I recently enjoyed reading an article from the ATM (Association of Teachers of Mathematics) regarding the geometric representation of surds (Colin Jackson and the all-attainment group). The all attainment group were looking at how to provide a geometric representation of addition and multiplication rules such as:

(1) √2 + √3 ≠ √5

(2) √2 × √3 = √6

It immediately occured to me that (1) would be difficult using pythagoras due to the different angles that √2 and √3 make with the horizontal.

Of course, you could get around this on Geogebra using the following applet.

Alternatively, you can create a new triangle with lengths of √2 and √3 and use the cosine rule to find the missing length.

Then using the triangle inequality √2 + √3 ≥ 3.315. Since √9 = 3 we can conclude that

√2 + √3 > √9 and therefore √2 + √3 ≠ √5

This can also be shown by looking at the roots of certain quadratic functions in the following way.

It definately depends what year group you’re doing this with as to how you approach it. Personally I think I’d opt for showing that √2 + √2 ≠ √4 by using pythagoras.

You could then go one step further with a slightly more complex diagram and show that

√2 × √2 = √4 = 2

If anyone else has any more ideas on this I’d be fascinated to know more.

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