Zero Breaks the Rules

I wanted to announce my first Campaign on Teespring: Zero. The idea of the campaign is to celebrate the importance of Zero in mathematics. Nowadays it is hard to imagine that Zero was not always considered a number and all the problems this thing might involve. The official description of the campaign is:

Show that you can break rules together with Zero. This has been one of the most controversial numbers in the history of mathematics. Show your passion for mathematics! Show your passion for Zero!

Since ancient time, not all civilization have understood and considered the existence of zero. This is especially because zero is closely linked to infinity and the idea of nothingness (or void). The Babylonians were the first to kind of recognize zero, but they have used it more like a place holder and not an independent number like 1, 2 or 9. The Egyptians, Greeks and then Romans were in a way scared of zero and its strange characteristics. They did not understand zero’s mathematical properties and they considered it different than the other numbers. Like Charles Seife says in the book Zero: The Biography of a Dangerous Idea: “A lone zero always misbehaves.” Zero was breaking the rules!!! To understand this better think of the axiom of Archimedes, which states that if you add something to itself enough times it will exceed any other number. This is true for all the natural numbers we know (and they knew at the time): 1 + 1 = 2; 2 + 2 = 4; 2 + 2 + 2 = 8. But zero was different because 0 + 0 = 0, it contradicted one of the most important axioms of the time. Also, think about subtraction: 2 – 1 = 1, but 2 – 0 = 2 (it changed nothing). If we go to multiplication and division things got even more stranger for them. Moreover, we all know the old problem of never dividing by zero. All the mathematics crumbles when we divide by zero. And they were shocked by this thing.

Zero was a concept that intervened in their religious concepts and it was hard to understand. Even after some mathematicians have shown the flows in doing mathematics without zero (the best example is Zeno’s Paradox) western philosophers still found it very hard to accept this. Taking this into consideration I have created this Zero campaign with the slogan: Zero breaks the rules. If you think this is a great idea, or if you just want to support my work in general, take a look at the campaign page. You will find many things there, from T-shirts to Sweatshirts, Hoodies, Tote Bag and Stickers.

 

If you to find out more about this number, check an older post of mine: Number zero.

Hope you enjoyed this post and that you are excited for the future ones. Have a great week. You can find me on Facebook, Tumblr, Google+, Twitter  and Instagram. I will try to post there as often as possible.

Don’t forget that maths is everywhere! Enjoy! 

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Birthday Paradox Question

Not a long time ago I got a very interesting question on The Birthday Paradox and I thought I will share it with you.

The question is as follows:

It’s almost a Mathematical certainty that in a group of 100 people at least 2 will have the same birthday, in my smallish group of friends (about 100) I have just noticed 4 pairs have the same birthday, Also one of the pairs has the same birthday as me. I know theoretically 367 people will definitely give us 1 pair, but what is the largest group among our friends WITHOUT 2 people sharing a common birthday?

With this question I thought I will say a couple of more things about The birthday Paradox. Everything about this is about probabilities, nothing more or less. To understand what I am talking about, here is an example: in a room of just 23 people there’s a 50-50 chance of two people having the same birthday and in a room of 75 there’s a 99.9% chance of two people matching. To understand more about how to calculate this, I recommend you read the article Understanding the Birthday Paradox. It ends with a very useful formula, which helps us understand more. Also, I have used that formula to try and get an answer for the above question. Here is the formula (with n = the number of people):

\displaystyle{p(n) = 1 - \left(\frac{364}{365}\right)^{C(n,2)} = 1 - \left(\frac{364}{365}\right)^{n(n-1)/2} }

My logic goes as follow. For example, in a group of just 23 people there’s a 50-50 chance of two people having the same birthday. Or you can say that there is a 50 – 50 chance of 2 people not having the same birthday. As the group gets smaller the probability decreases, such that in a group of 10 people the probability that 2 people have the same birthday is 12%. I think that because we talk about probability (%) there is not a precise answer.

Therefore, if we think about the largest group which has no coincidental birthdays, we want our probability to be 0, i.e. we want to find such that P(n) = 0.  If we take a look at the above formula, we observe that the 2nd term should be 1. Because that is an exponential function, we should know that the power should be 0 to get 1. Therefore, we go directly to the fact that n(n-1)/2 = 1, which gives us immediately n = 0 or n = 1. So I would say we need a group of just 1 person, because a group with 0 people is not a group anymore. Quite interesting that even in a group of 2 people there is a probability of 0.2% for them to have the same birthday.

I have to confess that I got quite excited about this question, but I have the feeling there should be more to it. If there is anyone that has any ideas let me know in the comment box bellow, I would love to know more.

If you want to read more about probability, I totally recommend the following 3 books: Probability: A Very Short Introduction (Very Short Introductions) by John Haigh, Probability: For the Enthusiastic Beginner by David J. Morin and Probability: An Introduction by Geoffrey Grimmett.
  

Hope you enjoyed this post and that you are excited for the future ones. Have a great week. You can find me on Facebook, Tumblr, Google+, Twitter  and Instagram. I will try to post there as often as possible.

Don’t forget that maths is everywhere! Enjoy! 

Taxi-Cab Geometry

I have read about this topic a while ago and when I saw a workshop on this at the One-Day Conference for the Teacher of Secondary Mathematics I wanted to participate. Taxi-Cab Geometry looks at things in a completely different way than what we are used to. If you are used with different metric spaces, than this is going to be just a quick game for you.

In our day to day life, we measure distance by virtue of the theorem of Pythagoras. The theorem provides a metric, but if we change the way we measure (in other words, if we change the metric), do we change the appearance of things? Is the square going to look like a square, or the circle like a circle? It sounds a little strange first, but you may not realize that the way we measure, or calculate, the space in which we live fixes the shapes that we see around us. Thus if we change the way we measure or calculate we may well change many other things.

Defining taxi-cab geometry is not very hard. The definition Wikipedia offers is quite good: Taxicab geometry, considered by Hermann Minkowski in 19th-century Germany, is a form of geometry in which the usual distance function of metric or Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences of their Cartesian coordinates. But the best way to see this type of geometry is to literally consider its name. Think about it as if you are a Taxi driver and the way you move into a town where the roads are the lines from a square paper, or more easily, think about it as if you live in a city with no round-abouts. To better understand the difference between Euclidean geometry (our day to day geometry) and Taxi-Cab geometry, take a look at the image bellow ( in taxicab geometry, the red, yellow, and blue paths all have the shortest length of 12. In Euclidean geometry, the green line has length approximate 8.49 and is the unique shortest path).

2000px-manhattan_distance-svg

During the workshop we played a little with finding distances between points and what different shapes would look like, which was quite interesting and relaxing. To get an idea, here is an image which shows you different shapes in taxi-cab geometry. Also, I would totally encourage you to try and drawn a circle, you will be surprised.

taxicab_shapes

After the workshop I was thinking about the fact that this type of geometry could be extremely useful in our society and it could be used more. It could really help us position some buildings, such as hospitals and police, so that they are at equal distances from different points of the city. Also, building schools could be another interesting application of this. I was quite interested by problems like the following:

taxicab-geometry-presentation-5-728

I hope I have made you more curios about this topic and its applications. If you want to find out more, I recommend Taxicab Geometry: An Adventure in Non-Euclidean Geometry (Dover Books on Mathematics) by Eugene F. Krause:
 

If you want to read more about my day at the conference, read One Day in Stirling. Also, if you are interested in some new way to teach mathematics in a more graphical way, I recommend reading about another workshop from the conference, Graphical Method for teaching Maths.

Hope you enjoyed this post and that you are excited for the future ones. Have a great week.  You can find me on FacebookTumblrGoogle+Twitter  and Instagram. I will try to post there as often as possible.

Don’t forget that maths is everywhere! Enjoy! 

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Graphical method for Teaching Maths

This post is inspired by one of the workshops I participated at the One-Day Conference for the Teachers of Secondary Mathematics. The workshop was about how Singapore students are taught to use some sort of graphical method to solve increasingly complex problems such as ratio, percentage change or simultaneous equations.

This method is called the Singapore Bar Model. The only important part in all of this is to make children use bar models to represent different types of problems. I have a couple of examples bellow:

  1. Two apples and a banana cost £1.95. Two apples and 3 bananas cost £3.45. How much does an apple cost?  This is a typical algebra – simultaneous equations – type questions. Generally students are taught to start writing the 2 equations. The complains I normally get from pupils at this stage is that they do not really know what to do first. This graphical method seems to access their intuition more and give a purpose to the algebra that comes after. The bar representation for this problem is easier than you think: 14273299_10210578089788651_1729199487_o
  2. The ratio of the number of Katie’s marbles to Emma’s marbles is 4:5. Katie is given 12 more marbles. Katie now has twice as many marbles as Emma. How many marbles did Katie have to begin with? Again another more algebraic question, which most of the students find quite hard to start it. Their intuition is a little lost at the start and even though they know what ratio is, starting the question is not straight forward. Using this bar method, the question gets very easy. The only problem I would say is reading the question paying attention to details and then making the bar: 14284962_10210578089468643_1639686826_o
  3. Alison gave 1/6 of her sweets to her Mum and 2/5 of the remainder to her Dad and has 6 sweets left. How many sweets did she have at first? This is another example which could go wrong if we go straight to algebra notation. Here is the bar model option: 14281500_10210578089068633_1221809777_n

We have done many other examples during the workshop, but there a couple of things which are very important. The method gives the students a way to use their intuition better by giving them the opportunity to represent it graphically. Moreover, it gives them a good way to start thinking about a problem. Also, it is a great way to start explaining the algebra that follows normally after using the drawings. If you want to find out more about this method, I recommend Bar Modeling: a Problem Solving Tool: Professional Development Book by Houghton Mifflin Harcourt : 

If you want to read more about my day at the conference, read One Day in Stirling.

Have you ever used this method? Or maybe you have similar that you use from time to time. Just let me know in the comment box bellow. I would love to find out more. Hope you enjoyed this post and that you are excited for the future ones. Have a great week.  You can find me on Facebook,TumblrGoogle+Twitter  and Instagram. I will try to post there as often as possible.

Don’t forget that maths is everywhere! Enjoy! 

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One Day in Stirling

Yesterday I was at my first Mathematical Association Secondary Education One-Day Conference at University of Stirling and it was a great experience. In this post I will make a short description of what I did there with further posts about the maths in the following 2 weeks. Enjoy!

First I have to say that this was the first time I went to Stirling and even though I stayed just for a couple of hours there, it impressed me. I really like it.

The conference started with a normal registration and coffee slot at 9:00 am and continued with an open lecture about what teacher could learn from the pupils presented by Jenny Golding. It was an interesting lecture with great examples of maths activities that could be used in the classroom and some quite interesting reactions and question from children. Next, the conference offered the opportunity to chose 4 workshops for the rest of the day.

For the first workshop I have chosen “Using the Singapore Bar in High Schools” by Andrew Jeffrey, where we got the opportunity to see how Singapore students are thought to bar model to solve increasingly complex problems, such as ratio, percentage change, simultaneous equations and more. The presenter offered some interesting tips and I cannot wait to try some out in my lessons. Also, he started with an interesting puzzle. Take a look and let me know what you think (U/D = up/down & L/R = left/right):

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Next on my list was “Further Rigour and Recreation” by Clive Chambers, where I have found out even more interesting historical facts about Pi. We also discussed about different ways to solve a problem in order to asses different topics. We ended the workshop with 3 puzzles to solve during lunch. I still need to solve one of them and I became a little obsessed with it.

First workshop after lunch break (lunch break = the perfect time to buy books) was “Taxi-Cab Geometry” by Tom Roper. I really needed something like this to remind me of my university days (metric space) and also to remind me that I should continue doing more higher maths. The workshop explored the way the world looks using the taxi-cab metric and gave us extra resources for further exploration.

The last workshop was “AM Approximate Countdown” by Chris Smith. I really wanted to see one of his workshops because I heard it is quite fun and presents interesting concepts that not everyone knows about. I wasn’t disappointed, it made me laugh and made me feel more enthusiast about my future lessons.

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It was great fun and I cannot wait to go to more Conferences during the year. In the future next 2 weeks I will write more posts about the workshops and what I have learnt from them.

Hope you enjoyed this post and that you are excited for the future ones. Have a great week.  You can find me on Facebook,TumblrGoogle+Twitter  and Instagram. I will try to post there as often as possible.

Don’t forget that maths is everywhere! Enjoy! 

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Mathematicians born in August

Until Christmas I have decided to substitute my monthly favorites with a post about 3 great mathematicians born in the previous month. So, because today is the 1st of September, I have decided to write a little about 3 great mathematicians born in August. Enjoy!

John Venn, born on the 4th of August 1834, was an English logician and philosopher noted for introducing the Venn diagram, used in the fields of set theory, probability, logic, statistics, and computer science. Previously, I have written a post on Venn Diagrams. When it comes to mathematics we use the Venn diagrams especially with sets. It is very easy to understand any set operations using this diagrams. Basically we learn the definition of union, intersection, symmetric difference and any other things using the diagrams.

Arthur Cayley, born on the 16th August 1821, was a British mathematician. He helped found the modern British school of pure mathematics. As a child, it seems that he enjoyed solving complex maths problems for amusement. He postulated the Cayley–Hamilton theorem—that every square matrix is a root of its own characteristic polynomial, and verified it for matrices of order 2 and 3. He was the first to define the concept of a group in the modern way—as a set with a binary operation satisfying certain laws. Formerly, when mathematicians spoke of “groups”, they had meant permutation groups. Cayley’s theorem is named in honour of Cayley. Also, I have mentioned one of his theorems in the post Maths in “Good Will Hunting”.

Portrait of Arthur Cayley

Scientific Identity, Portrait of Arthur Cayley

Jacques Tits, born on the 12th of August 1930 (age 86), is a Belgium-born French mathematician who works on group theory and Incidence geometry, and who introduced Tits buildings, the Tits alternative, and the Tits group. Before over thinking about his surname, he did incredible work in understanding Lie Groups – a very important part of group theory – and many other applications.

320px-Jacques_Tits_(2008)

Special mention for this month is Niels Abel. I did not want to write about him in this post because I have done that in the past: Niels Abel.

niels-henrik-abel

Hope you enjoyed this short post. Have a great week.  You can find me on Facebook, TumblrGoogle+Twitter  and Instagram. I will try to post there as often as possible.

Don’t forget that maths is everywhere! Enjoy! 

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Maths in “Good Will Hunting”

I have been trying to find some great maths related movies recently and I have found “Good Will Hunting”. It is an old movie (1997), but even though I have heard a lot about it I never watched it. So, I thought it is the time to give it a try. The movie follows a 20 year old laborer Will Hunting, an unrecognized genius who, as part of a deferred prosecution agreement after assaulting a police officer, becomes a client of a therapist and studies advanced mathematics with a renowned professor.

The movie is incredible and I loved it. You get to see how Will re-evaluates his relationships with people around him and how he confronts his past and decides about his future. Totally recommend this movie. I this post I don’t want to talk about the sentimental part, but I want to mention some interesting mathematics which appears in it.

The problem I am talking about is the one which appears at the beginning of the movie, when the professor gives his students a tricky task:

gwh-00079

The problem is not extremely easy to understand because it does involve quite a lot of university level maths: linear algebra (elementary theory of matrices, powers of matrices, Jordan normal-form), analysis (convergence in normed vector spaces, power series, convergence of power series), combinatorics  (generating function, counting, recurrence formulae) and graph theory (adjacency matrix, paths, powers of the adjacency matrix).

The problem mostly comes from the mathematics field called Graph Theory. This is  the study of graphs – mathematical structures which model pairwise relations between objects. A graph in this context is made up of vertices, nodes, or points which are connected by edges, arcs, or lines. We can say that graphs can be undirected (there is no distinction between the 2 vertices associated with each edge) and directed (its edges are directed from one vertex to another).

It turns up that in the end the problems is related to Cayley’s formula stating that the number of labeled trees on n nodes is nn-2. Then he lists 8 different unlabeled trees with 10 nodes. To make more light into this, you have to understand that a tree is an undirected graph in which any two vertices are connected by exactly one path. In case you were wondering, mathematics has also the notion of forest in this case: a disjoint union of trees.

For a more mathematical explanation, I advice you to read Mathematics in Good Will Hunting II: Problems from the Students Perspective. Also, Numberphile has a great movie on this problem:

Totally advice you to read more about this and maybe (why not?!) start reading about graph theory (click the image for more information):
   

Have a great week.  You can find me on FacebookTumblrGoogle+Twitter  and Instagram. I will try to post there as often as possible.

Don’t forget that maths is everywhere! Enjoy!

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