50 Mathematical Ideas You Really Need to Know

symbol (א or ‘aleph’ is from the Hebrew alphabet; the symbol is read as ‘aleph nought’). So, in mathematical language, we can wr ...

By displaying all the fractions in this way, potentially at least, we can construct a one-dimensional list. If we start on the t ...

Suppose you did not believe Cantor. You know that each number between 0 and 1 can be expressed as an extending decimal, for exam ...

08 Imaginary numbers We can certainly imagine numbers. Sometimes I imagine my bank account is a million pounds in credit and the ...

Engineering Even engineers, a very practical breed, have found uses for complex numbers. When Michael Faraday discovered alterna ...

number 4 + 32i. (2 + 3i) × (8 + 4i) = (2 × 8) + (2 × 4i) + (3i × 8) + (3i × 4i) With complex numbers, all the ordinary rules of ...

Adding and multiplying mates together always produces a real number. In the case of adding 1 + 2i and 1 −2i we get 2, and multip ...

The completeness of the complex number system becomes clearer when we think of what are called ‘the nth roots of unity’ (for mat ...

dimensional numbers as a possible continuation of the story – but 50 years after Hamilton’s momentous feat, they were proved imp ...

09 Primes Mathematics is such a massive subject, criss-crossing all avenues of human enterprise, that at times it can appear ove ...

Discovering primes Unhappily there are no set formulae for identifying primes, and there seems to be no pattern in their appeara ...

In 1792, when only 15 years old, Carl Friedrich Gauss suggested a formula P(n) for estimating the number of prime numbers less t ...

The unknown Outstanding unknown areas concerning primes are the ‘Twin primes problem’ and the famous ‘Goldbach conjecture’. Twin ...

1 are expressible as the sum of two squares in exactly one way (e.g. 17 = 1^2 + 42 ), while those of the form 4k + 3 (like 19) ...

10 Perfect numbers In mathematics the pursuit of perfection has led its aspirants to different places. There are perfect squares ...

than itself. So the number 26 is deficient because its divisors 1, 2 and 13 add up to only 16, which is less than 26. Prime numb ...

perfect number. It is easy to check it really is the sum of its divisors: 496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 +248. For the ...

prime. But it is those Mersenne numbers that are also prime that can be used to construct perfect numbers. Mersenne knew that if ...

223 – 1 = 8,388,607 = 47 × 178,481 Construction work A combination of Euclid and Euler’s work provides a formula which enables e ...

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