e’s are good – He’s Leonard Euler.
Having recently starting a topic on the exponential function, I was really struggling to find some good resources online – which is pretty surprising given that e is one of the most important and useful numbers in mathematics. So, here are some possible approaches.
1) e memorisation challenge.
This is always surprisingly popular – and a great starter which reinforces both that e is infinite and also that it’s just a number – so shouldn’t be treated like other letters when it comes to calculus.
5 minutes: How many digits of e can students remember?
Recital at the front. You can make this easier by showing them that 2. 7 1828 1828 45 90 45 they only need to remember 2.7 and then that 1828 repeated, followed by the angles in a triangle – 45, 90, 45. Good students can get 20 places plus – and for real memory champions here are the first 1000 digits .
2) Introduction to Leonard Euler
Euler is not especially well known outside of mathematics, yet is undoubtedly one of the true great mathematicians. As well as e being named after him (Euler’s number), he published over 800 mathematical papers on everything from calculus to number theory to algebra and geometry.
30-40 minutes – The Seven Bridges of Königsberg
This is one of Euler’s famous problems – which he invented a whole new branch of mathematics (graph theory) to try and solve. Here is the problem:
The city of Königsberg used to have seven bridges across the river, linking the banks with two islands. The people living in Königsberg had a game where they would try to walk across each bridge once and only once. You can chose where to start – but you must cross each bridge only once:
The above graphic is taken from the Maths is Fun resource on Euler’s bridge problem. It’s a fantastically designed page – which takes students through their own exploration of how to solve similar problems (or as in the case of the 7 bridges problem, understanding why it has no solution).
3) Learning about e
30 minutes – why e ?
This is a good activity for students learning about differentiation for the first time.
First discuss exponential growth (example the chessboard and rice problem ) to demonstrate how rapidly numbers grow with exponential growth – ie. if I have one grain of rice on the first square, two on the second, how many will I have on the 64th square?
Next, students are given graph paper and need to sketch y = 2^x y = e^x y = 3^x for between x = 0 and 3. Students can see that y = e^x is between y=2^x and y = 3^x on the graph, so why is e so much more useful than these numbers? By graphical methods they should find the gradient when the graphs cross the y axis. Look at how the derivative of e^x is still e^x – which makes it really useful in calculus. This is a nice short video which explains graphically why e was chosen to be 2.718…
4) The beauty of e.
10-30 minutes (depending on ability), discussion of some of the beautiful equations associated with e and Euler:
a) Euler Identity – frequently voted the most beautiful equation of all time by mathematicians, it links 5 of the most important constants in mathematics together into a single equation.
b) e as represented as a continued infinite fraction (can students spot the pattern? – the LHS is given by 2 then 1,2,1 1,4,1 1,6,1 etc.
c) e as the infinite sum of factorials:
d) e as the limit:
So, hopefully that should give some ideas for looking at this amazing number. (The post title will be lost on anyone not a teenager in England in the 1990s -to find out what you’re missing out on, here’s the song).
If you enjoyed this topic you may also like:
Cesaro Summation: Does 1 – 1 + 1 – 1 … = 1/2? – a post which looks at the maths behind this particularly troublesome series.
A Maths Snooker Puzzle – a great little puzzle which tests logic skills.