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Exploring Basic Inequalities with Triangles

Dive into the world of inequalities by understanding the basic concepts through visualizing Triangle Inequality. Explore relationships between numbers and geometric shapes to grasp fundamental inequalities and their applications in mathematics.

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Exploring Basic Inequalities with Triangles

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

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

  3. 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’.

  4. 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?

  5. 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?

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

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

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

  9. So AM GM  HM.

  10. Can we prove the AM-GMinequality 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.

  11. Now reflect on this diagram...

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

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