IB Maths and GCSE Maths Resources. Theory of Knowledge (ToK). Real life maths. Maths careers, Maths videos, Maths puzzles and Maths lesson resources.

The site is based around the idea of maths ToK – which tries to get students thinking about more than just syllabus content and to start exploring philosophical ideas, historical connections, cutting edge research and real life applications in mathematics.

Some of the content in the site includes:

  • An IB ToK Maths syllabus plan with a huge amount of ideas for incorporating maths ToK into lessons.
  • A large “Flipping the classroom” videos section for IB students.  These cover pretty much the entire IB HL and SL syllabus – with each topic taught in a short 10 minute or so video.  This should help students both prepare for lessons and also should be invaluable for revision.
  • A new School Code Challenge activity which allows students to practice their code breaking skills – each code hides the password needed to access the next level.
  • Over 200 ideas to help with students’ Maths Explorations – many with links to additional information to research.
  • A large number of posts on everything from imagining extra dimensions to modelling asteroid impacts.

If you would like to contact me, you can do so here. There’s loads of content – so please explore!

Unbelievable: 1+2+3+4…. = -1/12 ?

The above video by the excellent team at Numberphile has caused a bit of an internet stir – by providing a proof that 1+2+3+4+5+… = -1/12

It’s well worth watching as an example of what proof means – if something is proved which we “know” is wrong, then should we accept it as true?  The particular proof as offered in the video is certainly open to question – even if the end result: 1+2+3+4+5+… = -1/12 can actually be proved under certain definitions, using the Riemann Zeta function.

Grandi’s Series

The proof in the video requires that firstly we accept that the infinite summation, 1-1+1-1+1-1… = 1/2.  This series is known as the Grandi’s Series - and has been the cause of arguments in the mathematical community for centuries as to what the infinite summation should actually be.   One method (called Cesaro Summation) gives an answer of 1/2 – which is the answer accepted in the video.

Alternative interpretations of Grandi’s series would be to group the numbers as 1 + (-1+1) + (-1+1) +(-1+1)…. which you would expect to equal 1. Or, we could group the numbers as (1-1) + (1-1) + (1-1) … which you would expect to equal 0.  Therefore it would be also mathematically valid to say that the infinite summation 1-1+1-1…  has no sum.

Divergent Series are the invention of the Devil

For the proof in the video to be valid we have to therefore accept that the sum of Grandi’s series is 1/2.  We also need accept that it is possible to manipulate infinite series by “shifting them along by 1″ or by factorising. 

However as we have already seen in the case of Grandi’s series, infinite series don’t always follow normal arithmetic rules. Indeed, the 19th century Norwegian mathematician Niels Abel, warned that that, “Divergent series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever!”

Nevertheless it is an interesting method.  First they define 3 different infinite series:

S =  1 + 2 + 3 + 4 + 5 …..

S1 = 1 – 1 + 1 – 1 + 1 – 1 ….

S2 = 1 – 2 + 3 – 4 + 5…

Step 1:

The first step is to state that S1 = 1 – 1 + 1 – 1 + 1 – 1 …. = 1/2.

Step 2:

if S2 = 1-2+3-4+5…
then 2S2 = 1 -2+3-4+5…
                    +1-2+3-4…

Here we have “shifted along by one space” the second S2. This means that when we add the two sequences together we end up with:

2S2    = 1-1+1-1+1…  = 1/2
which gives S2 = 1/4.

Step 3:

Do S – S2 = 1 + 2 + 3 + 4 + 5 …..
                -(1 – 2 + 3 – 4 + 5…)
                = 4 + 8 + 12 + ….
                = 4(1 + 2 + 3….)
                = 4(S)

Now if S – S2 = 4S we can simply rearrange this equation and substitute the value of S2 = 1/4 which we found before to give: S = -1/12

As mentioned above this is not a very rigorous proof. There is a more rigorous (and complicated) method of proving this – which is the method used by Euler, and which employs the Riemann Zeta function. You can watch this method here:

You might notice when watching this proof that at the start of the video they use the infinite summation of a geometric sequence formula – which is only valid for absolute x less than 1. Then later on they substitute x = -1 into a result derived from it. This is OK because of analytical continuation (which is a method of extending the domain of a function beyond its usual domain). This idea starts to get really complicated – but if you’re interested in the basic idea look at the post on the Riemann Sphere below. The Riemann Sphere allows infinity to be included in the domain of the complex numbers.

If you enjoyed this post you might also like:

Mathematical Proof and Paradox: How you can “prove” things like 1 = 2. Can you spot the flaws in the logic?

The Riemann Hypothesis: How the Riemann Zeta function is fundamental to understanding the prime numbers – and how solving the Riemann Hypothesis is one of the greatest puzzles in mathematics.

The Riemann Sphere – an introduction to isomorphic mappings, which is a lot more interesting than it sounds!

Maths Studies IA Exploration Topics:

Make sure you read the Maths Studies guidance from the IB prior to starting your IA maths exploration – this linked site gives the full list of assessment criteria you will be judged against and also gives 9 full examples of investigations students have done.

Given the assessment criteria it’s probably easiest to conduct a data analysis investigation, though you can choose to explore other parts of the syllabus instead.  To get good marks make sure you carefully follow the marking criteria points given by the IB and try and personalise your investigation as much as possible.  Be innovative, choose something you are interested in and enjoy it!

Primary or Secondary data? 

The main benefit of primary data is that you can really personalise your investigation.  It allows you scope to investigate something that perhaps no-one else has ever done.  It also allows you the ability to generate data that you might not be able to find online.  The main drawback is that collecting good quality data in sufficient quantity to analyze can be time consuming.    You should aim for an absolute minimum of 50 pieces of data – and ideally 60-100 to give yourself a good amount of data to look at.

The benefits of secondary data are that you can gain access to good quality raw data on topics that you wouldn’t be able to collect data on personally – and it’s also much quicker to get the data.  Potential drawbacks are not being able to find the raw data that fits what you  want to investigate – or sometimes having too much data to wade through.

Secondary data sources:

1) The Census at School website is a fantastic source of secondary data to use.  If you go to the random data generator you can download up to 200 questionnaire results from school children around the world on a number of topics (each year’s questionnaire has up to 20 different questions).  Simply fill in your email address and the name of your school and then follow the instructions.

2) If you’re interested in sports statistics then the Olympic Database is a great resource.  It contains an enormous amount of data on winning times and distances in all events in all Olympics.  Follow links at the top of the page to similar databases on basketball, golf, baseball and American football.

3) If you prefer football, the the Guardian stats centre has information on all European leagues – you can see when a particular team scores most of their goals, how many goals they score a game, how many red cards they average etc.

4) The Guardian Datablog has over 800 data files to view or download – everything from the Premier League football accounts of clubs to a list of every Dr Who villain, US gun crime, UK unemployment figures, UK GCSE results by gender, average pocket money and most popular baby names.  You will need to sign into Google to download the files.

5) The World Bank has a huge data bank - which you can search by country or by specific topic.  You can compare life-expectancy rates, GDP, access to secondary education, spending on military, social inequality, how many cars per 1000 people and much much more.

6) Gapminder is another great resource for comparing development indicators – you can plot 2 variables on a graph (for example urbanisation against unemployment, or murder rates against urbanisation) and then run them over a number of years. You can also download Excel speadsheets of the associated data.

Example Maths Studies IA Investigations:

Some of these ideas taken from the excellent Oxford IB Maths Studies textbook.

Correlations:

1) Is there a correlation between hours of sleep and exam grades?
Studies have shown that a good night’s sleep raises academic attainment.
2) Is there a correlation between height and weight?
The NHS use a chart to decide what someone should weigh depending on their height. Does this mean that height is a good indicator of weight?
3) Is there a correlation between arm span and foot height?
This is also a potential opportunity to discuss the Golden Ratio in nature.
4) Is there a correlation between the digit ratio and maths ability?
Studies show there is a correlation between digit ratio and everything from academic ability, aggression and even sexuality.
5) Is there a correlation between smoking and lung capacity?
6) Is there a correlation between GDP and life expectancy?
Run the Gapminder graph to show the changing relationship between GDP and life expectancy over the past few decades.
7) Is there a correlation between numbers of yellow cards a game and league position?
Use the Guardian Stats data to find out if teams which commit the most fouls also do the best in the league.
8) Is there a correlation between Olympic 100m sprint times and Olympic 15000m times?
Use the Olympic database to find out if the 1500m times have go faster in the same way the 100m times have got quicker over the past few decades.
9) Is there a correlation between sacking a football manager and improved results?
A recent study suggests that sacking a manager has no benefit and the perceived improvement in results is just regression to the mean.
10) Is there a correlation between time taken getting to school and the distance a student lives from school?
11) Does eating breakfast affect your grades?
12) Is there a correlation between stock prices of different companies?
Use Google Finance to collect data on company share prices.
13) Does teenage drinking affect grades?
A recent study suggests that higher alcohol consumption amongst teenagers leads to greater social stress and poorer grades.
14) Is there a correlation between unemployment rates and crime?
If there are less work opportunities, do more people turn to crime?
15) Is there a correlation between female participation in politics and wider access to further education?
16) Is there a correlation between blood alcohol laws and traffic accidents?
17) Is there a correlation between height and basketball ability?
18) Is there a correlation between stress and blood pressure?

Normal distributions:

1) Are a sample of student heights normally distributed?
We know that adult population heights are normally distributed – what about student heights?
2) Are a sample of flower heights normally distributed?
3) Are a sample of student weights normally distributed?
4) Are a sample of student reaction times normally distributed?
Conduct this BBC reaction time test to find out.
5) Are a sample of student digit ratios normally distributed?
6) Are the IB maths test scores normally distributed?
IB test scores are designed to fit a bell curve. Investigate how the scores from different IB subjects compare.
7) Are the weights of “1kg” bags of sugar normally distributed?

Other statistical investigations

1) Does gender affect hours playing sport?
A UK study showed that primary school girls play much less sport than boys.
2) Investigation into the distribution of word lengths in different languages.
The English language has an average word length of 5.1 words.  How does that compare with other languages?
3) Do bilingual students have a greater memory recall than non-bilingual students?
Studies have shown that bilingual students have better “working memory” – does this include memory recall?
4) Investigation about the distribution of sweets in packets of Smarties.  A chance to buy lots of sweets!  Also you could link this with some optimisation investigation.

Modelling using calculus

1) How can you optimise the area of a farmer’s field for a given length of fence?
A chance to use some real life maths to find out the fence sides that maximise area.
2) Optimisation in product packaging.  
Product design needs optimisation techniques to find out the best packaging dimensions.

Probability and statistics

1) The probability behind poker games
2) Finding expected values for games of chance in a casino.
3) Birthday paradox:
The birthday paradox shows how intuitive ideas on probability can often be wrong.  How many people need to be in a room for it to be at least 50% likely that two people will share the same birthday? Find out!
4)  Which times tables do students find most difficult?
A good example of how to conduct a statistical investigation in mathematics.
5) Handshake problem
With n people in a room, how many handshakes are required so that everyone shakes hands with everyone else?

Other ideas

If you want to do an investigation with a bit more mathematical content then have a look at this page for over 100 ideas for Maths SL and HL students.

geometry

Visualising Algebra Through Geometry

This picture above is a fantastic example of how we can use geometry to visualise an algebraic expression.  It’s taken from Brilliant – which is a fantastic new forum for sharing maths puzzles.  This particular puzzle was created and uploaded by Arron Kau.  The question is, which of the following mathematical identities does this image represent?

geometry2

See if you can work it out!  I will put the answer in white text at the bottom of the post – highlight it to reveal the solution.

Another example of the power of geometry in representing mathematical problems is provided by Ian Stewart’s Cabinet of Mathematical Curiosities.  The puzzle itself is pretty famous:

A farmer wants to cross a river and take with him a wolf, a goat, and a cabbage. There is a boat that can fit himself plus either the wolf, the goat, or the cabbage. If the wolf and the goat are alone on one shore, the wolf will eat the goat. If the goat and the cabbage are alone on the shore, the goat will eat the cabbage. How can the farmer bring the wolf, the goat, and the cabbage across the river?

wolf goat cabbage

The standard way of solving it is trial and error with some logic thrown in.  However, as Ian Stewart points out, we can actually utilise 3 dimensional geometry to solve the puzzle.  We start with a 3D wolf-goat-cabbage (w,g,c) space (shown in the diagram).  All 3 start at (0,0,0).  0 represents this side of the bank, and 1 represents the far side of the bank.  The target is to get therefore to (1,1,1).  In (w,g,c) space , the x direction represents the wolf’s movements, the y direction the goat and z the cabbage.  Therefore the 8 possible triplet combinations are represented by the 8 vertices on a cube.

We can now cross out the 4 paths:

(0,0,0) to (1,00) as this leaves the goat with the cabbages

(0,0,0) to (0,0,1) as this leaves the wolf with the goat

(0,1,1) to (1,1,1) as the farmer would leave the goat and cabbage alone

(1,1,0) to (1,1,1) as the farmer would leave the wolf and goat alone.

which reduces the puzzle to a geometric problem – where we travel along the remaining edges – and the 2 solutions are immediately evident.

(eg.  (0,0,0) – (0,1,0) – (1,1,0) – (1,0,0) – (1,0,1)- (1,1,1)   )

What’s really nice about this solution is that it shows how problems seemingly unrelated to mathematics can be “translated” in mathematics – and also it shows how geometrical space can be used for problem solving.

Solution to the initial puzzle, highlight to reveal: The answer is the third option – 13 + 23…. = (1+2+….)2. This is quite a surprising identity. You can see it by seeing that there are (for example) 2 squares of length 2 – this gives you a total area of 2x2x2 = 23. Adding all the squares will give you the same area as a square of sides (1+2+3….)(1+2+3….) – hence the result.

 

 

Pierre_de_Fermat
Fermat’s Theorem on the sum of two squares

Not as famous as Fermat’s Last Theorem (which baffled mathematicians for centuries), Fermat’s Theorem on the sum of two squares is another of the French mathematician’s theorems.

Fermat asserted that all odd prime numbers p of the form 4n + 1 can be expressed as:

p = x^2 + y^2

where x and y are both integers.  No prime numbers of the form 4n+3 can be expressed this way.

This is quite a surprising theorem – why would we expect only some prime numbers to be expressed as the sum of 2 squares?  To give some examples:

13 is a prime number of the form 4n+1 and can be written as 32 + 22.
17 is also of the form 4n + 1 and can be written as 42 + 12.
29 = 52 + 22.
37 = 62 + 12.

Prime numbers of the form 4n + 3 such as 7, 11, 19 can’t be written in this way.

The proof of this theorem is a little difficult.  It is however easier to prove a similar (though not logically equivalent!) theorem:

All sums of x2 + y2 (x and y integers) are either of the form 4n + 1 or even.

In other words, for some n:

x2 + y2 = 4n + 1 or

x2 + y2 = 2n

We can prove this by looking at the possible scenarios for the choices of x and y.

Case 1:

x and y are both even (i.e. x = 2n and y = 2m for some n and m).  Then

x2 + y2 = (2n)2 + (2m)2
x2 + y2 = 4n2 + 4m2
x2 + y2 = 2(2n2 + 2m2)

which is even.

Case 2:

x and y are both odd (i.e. x = 2n+1 and y = 2m+1 for some n and m).
Then x2 + y2 = (2n+1)2 + (2m+1)2
x2 + y2 = 4n2+ 4n + 1 + 4m2 + 4m + 1
x2 + y2 = 4n2+ 4n + 4m2 + 4m + 2
x2 + y2 = 2(2n2 + 2m2 + 2m + 2n + 1).
which is even.

Case 3:

One of x and y is odd, one is even. Let’s say x is odd and y is even. (i.e. x = 2n+1 and y = 2m for some n and m).
Then x2 + y2 = (2n+1)2 + (2m)2
x2 + y2 = 4n2+ 4n + 1 + 4m2
x2 + y2 = 4(n2+m2+n) + 1
which is in the form 4k+1 (with k = (n2+m2+n) )

Therefore, the sum of any 2 integer squares will either be even or of the form 4n+1. Unfortunately this does not necessarily imply the reverse: that all numbers of the form 4n+1 are the sum of 2 squares (which would then prove Fermat’s Theorem). This is because,

A implies B
Does not necessarily mean that
B implies A

For example,

If A is “cats” and B is “have 4 legs”
A implies B (All cats have 4 legs)
B implies A (All things with 4 legs are cats).

A is logically sound, whereas B is clearly false.

This is a nice example of some basic number theory – such investigations into expressing numbers as the composition of 2 other numbers have led to some of the most enduring and famous mathematical puzzles.

The Goldbach Conjecture  suggests that every even number greater than 2 can be expressed as the sum of 2 primes and has remained unsolved for over 250 years. Fermat’s Last Theorem lasted over 350 years before finally someone proved that a2 + b2 =c2 has no positive integers a, b, and c which solve the equation for n greater than 2.

If you liked this post you might also like:

The Goldbach Conjecture – The Goldbach Conjecture states that every even integer greater than 2 can be expressed as the sum of 2 primes.  No one has ever managed to prove this.

Mathematical Proof and Paradox - how we can “prove” the impossible

mathematician lament

This is a fantastic passage – which is part of the Mathematician’s Lament by Paul Lockhart.  He goes into a lot more detail in the pdf (available to read here ).  He really highlights some of the absurdities in how mathematics is both viewed and taught in society.

A musician wakes from a terrible nightmare. In his dream he finds himself in a society where music education has been made mandatory. “We are helping our students become more competitive in an increasingly sound-filled world.”

Educators, school systems, and the state are put in charge of this vital project. Studies are commissioned, committees are formed, and decisions are made— all without the advice or participation of a single working musician or composer. Since musicians are known to set down their ideas in the form of sheet music, these curious black dots and lines must constitute the “language of music.”

It is imperative that students become fluent in this language if they are to attain any degree of musical competence; indeed, it would be ludicrous to expect a child to sing a song or play an instrument without having a thorough grounding in music notation and theory. Playing and listening to music, let alone composing an original piece, are considered very advanced topics and are generally put off until college, and more often graduate school.

As for the primary and secondary schools, their mission is to train students to use this language— to jiggle symbols around according to a fixed set of rules: “Music class is where we take out our staff paper, our teacher puts some notes on the board, and we copy them or transpose them into a different key. We have to make sure to get the clefs and key signatures right, and our teacher is very picky about making sure we fill in our quarter-notes completely. One time we had a chromatic scale problem and I did it right, but the teacher gave me no credit because I had the stems pointing the wrong way.”

In their wisdom, educators soon realize that even very young children can be given this kind of musical instruction. In fact it is considered quite shameful if one’s third-grader hasn’t completely memorized his circle of fifths. “I’ll have to get my son a music tutor. He simply won’t apply himself to his music homework. He says it’s boring. He just sits there staring out the window, humming tunes to himself and making up silly songs.”

In the higher grades the pressure is really on. After all, the students must be prepared for the standardized tests and college admissions exams. Students must take courses in Scales and Modes, Meter, Harmony, and Counterpoint. “It’s a lot for them to learn, but later in college when they finally get to hear all this stuff, they’ll really appreciate all the work they did in high school.”

Of course, not many students actually go on to concentrate in music, so only a few will ever get to hear the sounds that the black dots represent. Nevertheless, it is important that every member of society be able to recognize a modulation or a fugal passage, regardless of the fact that they will never hear one.

“To tell you the truth, most students just aren’t very good at music. They are bored in class, their skills are terrible, and their homework is barely legible. Most of them couldn’t care less about how important music is in today’s world; they just want to take the minimum number of music courses and be done with it. I guess there are just music people and non-music people. I had this one kid, though, man was she sensational! Her sheets were impeccable— every note in the right place, perfect calligraphy, sharps, flats, just beautiful. She’s going to make one hell of a musician someday.”

Waking up in a cold sweat, the musician realizes, gratefully, that it was all just a crazy dream. “Of course!” he reassures himself, “No society would ever reduce such a beautiful and meaningful art form to something so mindless and trivial; no culture could be so cruel to its children as to deprive them of such a natural, satisfying means of human expression. How absurd!”

The whole pdf is well worth a read.

If you enjoyed this you might also like:

Maths and Marking - Why are the current accepted school methods lagging behind evidence of what best raises attainment?

differential

Real life use of Differential Equations

Differential equations have a remarkable ability to predict the world around us.  They are used in a wide variety of disciplines, from biology, economics, physics, chemistry and engineering. They can describe exponential growth and decay, the population growth of species or the change in investment return over time.  A differential equation is one which is written in the form dy/dx = ……….  Some of these can be solved (to get y = …..) simply by integrating, others require much more complex mathematics.

Population Models

One of the most basic examples of differential equations is the Malthusian Law of population growth dp/dt = rp shows how the population (p) changes with respect to time.  The constant r will change depending on the species.  Malthus used this law to predict how a species would grow over time.

More complicated differential equations can be used to model the relationship between predators and prey.  For example, as predators increase then prey decrease as more get eaten. But then the predators will have less to eat and start to die out, which allows more prey to survive.  The interactions between the two populations are connected by differential equations.

differential2

The picture above is taken from an online predator-prey simulator .  This allows you to change the parameters (such as predator birth rate, predator aggression and predator dependance on its prey).  You can then model what happens to the 2 species over time.  The graph above shows the predator population in blue and the prey population in red – and is generated when the predator is both very aggressive (it will attack the prey very often) and also is very dependent on the prey (it can’t get food from other sources).  As you can see this particular relationship generates a population boom and crash – the predator rapidly eats the prey population, growing rapidly – before it runs out of prey to eat and then it has no other food, thus dying off again.

differential 2

This graph above shows what happens when you reach an equilibrium point – in this simulation the predators are much less aggressive and it leads to both populations have stable populations.

differential 3

There are also more complex predator-prey models – like the one shown above for the interaction between moose and wolves.  This has more parameters to control.  The above graph shows almost-periodic behaviour in the moose population with a largely stable wolf population.

Some other uses of differential equations include:

1) In medicine for modelling cancer growth or the spread of disease
2) In engineering for describing the movement of electricity
3) In chemistry for modelling chemical reactions
4) In economics to find optimum investment strategies
5) In physics to describe the motion of waves, pendulums or chaotic systems.

With such ability to describe the real world, being able to solve differential equations is an important skill for mathematicians.  If you want to learn more, you can read about how to solve them here.

If you enjoyed this post, you might also like:

Langton’s Ant – Order out of Chaos How computer simulations can be used to model life.

Does it Pay to be Nice? Game Theory and Evolution. How understanding mathematics helps us understand human behaviour

 

platonic

Is maths invented or discovered?  One of the most interesting questions to investigate with regards to maths Theory of Knowledge (ToK) is the relationship between maths and reality. Why does maths describe reality? Are the mathematical equations of Newton and Einstein inventions to describe reality, or did they exist prior to their discovery? If equations exist independent of discovery, then where do they exist and in what form?  The below passage is a brief introduction to some of the ideas on this topic I wrote a while back.  Hopefully it will inspire some further reading!

Mathematics and Reality

We live in a mathematical universe. Mathematics describes the reality we see, the reality that we can’t, and the reality that we suppose. Mathematical models describe everything from the orbital path of Jupiter’s moons, to the flight of a football through the air, from the spiral pattern of a shell to the evolution of honey bee hives, from the chaotic nature of weather, to the expansion of the universe.

But why should maths describe reality? Why should there be an equation linking energy and mass, or one predicting the decay of a radioactive atom or one even linking three sides of a triangle? We take the amazing predictive powers of mathematics for granted, and yet these questions lead onto one the most fundamental questions of all – is mathematics a human invention, created to understand the universe, or do we simply discover the equations of mathematics, which are themselves woven into the fabric of reality?

The Second Law of Motion which links force, mass and acceleration, drawn up by Sir Isaac Newton in 1687, works just as well on the surface of Mars as it does on Earth. Einstein’s equations explaining the warping of space time by gravity apply in galaxies light years away from our own. Heisenberg’s uncertainty principle, which limits the information we can know simultaneously about a subatomic particle applied as well in the post Big Bang universe of 13.7 billion years ago as it does today. When such mathematical laws are discovered they do not simply describe reality from a human perspective, but a more fundamental, objective reality independent of human observation completely.

Anthropic reasoning

Anthropic reasoning could account for two of the greatest mysteries of modern science – why the universe seems so fine-tuned for life and the “unreasonable effectiveness of mathematics” in describing reality.

The predictive power of mathematics might itself be necessary for the development of any advanced civilisation. If we lived in a universe in which mathematics did not describe reality – i.e. one in which we could not use the predictive mathematical models either explicitly or implicitly then where would mankind currently be?

At the core of mathematical models are an ability to predict the consequences of actions in the natural world. A hunter gatherer on the African savannah is implicitly using a parabolic flight model when throwing a spear, if mathematical models do not describe reality, then such interactions are inherently unpredictable – and the evolutionary premium on higher cognition which has driven human progress would have been significantly diminished. Our civilisation, our progress, our technology is all founded on the mathematical models that allow us to understand and shape the world around us.

Anthropic reasoning requires that the act of conscious questioning itself is taken into account. In other words, it is certain that we would live in a universe both fine-tuned for mathematics and fine-tuned for life because if our universe was not, we would not be an advanced civilisation able to consider the question in the first place.

This reasoning does however require that we simply accept what appear to be the vanishingly small probabilities that such a universe would be created by chance. For example, Martin Rees, in his book, “Just Six Numbers” looks at six mathematical constants which were they to alter even slightly would create a universe which could not support life.

Whilst tossing a coin and getting 20 heads in a row is unbelievably unlikely, if you repeatedly do this millions of times, then such an occurrence becomes practically assured. Therefore using this mathematical logic, any vanishingly small probabilities can be resolved. The universe is the way that we observe it, precisely because it is a universe taken from the set of all universes in which we can observe it.

Mathematics as reality

An even more intriguing possibility is that maths doesn’t merely describe reality – but that maths itself is the reality. When we view a website, what we are actually viewing is the manifestation of the website source code – which provides all the rules that govern how that page looks and acts. The source code does not simply describe the page, but it is what generates the page in the first place – it is the underlying reality that underpins what we observe. Using this same reasoning could explain why our continued search for a Theory of Everything continually discovers new mathematical formulae to explain the universe – because what we are discovering is part of the universal source code, written in mathematics.

MIT physicist Max Tegemark, describes this view as “radical Platonism.” Plato contended that there exists a perfect circle – in the world of ideas – which every circle drawn on Earth is a mere imitation of. Radical Platonism takes this idea further with the argument that all mathematical structures really exist – in physical space. Therefore there is a mathematical structure isomorphic to our own universe – and that is the universe we live in.

Whilst this may seems rather far fetched, it is worth noting that in quantum mechanics it is difficult to distinguish between mathematical equations and reality. It is already clear that mathematical equations -wave functions – describe reality at the subatomic level. At this level the spatial existence of particles is described not in terms of classical co-ordinates, but in terms of a probability density function. What is still not clear after decades of debate is whether this wave function merely describes reality (e.g. the Copenhagen interpretation), or if this wave function itself is what really exists (e.g. the Many Worlds interpretation). The latter interpretation would necessitate that at its fundamental level mathematical equations are indeed reality.

It is clear that there is a remarkable relationship between mathematics and reality, indeed this relationship is one of the most fundamental mystery in science. We live in a mathematical universe. Whether that is because of nothing more than a statistical fluke, or because of the necessary condition that advanced civilisations require mathematical models or because the universe itself is a mathematical structure is still a long way from being resolved. But simply asking the question, “Why these equations and not others?” takes us on a fantastic journey to the very bounds of human imagination.

leonard

e’s are good – He’s Leonard Euler.

Along with pi, e is one of the most important constants in mathematics. It is an irrational number which carries on forever. The first few digits are 2.71828182845945…

Leonard Euler

e is sometime named after Leonard Euler (Euler’s number).   He wasn’t the first mathematician to discover e – but he was the first mathematician to publish a paper using it.    Euler is not especially well known outside of mathematics, yet he is undoubtedly one of the true great mathematicians.  He published over 800 mathematical papers on everything from calculus to number theory to algebra and geometry.

Why is e so important? 

Lots of functions in real life display exponential growth. Exponential growth is used to describe any function of the form ax where a is a constant. One example of exponential growth is the chessboard and rice problem, (if I have one grain of rice on the first square, two on the second, how many will I have on the 64th square?) This famous puzzle demonstrates how rapidly numbers grow with exponential growth.

Sketch
y = 2x
y = ex
y = 3x

for between x = 0 and 3. You can see that y = ex is between y=2x and y = 3x on the graph, so why is e so much more useful than these numbers? By graphical methods you can find the gradient when the graphs cross the y axis. For the function y = ex this gradient is 1. This is because the derivative of ex is still ex – which makes it really useful in calculus.

The beauty of e.

e appears in a host of different and unexpected mathematical contexts, from probability models like the normal distribution, to complex numbers and trigonometry.

Euler’s Identity is frequently voted the most beautiful equation of all time by mathematicians, it links 5 of the most important constants in mathematics together into a single equation.

euler4

Infinite fraction: e can be represented as a continued infinite fraction can students you spot the pattern? – the LHS is given by 2 then 1,2,1 1,4,1 1,6,1 etc.

e=2+{\cfrac  {1}{1+{\cfrac  {1}{{\mathbf  2}+{\cfrac  {1}{1+{\cfrac  {1}{1+{\cfrac  {1}{{\mathbf  4}+{\cfrac  {1}{1+{\cfrac  {1}{1+\ddots }}}}}}}}}}}}}}=1+{\cfrac  {1}{{\mathbf  0}+{\cfrac  {1}{1+{\cfrac  {1}{1+{\cfrac  {1}{{\mathbf  2}+{\cfrac  {1}{1+{\cfrac  {1}{1+{\cfrac  {1}{{\mathbf  4}+{\cfrac  {1}{1+{\cfrac  {1}{1+\ddots }}}}}}}}}}}}}}}}}}.

Infinite sum of factorials: e can also be represented as the infinite sum of factorials:

e=\sum_{n=0}^\infty \frac{1}{n!}

A limit: e can also be derived as the limit to the following function.  It was this limit that Jacob Bernoulli investigated – and he is in fact credited with the first discovery of the constant.

\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n,

Complex numbers and trigonometry :  e can be used to link both trigonometric identities and complex numbers:

e^{{ix}}=\cos x+i\sin x,\,\!

You can explore more of the mathematics behind the number e here.

If you enjoyed this post you might also like:

Ramanujan’s Beauty in Mathematics Some of the amazingly beautiful equations of Ramanujan.

poker

This post is based on the fantastic PlusMaths article on bluffing- which is a great introduction to this topic.  If you’re interested then it’s well worth a read.  This topic shows the power of mathematics in solving real world problems – and combines a wide variety of ideas and methods – probability, Game Theory, calculus, psychology and graphical analysis.

You would probably expect that there is no underlying mathematical strategy for good bluffing in poker – indeed that a good bluffing strategy would be completely random so that other players are unable to spot when a bluff occurs.  However it turns out that this is not the case.

As explained by John Billingham in the PlusMaths article, when considering this topic it helps to really simplify things first.  So rather than a full poker game we instead consider a game with only 2 players and only 3 cards in the deck (1 Ace, 1 King, 1 Queen).

The game then plays as follows:
1) Both players pay an initial £1 into the pot.
2) The cards are dealt – with each player receiving 1 card.
3) Player 1 looks at his card and can:
(a) check
(b) bet an additional £1
4) Player 2 then can respond:
a) If Player 1 has checked, Player 2 must also check.  This means both cards are turned over and the highest card wins.
b) If Player 1 has bet £1 then Player 2 can either match (call) that £1 bet or fold.  If the bets are matched then the cards are turned over and the highest card wins.

So, given this game what should the optimal strategy be for Player 1? An Ace will always win a showdown, and a Queen always lose – but if you have a Queen and bet, then your opponent who may only have a King might decide to fold thinking you actually have an Ace.

In fact the optimal strategy makes use of Game Theory – which can mathematically work out exactly how often you should bluff:

poker2

This tree diagram represents all the possible outcomes of the game.  The first branch at the top represents the 3 possible cards that Player 2 can be dealt (A,K,Q) each of which have a probability of 1/3.  The second branch represents the remaining 2 possible cards that Player 1 has – each with probability 1/2.  The numbers at the bottom of the branches represent the potential gain or loss from betting strategies for Player 2 – this is calculated by comparing the profit/loss relative to if both players had simply shown their cards at the beginning of the game.

For example, Player 2 has no way of winning any money with a Queen – and this is represented by the left branch £0, £0.  Player 2 will always win with an Ace.  If Player 1 has a Queen and bluffs then Player 2 will call the bet and so will have gained an additional £1 of his opponents money relative to a an initial game showdown (represented by the red branch).  Player 1 will always check with a King (as were he to bet then Player 2 would always call with an Ace and fold with a Queen) and so the AK branch also has a £0 outcome relative to an initial showdown.

So, the only decisions the game boils down to are:

1) Should Player 1 bluff with a Queen? (Represented with a probability of b on the tree diagram )
2) Should Player 2 call with a King?  (Represented with a probability of c on the tree diagram ).

Now it’s simply a case of adding the separate branches of the tree diagram to find the expected value for Player 2.

The right hand branch (for AQ and AK) for example gives:

1/3 . 1/2 . b . 1
1/3 . 1/2 . (1-b) . 0
1/3 . 1/2 . 0

So, working out all branches gives:

Expected Value for Player 2 = 0.5b(c-1/3) – c/6
Expected Value for Player 1 = -0.5b(c-1/3) + c/6

(Player 1′s Expected Value is simply the negative of Player 2′s. This is because if Player 2 wins £1 then Player 1 must have lost £1). The question is what value of b (Player 1 bluff) should be chosen by Player 1 to maximise his earnings?  Equally, what is the value of c (Player 2 call) that maximises Player 2′s earnings?

It is possible to analyse these equations numerically to find the optimal values (this method is explained in the article), but it’s more mathematically interesting to investigate both the graphical and calculus methods.

Graphically we can solve this problem by creating 2 equations in 3D:

z = 0.5xy-x/6 – y/6

poker7

z = -0.5xy+x/6 + y/6

poker8

In both graphs we have a “saddle” shape – with the saddle point at x = 1/3 and y = 1/3.  This can be calculated using Wolfram Alpha. At the saddle point we have what is known in Game Theory as a Nash equilibrium – it represents the best possible strategy for both players.   Deviation away from this stationary point by one player allows the other player to increase their Expected Value.

Therefore the optimal strategy for Player 2 is calling with precisely c = 1/3 as this minimises his loses to -c/6 = -£1/18 per hand.  The same logic looking at the Expected Value for Player 1 also gives b = 1/3 as an optimal strategy.  Player 1 therefore has an expected value of +£1/18 per hand.

We can arrive at the same conclusion using calculus – and partial derivatives.

z = 0.5xy-x/6 – y/6

For this equation we find the partial derivative with respect to x (which simply means differentiating with respect to x and treating y as a constant):

zx = 0.5x – 1/6

and also the partial derivative with respect to y (differentiate with respect to y and treat x as a constant):

zy = 0.5y -1/6

We then set both of these equations to 0 and solve to find any stationary points.

0 = 0.5x – 1/6
0 = = 0.5y -1/6
x = 1/3 y = 1/3

We can then see that this is a saddle point by using the formula:

D = zxx . zyy – (zxy)2

(where zxx means the partial 2nd derivative with respect to x and zxy means the partial derivative with respect to x followed by the partial derivative with respect to y. When D < 0 then we have a saddle point).

This gives us:

D = 0.0 – (0.5)2 = -0.25

As D < 0 then we have a saddle point – and the optimal strategy for both players is c= 1/3 and b = 1/3.

We can change the rules of the game to see how this affects the strategy.  For example, if the rules remain the same except that players now must place a £1.50 bet (with the initial £1 entry still intact) then we get the following equation:

Player 2 Expected Value = b/12(-1+7c) – 3c/12

This has a saddle point at b = 3/7, c = 1/7.  So the optimal strategy is 3/7 bluffing and 1/7 calling.  If Player 2 calls more than 3/7 then Player 1 can never bluff (b = 0), leaving Player 2 with a negative Expected Value.  If Player 2 calls less than 3/7 then Player 1 can always bluff (b = 1).

If you enjoyed this you might also like:

The Gambler’s Fallacy and Casino Maths - using maths to better understand casino games

Game Theory and Tic Tac Toe - using game theory to understand games such as noughts and crosses

Joh-RiemannSphere01

The Riemann Sphere

The Riemann Sphere is a fantastic glimpse of where geometry can take you when you escape from the constraints of Euclidean Geometry – the geometry of circles and lines taught at school.  Riemann, the German 19th Century mathematician, devised a way of representing every point on a plane as a point on a sphere.  He did this by first centering a sphere on the origin – as shown in the diagram above.  Next he took a point on the complex plane (z = x + iy ) and joined up this point to the North pole of the sphere (marked W).  This created a straight line which intersected the sphere at a single point at the surface of the sphere (say at z’).  Therefore every point on the complex plane (z) can be represented as a unique point on the sphere (z’) – in mathematical language, there is a one-to-one mapping between the two.  The only point on the sphere which does not equate to a point on the complex plane is that of the North pole itself (W).  This is because no line touching W and another point on the sphere surface can ever reach the complex plane.  Therefore Riemann assigned the value of infinity to the North pole, and therefore the the sphere is a 1-1 mapping of all the points in the complex plane (and infinity).

Riemann 2

So what does this new way of representing the two dimensional (complex) plane actually allow us to see?  Well, it turns on its head our conventional notions about “straight” lines.  A straight line on the complex plane is projected to a circle going through North on the Riemann sphere (as illustrated above).  Because North itself represents the point at infinity, this allows a line of infinite length to be represented on the sphere.

riemann sphere

Equally, a circle drawn on the Riemann sphere not passing through North will project to a circle in the complex plane (as shown in the diagram above).  So, on the Riemann sphere – which remember is isomorphic (mathematically identical) to the extended complex plane, straight lines and circles differ only in their position on the sphere surface.  And this is where it starts to get really interesting – when we have two isometric spaces there is no way an inhabitant could actually know which one is his own reality.  For a two dimensional being living on a Riemann sphere,  travel in what he regarded as straight lines would in fact be geodesic (a curved line joining up A and B on the sphere with minimum distance).

By the same logic, our own 3 dimensional reality is isomorphic to the projection onto a 4 dimensional sphere (hypersphere) – and so our 3 dimensional universe is indistinguishable from a a curved 3D space which is the surface of a hypersphere.  This is not just science fiction – indeed Albert Einstein was one to suggest this as a possible explanation for the structure of the universe.  Indeed, such a scenario would allow there to be an infinite number of 3D universes floating in the 4th dimension – each bounded by the surface of their own personal hypersphere.  Now that’s a bit more interesting than the Euclidean world of straight lines and circle theorems.

If you liked this you might also like:

Imagining the 4th Dimension. How mathematics can help us explore the notion that there may be more than 3 spatial dimensions.

The Riemann Hypothesis Explained. What is the Riemann Hypothesis – and how solving it can win you $1 million

Are You Living in a Computer Simulation? Nick Bostrom uses logic and probability to make a case about our experience of reality.

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