Algebra

# B.E.D.M.A.S.

Almost everything that surrounds us is a result of a process, and every process has certain rules. Rules make this world go round, and even though I’m a person who tends to live outside them, I still have to follow some of them in order to get by. Today I’m gonna talk about the first rules we learn in Maths: the BEDMAS (Brackets, Exponents, Division & Multiplication, Adition & Substraction) aka BODMAS aka PEDMAS (depends on the synonyms for Brackets and Exponents).

The rules simply say that first we start with things inside the brackets, then move onto exponents. Multiplication and division are all done at the same time, starting from the left and moving to the right. Then comes adition and substraction – again, these are done in the same time, starting from the left and moving to the right. Higher functions, such as logarithms and trigonometric functions, happen at the exponent level.

The levels make sense if you think about the operations that are performed. Adition is the most basic operation – the first we learn. Multiplication is really just successive aditions; that is to say, ‘two times five’ is really two added to itself five times. Next, an exponent just represents successive multiplications.

So, as you can see, as we move through the order of operations, we move towards ever more simple operations.

Sometimes we have to deal with nested brackets. Well, it’s no worry, we deal with them at a time, starting from the inside out.

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# 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:

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.

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.

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:

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

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

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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# 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 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.

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# 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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# Types of Numbers, part I

Well it was about time for a Maths related post, after all those introductions, right? My first one will introduce the very basics, the simbols we cannot live without: numbers. A number is just a number, right? Well, not really! Numbers are much like people: they belong to different groups. Just like a high school will have cool kids, the geeks and so on, so do numbers. In fact, some numbers are squares, some are perfect and one is even golden!

First of all, we should talk about number sets. Understanding the differences between the main types of numbers makes Maths way easier to get. I have to admit sets weren’t my friends when i met them, but my awesome personality made them like me. I came to the conclusion that numbers sets evolved with society and its needs.

Let’s go back  to 1000 BC and say that we’re growing trees. We count 1,2,3,4,5,.. apples, which actually represents the most basic of number system, the ‘natural’ or ‘counting’ numbers. These numbers worked very well for many years until the man got bored of what he had and looked in his neighbour’s yard and wanted his apples. But he wanted more apples than he had to exchange so his neighbour said ‘it’s ok, you can have them on debt’, which actually meant you had -n apples. Bam! Suddenly we got negative numbers. The positive and negative whole numers are called ‘integers’. The integer set is …, -3, -2, -1, 0, 1, 2, 3,.. and can also be expressed as 0, ±1, ±2, ±3,… and so on.

As society evolved, it had various products to consume so people couldn’t eat a whole apple or a whole bread, and they started to cut it into pieces and eat portions of it. Half an onion with 2 and a half potatoes , one and a half tomatoes and 3,5 l of water make an excellent soup!  These are fractions, or rational numbers. This set includes any number that can be expressed in the form ab, where a and b are integers and b cannot equal zero – dividing by zero is a big no-no! Another way to express this is to say that rational numbers can be any ‘terminating’ or ‘repeating’ decimal. For example, 1/4 is equal to 0,25 (‘terminating’ decimal) and 1/3 equals 0,(3) (‘repeating’ decimal). But the debt idea is still valid so the rational numbers can be negative too.

As years passed by, the industry has appeared and with it more complicated operations which didn’t stop at division. All these other numbers that appeared and that couldn’t be expressed as a fraction, are called ‘irrational’ numbers. Two good examples of irrational numbers are π and √2 (square root of 2); these are weird numbers because they continue forever, without repeating or terminating.

Now if we take a final look at these sets we can see so many relations between them. The natural set is included in the integer set. The integer set is included in the rational numbers set. I always imagined these sets as Russian dolls, the irrational set being separated from the others.

Another important thing we didn’t talk about is zero’s place. He’s not a part of the natural set because it’s not natural to count nothing, but he is a whole number. We use zero on a daily basis, but few of us know its significance. Zero is a vital part of our place-value system. Without it 206 and 26 would look very similar indeed. Although this might seem obvious to us now, the theoretical leap required to develop a symbol that represents nothing is very impressive –  and neither the ancient Greeks nor the Romans had a representation of zero. The Indian mathematician Grahmagupta was the author of the first text to treat zero as a number. It’s sometimes said that you cannot ponder the infinite until you have pondered zero. In fact, this pondering of zero and the infinite is a big part of calculus – the nightmares of almost every student.In essence, calculus is used in science, economics and engineering to look at the infinitely large and small. So, not to put too fine a point on it, the appearance of zero was a huge moment in the history of Mathematics.

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# 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.

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# 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. 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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