50 Mathematical Ideas You Really Need to Know

11 Fibonacci numbers In The Da Vinci Code, the author Dan Brown made his murdered curator Jacques Saunière leave behind the firs ...

Fibonacci wanted to know how many rabbit pairs there would be at the end of the year. The generations can be shown in a ‘family ...

The result of each of these sums will form a sequence as well, which we can place under the original sequence, but shifted along ...

The value of £4, as we draw the coins out of the purse, can be any of the following ways, 1 + 1 + 1 + 1; 2 + 1 +1; 1 + 2 + 1; 1 ...

The cattle population Despite the wealth of knowledge known about the Fibonacci sequence, there are still many questions left to ...

1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, 88, 129, 189, 277, 406, 595,... Thus the generation skips a value so for example, 41 ...

12 Golden rectangles Rectangles are all around us – buildings, photographs, windows, doors, even this book. Rectangles are prese ...

desirable property, one that does not occur for arbitrary paper sizes. If an A-size piece of paper is folded about the middle, t ...

which can be multiplied out to give x^2 = x + 1. An approximate solution is 1.618. We can easily check this. If you type 1.618 i ...

Now let’s see if we can build a golden rectangle. We’ll begin with our square MQSR with sides equal to 1 unit and mark the midpo ...

Leonardo da Vinci. In the Renaissance, the golden section achieved near mystical status – the astronomer Johannes Kepler describ ...

NP/MN (because triangles MRJ and MNP are similar), so y/1 = 1/x which means x × y = 1 and we say x and y are each other’s ‘recip ...

13 Pascal’s triangle The number 1 is important but what about 11? It is interesting too and so is 11 × 11 = 121, 11 × 11 × 11 = ...

The Pascal pattern is generated from the top. Start with a 1 and place two 1s on either side of it in the next row down. To cons ...

Almost diagonals in Pascal’s triangle Properties The first and most obvious property of Pascal’s triangle is its symmetry. If we ...

numbers: 1, 1 , 2, 3 , 5, 8 , 13, 21 , 34, 55 , 89, 144 ,... Even and odd numbers in Pascal’s triangle Pascal combinations The P ...

0s and 1s In Pascal’s triangle, we see that the inner numbers form a pattern depending on whether they are even or odd. If we su ...

The Leibniz harmonic triangle (1 + x)−2 = 1 – 2x + 3x^2 – 4x^3 + 5x^4 – 6x^5 + 7x^6 – 8x^7 +... The Leibniz harmonic triangle Th ...

In the words of the old song, ‘the knee bone’s connected to the thigh bone, and the thigh bone’s connected to the hip bone’. So ...

14 Algebra Algebra gives us a distinctive way of solving problems, a deductive method with a twist. That twist is ‘backwards thi ...