Fun with infinity

I’m lucky enough to teach two top set year 8 classes (year 7 last year) who are at least semi-geeky about their maths. Last summer I did a lesson with them on the nature and craziness of infinity, with the able assistance of the amazing Vi Hart (or at least her Youtube vids), and Jordan Ellenberg’s book: How Not to Be Wrong. They loved it, so I’ve reproduced some of the resources below.

Vi Hart videos:

The infinite series paradox (1):

What is the sum of: 1 + 2 + 4 + 8 + 16 + …?

To answer this, first multiply the sum by 2: 2(1 + 2 + 4 + 8 + 16 + …) = 2 + 4 + 8 + 16 + …

Now: 2(1 + 2 + 4 + 8 + 16 + …) – 1(1 + 2 + 4 + 8 + 16 + …) = 2 + 4 + 8 + 16 + … – 1 – 2 – 4 – 8 – 16 – … = -1

But the LHS simplifies to: 1 + 2 + 4 + 8 + 16 + … = -1

Thus we seem to have proved that the infinite sum of ever-increasing terms is -1. Huh.

The infinite series paradox (2):

What is the value of the infinite sum 1 – 1 + 1 – 1 + 1 – 1 + …?

Answer 1: (1 – 1) + (1 – 1) + (1 – 1) + … = 0 + 0 + 0 + … = 0

So far so good. But hang on…

Answer 2: 1 – (1 – 1) – (1 – 1) – (1 – 1) … = 1 – 0 – 0 – 0 … = 1. Huh.

Answer 3: Suppose T is the value of our mystery sum: T = 1 – 1 + 1 – 1 + 1 – 1 + …

Take the negative of both sides: – T = – 1 + 1 – 1 + 1 …

But this is just what you’d get if you take 1 away from T i.e. T – 1.

So T – 1 = -T, which is only satisfied when T = 1/2. Eek.

I find answer 3 both most counterintuitive and most satisfying. Brilliantly, some of my year 7s came up with this solution themselves.

 

Relevance to the classroom:

This stuff isn’t going to be on any curriculum, and one observer in my classroom was heard afterwards to mutter ‘he just let them doodle!’ But for some of our most gifted mathematicians, we have to squeeze in real maths around the staid content of the GCSE curriculum – it might just inspire them to carry on with the subject to A-level and beyond. Plus everyone loves to have their brains messed with.

Abraham Wald and the bullet holes

So I’m constantly on the hunt for ways to relate maths to real-life, and to find hooks for topics. Over the summer I read Jordan Ellenberg’s book ‘How Not to Be Wrong’ and the following story is lifted straight from that book. It’s a great intro to a probability topic.

The premise
You’re in the air force during World War 2. Your job is to stop your planes getting shot down by enemy fighters, by armouring them. But armour makes a plane heavier, less manoeuvrable and uses more fuel. So you have to compromise between weight and protection from bullets.

The data
You are presented with the table below, which gives average bullet holes on planes returning from missions:

Section of plane Bullet holes per square metre
Engine 1.11
Fuselage 1.73
Fuel system 1.55
Rest of the plane 1.8

The question:

Where is the best place to put extra armour?

Continue reading