that demonstrates that counting occurs in other creatures. Maths has a very long history, not very known by many, and though I had a passion for it all my life, I only read about it’s history during college. And not even then, it’s roots were so clear to me. And after reading some books I could only draw the conclusion that if you want to study the history of maths, you have to study the history of civilization. Although many important results of mathematics go way back to Renaissance, we have to wonder what made them possible? Who decided which language to use, what simbols to use as numbers, or what were the numbers by the way ? Fibonacci was the one who introduced Hindu-Arabic numerals to Europe in the thirteenth century, and freed the mathematicians from the constraints of Roman numerals. ancient math So basically, Maths has been developed everywhere in the world, but the speed of it’s advance was not the same for each region. Ideas have been discovered and lost, and then refound. The modern mathematics borrows ideas from many different places, but the combination between Arabic and Persian Maths with the Greek and Indian gives what we learn today. Though it was discovered in different times in different parts of the world, the results were basically the same, and thanks to the Renaissance mathematicians we have the results that everybody uses now.

Maths hasn’t been so developed at it’s beginning, obviously, but it raised many interesting questions, and gave freedom to think. Nowadays, Maths results are rarely given, but it became used in many other new fields as Computer Science. But you don’t have to be a computer whiz or a math genius to appreciate the beauty of numbers. You don’t have to understand the equation that stands behind everyday things, you only have to become more aware of Mathematics’ influence in the world around you.

And as the great Rene Descartes said, “With me everything turns into mathematics”, make it possible with you and the world will be bigger, more meaningful and more beautiful.

 

photo from: http://graphics8.nytimes.com

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