Transcendental Numbers

Numbers are pretty amazing really, and transcendental numbers are some of the most amazing numbers of them all.  Proof by contradiction can sometimes be used to prove that a particular number is irrational, but proving that numbers are transcendental is a lot more complicated.

Make sure you can prove that root 2 and root 3 are irrational using proof by contradiction. You should also be able to use proof by contradiction to prove that there are infinitely many prime numbers.

In your A level Maths syllabus you will learn about irrational numbers and rational numbers and how to prove that certain numbers are rational or irrational.  In the new spec A level maths exam, you may be asked to use proof by contradiction to prove that numbers such as root 2 and root 3 are irrational.

But did you know that there are different types of irrational numbers? Mathematicians can even measure how irrational a number is using this gorgeous formula:

Some very special irrational numbers, introduced by Leibnitz around 1700, are called transcendental numbers. These are the real numbers which are not algebraic.  What’s an algebraic number? It’s a number which is the solution of a polynomial equation with rational coefficients. This isn’t as confusing as it sounds, the video explains all! 

Cantor (an amazing mathematician) proved that there are more transcendental numbers than algebraic numbers!  Even though there are infinitely many of both of them! Yep – some infinities are bigger than other infinities. Fact.

Well known transcendental numbers include the golden ratio as well as sin(x) cos(x) and tan(x) for algebraic non-zero values of x. It is amazing but even though mathematicians have proved that and e are transcendental, we still have no idea whether + e is transcendental or not.  

Maybe future-you will be a mathematician and prove it one way or the other…

KEY POINTS: A level Maths + Maths Degree + Mechanics  + Statistics + Proof  + Calculus + Geometry

EXAM BOARDS: AQA, EDEXCEL, OCR, WJEC