# pure math

## Area of a circle – proof by exhuastion

Once has been defined as the ratio circumference to diameter, the area of a circle must be . A proof relies on an infinite, limiting process which paves the way to some calculus-like ideas. A circle is cut up into 6 sectors which are then rearranged into a near rectangle, What doesn’t look too close …

## Integration by Substitution

Current UK exam textbooks pass over proofs and mathematical discussions in a hurry to show the ‘how to’ of exam questions. Integration by substitution is a little more than just backwards chain-rule and deserves a fuller treatment. Try this, Let, then, Suppose that there exists a function g, of another variable , such that and …

## Differentiation From First Principles

The gradient of a smooth curve, , at a point is the gradient of the tangent to the curve at the point . Point is on the curve and is a neighbouring point whose value is displaced a small quantity, . The idea behind differentiation is that as becomes very small, the gradient of tends …

## Parallelograms generalise Pythagoras to any triangle

There’s always space on the inter-web for another proof of Pythagoras’s Theorem.  Here’s one that uses the following equal areas property of parallelograms. This kind of area chopping and shape translation is a feature of Euclidean geometry and our senses support it’s veracity at the order of size of the classroom. The squares on the …

Properties of conic section have been known since classical times.  This isosceles triangle in a parabola could still appear in school exams (FP1, STEP) in UK.

jpedmaths: Transformations for IGCSE summary – too late for this year. Year 10?? Nice colours though. Transformations for IGCSE maths – if any of my students don’t know them (and recent work suggest some don’t) then learn them now.

Prime numbers hang like beads on a necklace (except that would be cosh …) on this quadratic.

The Natural Logarithm is an integration defined function. The shaded area under the graph is the value of the natural logarithm of the upper limit in the integration.