Activity 1-2: Inequalities www.carom-maths.co.uk Activity 1-2: Inequalities

What inequalities do you know? What do you think the most basic inequality of all might be? Maybe … the Triangle Inequality.

a < b < c  A < B < C Notice that a triangle has another basic inequality; a < b < c  A < B < C The length of any one side of a triangle is less than the sum of the other two. a < b + c, b < a + c, c < a + b. Travelling from A to B direct is shorter than travelling from A to B via C; we are saying ‘the shortest distance between any two points is a straight line’.

Standard inequalities like these are of great use to the mathematician. More arise from this question:  How do we find the average of two non-negative numbers a and b?

How are these ordered? Does the order of size depend on a and b? Task: try to come up with a proof that AM ≥ GM for all non-negative a and b. When does equality hold? Now try to show that GM ≥ HM for all non-negative a and b.

We can see that equality only holds in each case when a = b.

We can often come up with a diagram that demonstrates an inequality. What inequality does the following diagram illustrate?

How about this? Hint: calculate OA, AB, AC.

So AM  GM  HM.

Can we prove the AM-GM inequality for three numbers? That is, if a, b, c > 0, does 3abc ≤ a3 + b3 + c3 hold? First reflect on this diagram. So we have that ab + bc + ac  a2 + b2 + c2.

Now reflect on this diagram...

With thanks to Claudi Alsina and Roger B. Nelsen, authors of When Less is More; Visualising Basic Inequalities. Carom is written by Jonny Griffiths, hello@jonny-griffiths.net