50 Mathematical Ideas You Really Need to Know

18 Sets Nicholas Bourbaki was a pseudonym for a self-selected group of French academics who wanted to rewrite mathematics from t ...

Venn. Euler used diagrams like these even earlier. The set A ∩ B consists of elements which are members of A and B and is called ...

¬(A ∩ B) = (¬A) ∪(¬B) The paradoxes There are no problems dealing with finite sets because we can list their elements, as in A = ...

with a problem of their own – they didn’t have to explain what straight lines were, but only how they should be dealt with. In t ...

The problem was so important that German mathematician David Hilbert placed it at the head of his famous list of 23 outstanding ...

19 Calculus A calculus is a way of calculating, so mathematicians sometimes talk about the ‘calculus of logic’, the ‘calculus of ...

about to let a stone drop. What will happen? The advantage of a thought experiment is that we do not actually have to be there i ...

We might be tempted to make the small extra time equal to zero. But in our thought experiment, the stone has not moved at all. I ...

where v is the instantaneous velocity of the stone at time x. For example, the instantaneous velocity of the stone after 1 secon ...

Integration The first application of integration was to measure area. The measurement of the area under a curve is done by divid ...

we integrate the derivative du/dx = 2 x we also get the original u = x^2. Differentiation is the inverse of integration, an obse ...

20 Constructions Proving a negative is often difficult, but some of the greatest triumphs in mathematics do just that. This mean ...

Trisecting the angle Here is a way to divide an angle into two equal smaller angles or, in other words, bisect it. First place t ...

The duplication of the cube is a similar problem known as the Delian problem. The story goes that the natives of Delos in Greece ...

Mathematicians generally believed that π was a transcendental but this ‘riddle of the ages’was difficult to prove until Ferdinan ...

Repeat this with the compass point at B using the same radius. The intersection point of these two arcs is at P. As AP = AB and ...

this one is not a prime number, because 4,294,967,297 = 641 × 6,700,417. If we put n = 6 or 7 into the formula the results are h ...

21 Triangles The most obvious fact about a triangle is that it is a figure with three sides and three angles (tri-angles). Trigo ...

called the ‘triangle inequality’ and is important in abstract mathematics. The Epicureans, with their down-to-earth approach to ...

century than a Euclidean proof from 300 BC. This is a proof ‘without words’. In the figure the square with side a + b can be div ...