activity 1-2: inequalities
Post on 20-Jan-2016
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Activity 1-2: Inequalities
www.carom-maths.co.uk
What inequalities do you know?
What do you think the most basic inequality of all might be?
Maybe …
the Triangle Inequality.
The length of any one side of a triangle is less than the sum of the other two.
Notice that a triangle has another basic inequality;
a < b < c A < B < C
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 theseare 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. When does equality hold?
We can see that equality only
holds in each casewhen 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.
First reflect on this diagram.
So we have that ab + bc + ac a2 + b2 + c2.
Can we prove the AM-GM inequality for three numbers?That is, if a, b, c > 0, does 3abc ≤ a3 + b3 + c3 hold?
Now reflect on this diagram...
Carom is written by Jonny Griffiths, hello@jonny-griffiths.net
With thanks to Claudi Alsina and Roger B. Nelsen,authors of When Less is More; Visualising Basic Inequalities.
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