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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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, Analysis

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.

russian dolls

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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Algebra, Analysis, General, Geometry

An Introduction.

Words have never come easy for me, they only made sense when I’ve read them written by someone else. This is one of the reasons I’ve been attracted to Science. Unfortunately Science covers a wide range of subjects such as Physics, Chemistry and many others related to them. I’m an Aries, so obviously I wanted to top all those classes, and I did, but couldn’t discover them in depth. So I focused on studying the language that connects all the Science fields: Maths. One can study Physics/Chemistry all his life, but without this irreplaceable tool you get nowhere. Even Einstein had the help of Max Planck (well-known german physicist) when he developed the Relativity Theory, because it was known that Maths wasn’t one of his strenghts.

Mathematics means many things to many people. Most see it as a daunting subject, whether you try to solve an ecuation or add up the bills, almost every time you end up with a headache. But for the others it represents the beauty of the universe. An English Matematician and philosopher, described it as “the most original creation of the human spirit”.

Maths has been described in many ways: the science of numbers and magnitude, the science of patterns and relationships or the language of science. Galileo claimed that “The Laws of Nature are written in the language of mathematics”. Very very true! Just by seeing the award winning movie “A Beautiful Mind”, we can see how Maths apply in the day-to-day activities. Unfortunately, neither the media or the books cover the most important part of it: it’s root, it’s history, the place it came from! I’ve been asked more than once by my pupils who discovered the numbers and the operations, and I was ashamed to admit that I didn’t have a clue, no one ever told me and I didn’t even ask myself. So my future posts will come to help your curiosities and maybe widen your horizons. Cheers 😉

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