Who we are: Maths Society 2018
Factorial - Summer 2018
Proving 3=0 & Extraneous Solutions
By now, you will all have seen the posters around the maths classrooms warning us against classic maths mistakes: forgetting the two solutions of x2 = 64, thinking that (a+b)2 is equal to a2+b2 (it is a2 + b2 + 2ab), to name a couple. So, while it might be safe to conclude that we are more thorough and meticulous mathematicians, try this ‘proof’ on for size:
Let x be a solution of x2 + x + 1 = 0.
We divide by 0, as x cannot be 0: x + 1 + 1/x = 0.
Rearrange initial equation: -x2 = x + 1.
Substitute into second equation: -x2 + 1/x = 0.
Multiply by x: -x3 + 1 = 0; x3 = 1.
As 1 is a solution, substitute 1 into initial equation: (1)2 + (1) + 1 = 0; 3=0.
Seems rigorous enough - no sign errors or dividing by 0 errors, so where is the mistake? If we follow through the proof carefully, we can keep track of the number of solutions.
x2 + x + 1 = 0 is a quadratic and has two complex (numbers including the imaginary number i) solutions, using the quadratic formula. These are (-1+i√3)/2 and (-1-i√3)/2.
x + 1 + 1/x = 0 has the same two solutions. We are able to divide by 0 here as 0 is clearly not a solution: (0)2 + 0 + 1 is not 0.
x+1=-x2 is found simply by subtracting x2 from both sides.
-x2+1/x = 0 has the same two solutions as before, but if we try x = 1, we realise that this is also a solution. By substituting, we have created what is known as an extraneous solution. 1 clearly does not give 0 in the first equation!
Generally, when we solve equations, we try to rewrite them in a way that is easiest for us to find the solutions. However, in doing so, we can sometimes introduce or miss solutions to equations. The number of solutions to an equation is equal to the degree (highest power of x), and so once we substitute, we go from a quadratic with two solutions to a cubic, and introduce the 3rd solution of x=1.
As another example, consider x + 2 = 5.
Multiply both sides by x. x(x+2) = 5x; x2 -3x = 0; x(x-3) = 0. x is 0 or 3.
This creates another solution of x = 0, which does not work in the first equation.
Overall, the message to take away from this is that when manipulating equations, be sure that you are aware of any solutions you may add or miss! Always be sure to check that your answers satisfy the initial equation given.
By Victoria Adje
Have you ever heard the term “Actuarial Science” and wondered whatever could it be? Well, if so, don’t fret because you’re about to find out all about it.
You might know that actuarial means to do with corporate risks and uncertainty, so from this, you’d think this discipline would be pretty exciting and you’d be right! Actuarial science involves using maths (mainly Statistics) to anticipate risks in fields like Finance and Insurance; nowadays, this involves using statistical modelling software to calculate the likelihood of particular events and potential risks.
For example, for a big company who are planning to invest lots of money in building an extravagant hotel on the coast of Wales, they might employ an Actuary to calculate the statistical probability of natural disasters in this area to inform the company of these possible financial risks and costs. The discipline is actually very in demand right now, with the US Bureau of Labour Statistics projecting that it will see a growth of 26% from 2012 to 2022.
After hearing all that, you probably have a lot of questions, so here are some.
Frequently Asked Questions about Actuarial Science
Q: What’s the difference between Accountants and Actuaries?
A: About 25 grand, normally. Also, while actuaries use statistics to calculate risks, the jobs of accountants and audits involve managing, analysing, and presenting about all the financial operations of a business.
Q: What kind of person do I need to be to become an actuary?
A: Well, you’ll need to like crunching numbers and be able to apply mathematics to real life situations. In addition, you’ll have to be a good communicator, as this job will involve communicating pretty difficult topics to non-specialists.
Q: This sounds just like me- how can I become an actuary?
A: That’s a great question- you’ll obviously need an A-level in Mathematics and will need to complete a highly mathematical degree. Following this, you’ve got to prepare for and complete an exam so you can have the necessary qualifications: these might be the CERA Chartered Enterprise Risk Actuary or the CAA (Certified Actuarial Analyst)
Q: Why does a heavy metal fan want to become an actuary?
A: So, he can get paid for predicting death and destruction.
By Saachi Pahwa
3 Easy Microbit Projects!
If you find yourself bored over the holidays, pick up a Microbit and learn to code using these three easy projects!
You will need a Microbit (obviously!) and a laptop to code on. You need to connect the Microbit to the computer and open up the micro-python program via the BBC Microbit Website.
from microbit import *
In the brackets of music.play(), you can also insert different songs:
You can also make your own music:
tune = ["C4:4", "D4:4"]
The letter and numbers inside the quotation marks represent the name of the note, the octave, and the length of the note.
Try your own combinations!
This will display “Hello World”!
To display your own message, replace the text in the speech marks!
Have fun coding!
By Laila Arain
Why is the volume of a cone a third of a cylinder?
We all know that the volume of a right-cone is a third that of a cylinder with the same base and height (or, at least, we've seen it on some formula booklet or other). But why?
Well, weirdly, this proof begins with graphs. If we represent the centre-line of the cone and cylinder as the x-axis, we can draw the slope in 2-dimensions as a normal function:
For a cylinder, ;
For a cone,
Where r is the radius of the cone/cylinder and h is the height. The equation of the cone’s slope is worked out using the trusty y = mx + c, as the gradient is the radius over the height and the point of the cone is at (0,0) (see Diagram).
As A-level and IB students will know, the volume of revolution of a graph is equal to the definite integral of y2π. For the uninitiated amongst us, that means that the volume formed by rotating the equation of the line (aka the volume of our cone and cylinder) is found through integrating the function.
With the cylinder, this gives us:
Which we should all recognise as the volume of a cylinder.
So, applying this to a cone, we get:
N.B. this is done just through simple integration. If you want to know more on the individual steps, feel free to ask any of us in Maths Society at Maths Clinics (or just pressure your teacher into showing you)
And what you have probably already noticed, is that the volume we have just calculated is, in fact, one third of the volume of our cylinder. Next time you get a question in a maths test on the volume of a cone, instead of answering, just write this instead, it’s sure to impress your teacher*!
*Please do not do this, it is a terrible idea, and Maths Society in no way accepts responsibility if you fail a test due to this trick.
By Imogen Woods-Wilford
Is it possible to find two irrational numbers a and b such that ab is rational?
Rational and irrational numbers have provided a multitude of complex and interesting problems since the discovery of the first irrational number, √2, in the 5th century BC. One such example is the particular problem ‘Is it possible to find two irrational numbers a and b such that a^b is rational?’ At first glance, this may seem to require a complicated proof, however, a relatively simple solution can be found.
Suppose a=√2 and b=√2, then a^b=√2^√2.The question that arises from this solution is the rationality of the number √2^√2. Assuming this number is rational, the problem has been solved! However, if the alternative is true and √2^√2 is irrational, this number can be used as a. Hence, ab = (√2^√2)^√2 which is equal to 2, an undoubtedly rational number.
Although there is a solution regardless, the ambiguity of the rationality of √2^√2 still remains. According to the Gelfond-Schneider Theorem of 1934, √2^√2 is irrational. This is because ‘if a and b are algebraic numbers with a ≠ 0, a ≠ 1, and b irrational, then any value of ab is a transcendental number.’
Therefore, √2^√2 is a transcendental number, a number which is not the root of a nonzero polynomial equation.
Overall, this proof demonstrates it is possible to find two irrational numbers a and b such that ab is rational, even without knowledge of the complex mathematics behind it.
Values other than √2 and √2^√2 for a and b also exist. Can you find them?
By Jessica Kohli
Who got there first, Newton or Leibniz?
You know that feeling when you get an implicit/quotient-rule/chain-rule differentiation question? Yeah… well you’ve got Newton to thank for that. Or wait… is it Leibniz?
The Calculus Controversy was the most well-remembered controversy in the 17th Century and the protagonists were Isaac Newton and Gottfried Wilhelm Leibniz. They both claimed that they invented calculus but who actually got there first?
Early in 1665 Newton had the fundamental theorem of calculus which was summarised in an essay written in 1966. Leibniz first began working on his calculus in 1674 but didn’t publish anything until two decades later. Though both mathematicians invented their own theories of calculus independently of each other, the fundamental ideas were similar despite Newton’s lack of notation. The major difference between their theories was the approach. Newton’s approach had more to do with physics and Leibniz approached it more geometrically.
However, we can never really be sure of who invented calculus as neither man published his work straight away. Manuscripts that were published in 1736, nearly a decade after Newton’s death, referred to what we recognise now as differential and integral calculus. Leibniz first got significant results with his calculus in 1675 but didn’t publish anything until 1684. Newton and Leibniz often corresponded by mail, so it is more than likely that they discussed their theories of calculus and maybe derived ideas off of each other (pun intended). It led to Newton accusing Leibniz of plagiarism by using hic contact in the royal society and his notation to defame him.
Regardless of who invented it first, Newton is more widely accredited with differentiation and Leibniz with integration. So, I guess it is Newton that you have to thank :)
By Patrycja Lakomie
Factorial - Michaelmas 2017
How Come Imaginary Numbers Are Useful in Life If They Aren’t Real?
A number is “imaginary” when it is expressed in terms of the square root of -1, usually denoted as i. However, being “imaginary” is not to say that it does not exist or have any use. In fact, it can be argued that no number truly exists – they are merely a concept invented by mathematicians. Yet I’m sure you’ll agree that numbers are still an extremely important tool in our lives. In the early ages of mathematics, the ancient Greeks worked solely with whole numbers. These were thought to be the only type of numbers. With the idea of ‘debt’, however, negative numbers were introduced. This essentially extended the number line to the left, providing us with a larger toolbox to work with.
Imaginary numbers can be thought of in a similar way – they add another dimension to our number system, kind of like extending the number line upwards, which allows us to solve even more problems, like solving x2+1=0.
With this in mind, we can hopefully be less unsettled by the concept of imaginary numbers, and begin to understand that they can be utilised just like any other type of number. For example, a complicated electronic signal can be made much simpler by splitting it into the smaller components which make it up. This is known as Fourier transform, and is done with the help of complex numbers, which are numbers containing a mixture of real and imaginary parts. Fourier transforms have applications in modelling the flow of fluids through pipes, or the movement of oscillating objects (including some stars), or the positions of particles in quantum mechanics. All these theories provide insight into things from the real world, such as how to pump oil in oil rigs, how earthquakes shake buildings, and how electronic devices work on a quantum level… it even helps understand how your own brain processes light and sound!
Clearly, complex numbers are therefore invaluable to mathematicians, scientists, and engineers alike. Without them, we would not be living in the same world.
Maths in the News
On the 24th August 2017, ‘The Independent’ published an article called, “Babylonians developed trigonometry 'superior' to modern day version 3,700 years ago.” The article explains how Hipparchus, a Greek astronomer who lived in about 120 BC, is traditionally regarded as the founder of trigonometry. But two professors discovered that a Babylonian table called Plimpton 322 predates Hipparchus by more than 1,000 years. Plimpton 322 was discovered in southern Iraq by the early 1900s by archaeologist, diplomat and antique dealer Edgar Banks. The tablet has numbers written in cuneiform script in four columns and 15 rows. There were suggestions in the 1980s that the numbers showed knowledge of trigonometry, but this had been dismissed more recently. However Dr Mansfield said their research revealed it was a “novel kind of trigonometry” that was based on ratios, rather than angles and circles.
On the 15th July 2017, ‘BBC News’ published an article called “Maryam Mirzakhani, first woman to win maths' Fields Medal, dies.” Maryam Mirzakhani was the first woman, and first Iranian to receive the prestigious Fields Medal for mathematics. Nicknamed the "Nobel Prize for Mathematics," the Fields Medal is only awarded every four years to between two and four mathematicians under 40. It was given to Prof Mirzakhani in 2014 for her work on complex geometry and dynamical systems. Born in 1977, Prof Mirzakhani was brought up in post-revolutionary Iran and won two gold medals in the International Mathematical Olympiad as a teenager. She earned a PhD at Harvard University in 2004, and later worked at Princeton before securing a professorship at Stanford in 2008.
IB vs A Level Maths
When studying mathematics in the Sixth Form, there is a range of different options to choose from. Below the Maths Society have created an overview of each course in the hope of helping those interested:
A LEVEL FURTHER MATHEMATICS: 2 Pure Mathematics papers, covering topics such as complex numbers, hyperbolic functions, differential equations, matrices, polar coordinates. 2 additional papers from a choice of Further Pure, Statistics, or Mechanics. 12 periods per week, 4 exams at the end of Y13.
AS LEVEL FURTHER MATHEMATICS: 1 Pure Mathematics paper, and another paper from a choice of Further Pure, Statistics, or Mechanics. 8 periods per week, 2 exams at the end of Y12 (distinct from A Level Further exams).
A LEVEL MATHEMATICS: 2 Pure Mathematics papers, covering topics such as proof, algebra, trigonometry, calculus, vectors. 1 Applied Mathematics paper, covering Statistics and Mechanics. All exams are taken at the end of Y13, and lessons take up 8 periods per week.
AS LEVEL MATHEMATICS: 1 Pure paper, 1 Applied paper (weighted 63%, 37% respectively). 2 exams are taken at the end of Y12.
IB higher leveL: This takes up 8 periods each week. the depth is comparable to Further Maths and the list of topics is slightly different. There is a calculator, non-calculator and an options paper (where the class decides what they want to study) as well as an exploration: many IB students say their maths explorations are the high point of their time here.
IB Standard level: this is the IB equivalent of A level maths. It is made up 20% of coursework, one calculator paper and one non-calculator paper. Coursework is completed on a topic of interest related to one’s choice of university course (such as the link between gender inequality and murder). Maths takes up 4 periods each week.
IB STUDIES: Students complete exams and coursework, with 4 periods of Maths each week. the focus of Maths Studies is on topics which are useful in the ‘real world’, like percentages, logic, and statistics. The course is a lot of fun and suits humanities specialists really well.
Fun Maths Websites
Here are three maths websites you can try out if you get bored, or are looking for a “productive” way to procrastinate :))
Brilliant.org: With thousands of fun exercises on anything from number theory, to machine learning, to quantum mechanics, ‘Brilliant.org’ is great for anyone who has an interest in mathematics and science. You can work through engaging and interactive online courses, which are expertly written by leading instructors and researchers to help you master foundational concepts to topics far beyond the standard curriculum. The weekly challenges (complete with discussions of the solutions from the community) are also a great way brush up on problem solving skills!
Who Discovered Pi?
There are many things we have not yet discovered about pi: how it was found, and indeed, who discovered it first. The most we can do is trace it back. Geometers from ancient civilizations, dating back to around 1900BC, were aware that the circumference of a circle is just over 3 times the diameter, however they could not accurately calculate pi.
The first known individual to calculate pi, to a relatively good level of accuracy, was Archimedes. He drew a square on the outside of a circle, and then another square on the inside of a circle. The perimeters, of these squares, created upper and lower bounds for the circumference of the circle, therefore deducing that pi was between 2.83 and 4. By using shapes with more and more sides and therefore closer to a circle [see below], mathematicians were able to find more accurate measurements for pi. Archimedes persevered and was eventually able to use 96-sided shapes, to deduce that pi was between 3.141 and 3.143.
Many mathematicians used this technique to refine the value of pi further. In the 5th century AD Zu Chongzhi used 12288-sided polygons to estimate pi to 7 decimal places: no-one improved on this for 1000 years! In the seventeenth century, Ludolph van Ceulen used 4-billion-billion sided shapes to calculate pi to 35 decimal places, and when he died his tombstone gave tribute to this.
There is no one particularly credited with “discovering pi”, however. Its value has been estimated by a series of mathematicians, each carrying forth the previous work to better the accuracy of this very important constant.