1 / 13

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. Notice that a triangle has another basic inequality ;. a < b < c  A < B < C.

donald
Download Presentation

Activity 1-2: Inequalities

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related