50 Mathematical Ideas You Really Need to Know

The Italian connection The theory of cubic equations was fully developed during the Renaissance. Unfortunately it resulted in an ...

could not produce a formula which was generally applicable to equations involving x^5 , the ‘quintic’ equations. What was so spe ...

acknowledged today by a plaque. With the date scored into his mind, the subject became Hamilton’s obsession. He lectured on it y ...

15 Euclid’s algorithm Al-Khwarizmi gave us the word ‘algebra’, but it was his ninth-century book on arithmetic that gave us the ...

All we have to do is carry out the algorithm in sequential steps one after the other. The only thing missing in this recipe, usu ...

The number 2 divides both 18, and 84, but so does the number 3. So 6 will also divide both numbers. Is there a larger number th ...

going to concentrate on finding the gcd. We have already calculated gcd(18, 84) = 6 but to do it we needed to know the divisors ...

Greek mathematician Diophantus of Alexandria. Let’s imagine Great Aunt Christine is going for her annual holiday to Barbados. Sh ...

2 × 5 × 7 = 70. If Angus has between 150 and 250 bottles then the theorem nails the solution down to 213 bottles. Not bad for a ...

16 Logic ‘If there are fewer cars on the roads the pollution will be acceptable. Either we have fewer cars on the road or there ...

In this case, the individual statements are plainly nonsensical if we are using the usual connotations of the words. Yet both in ...

more than 2000 years and held an important place in undergraduate studies in medieval universities. Aristotle’s logic – his theo ...

If we have another proposition b such as ‘Ethel is a cat’ then we can combine these two propositions in several ways. One combin ...

Is the argument valid or not? Let’s assume the conclusion P is false, but that all the premises are true. If we can show this fo ...

boundary as to what is in and what is out is left fuzzy. Mathematics allows us to be precise about fuzziness. Logic is far from ...

17 Proof Mathematicians attempt to justify their claims by proofs. The quest for cast iron rational arguments is the driving for ...

Let’s start by being sceptical – this is a method of proving a statement is incorrect. We’ll take a specific statement as an exa ...

or, rewriting using brackets, 6 × 6 = 2 × (3 + 3 + 3 + 3 + 3 + 3) This means 6 × 6 is a multiple of 2 and, as such, is an even n ...

years. This specific technique (not to be confused with scientific induction) is widely used to prove statements involving whole ...

solution. They are called ‘Constructivists’ (of varying shades) who say the proof method fails to provide ‘numerical meaning’. T ...