Great Mathematicians 7 – Pythagoras

Statue of Pythagoras on Samos (image from visitgreece.com) Much of what we know about Pythagoras is conjecture. We do know that he was born on the Greek island of Samos in about 570 BC and died in about 495 BC. Around about 530 BC he founded what can only be called a rather bizarre sect in Croton, in Italy: it was semi-mystical, and Pythagoras imposed strict vegetarianism, communal living, secret rites and strange rules - such as instructions to his ...

Understanding the Factor Theorem

You have probably learnt how to factorise a quadratic expression: for example, x2 - 2x - 8 factorises to give (x - 4)(x + 2). This is useful because it enables us to solve the equation x2 - 2x - 8 = 0. How? Because if we use the factorised form we get (x - 4)(x + 2) = 0, and if two numbers multiply to give zero then one of the number must be zero. So either x - 4 = 0, ...

It’s all in the detail

Mathematics, as you will know from Theory of Knowledge, is at the top of the tree of knowledge. It is self-referential - that is, its theorems do not need to be proven by reference to the real world, but instead by starting with other axioms and theorems. Of course, maths has many real-world applications, and if you are starting out on the new Applications and Interpretations course, then you will spend a lot of time solving real-word problems. Many of the ...

The new IB Diploma Maths syllabus

All IB Diploma students have to follow a maths course at either standard or higher level. Until now, there were three choices: Maths HL, a tough(ish) course for good mathematicians; Maths SL, better for those who didn't want the standard of maths required at the higher level, but still needed, or enjoyed, a course with a reasonable level of mathematical content; and Maths Studies SL, for those, frankly, who weren't mathematically orientated, and which contained topics such as financial maths ...

Optical Illusions

There are many types of optical illusion but, in every case, the brain is being fooled by what the eyes see. In a short blog such as this I can only show a small set of examples from a limited number of categories but, if you are as fascinated by these as I am, a short web search will reveal countless more. It's also the case that some optical illusions can only be created using paper, and won't work on-screen. Illusory ...

Pythagorean Triples

We are all familiar with Pythagoras' Theorem: that if a triangle with sides of length a, b, and c is right-angled, then a2 + b2 = c2. (You can find a proof of the theorem in an earlier blog here). If the sides all have integer values, then the numbers a, b, c form a "Pythagorean triple" - the simplest of which is 3, 4, 5 since 32 + 42 = 52. Further triples can be formed by simple multiplication: thus, 6, 8, 10 ...

An Easy Mathematical Trick

Start by writing the number 1089 on a piece of paper and put it in your pocket. Now get someone to choose a three digit number where the last digit is at least 2 less than the first digit. Turn it round to form a new three digit number, then subtract it from the first one. For example, 481 − 184 = 297. Now turn the new number round to form a fourth three digit number, and add it to the third. 297 + ...

Hypothesis testing made easy

Carrying out a hypothesis test often causes confusion. Here's how it works. Some hypothesis tests start with a known fact, such as "25% of patients treated for a particular disease will suffer side effects." A drug company may then claim that "a new treatment reduces the number of patients suffering side effects." The original figure, the status quo, is known as the "null hypothesis" and given the symbol H0. The new claim is called the "alternative hypothesis" and given the symbol ...

Does division by zero = infinity?

Why should it? Well, try this: 5 ÷ 10 = 0.55 ÷ 1 = 55 ÷ 0.1 = 505 ÷ 0.01 = 5005 ÷ 0.001 = 5000 As we divide by smaller and smaller numbers the result gets ever bigger. Logically, then, as the divisor tends to (ie gets closer to) zero, so the result tends to infinity. But this is not the same as saying that division by zero actually is infinity, is it? What about drawing a graph with ...

A Mathematical Card Trick

Card tricks fall into several categories, and my favourite are those which look amazing but actually have some pretty simple maths behind them. Here's one of the simplest! 1.     Take an ordinary pack of cards and shuffle it. Deal out 26 cards, face up, and remember the 7th card. 2.    When you've dealt out the 26 cards, pick up the pile and turn them face down on the table, to one side. (If you're brave, shuffle them first, but make ...