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 Question from Dawn, a teacher: Yesterday my advanced 7th grade pre-algebra students were finding the distance between two points on a coordinate grid using the distance formula. A student asked me how come we or math I guess is able to represent a finite distance with an irrational number. I found this to be a very interesting question. I and the 7 other math teachers on my staff had discussed this and we have a variety of thing to say to the kid, but nothing that amounts to an actual answer. The students have studied what it means for a number to be irrational, and they cannot understand how that can "measure" a line that obviously stops. Got any suggestions?

We have four responses for you

Dawn,

Well the easiest way to look at it is as the Pythagoreans did (and it upset some of them too!). A rational number is a ratio of two integers while an irrational is not such a ratio. Now when we look at a right angle triangle, both of the shorter sides having length 1, the hypotenuse has length (by Pythogras' Theorem) equal to square root of 2, which is an irrational number. So, that clearly finite distance has an irrational length, the fact that the decimal representation is infinite has nothing to do with the finitenes of the length. The same would be true of a line of length 1/3, which as a decimal is .333333... .

Penny

Hi Dawn.

I would suggest referring him first to the proof that the square root of 2 is irrational.

The outline is this: Consider a 1 by 1 square with a diagonal drawn on it. What is the measure of the length of the diagonal?

According to Pythagoras, it is √2 which we are TOLD is irrational. Can you "measure" the finite length of the diagonal? Clearly yes. It is right there. But is it rational or irrational? What does that mean?

It means you cannot express the number as a ratio of two integers in lowest form (the numerator and denominator have no common factors).

So let's prove this using reductio ad absurdum (that is, proof by contradiction).

If √2 is rational then you can write it as m/n for integers m, n which have no common factors. Thus m = n √2 and by squaring both sides, m² = 2n². This shows us m must be even. So then there is another integer p such that m = 2p. Therefore 4p² = 2n² which simplifies to 2p² = n² which in turn shows us that n is also even. But if n and m are even, they have a common factor of 2! This is the logical contradiction.

By showing the contradiction, we must retrace back to our assumption and invalidate it. Thus √2 is not a rational number, yet it really exists as a measurable finite quantity.

It seems like we can express anything with some rational number, but clearly we cannot express some of them. Are irrational numbers rare? Perhaps maybe there are just a few, like square roots and pi and such.... Well, no. There are a lot of them! In fact, far more than rational numbers.

This may sound astounding, because there are an infinite number of integers, therefore an infinite number of ratios of integers (rational numbers). So I'm saying there are vastly more irrational numbers.

You can use this for your students as a segue to discussion of cardinality if you like, perhaps by first discussing how many prime numbers there are and comparing that to how many integers there are.

It is a good area of exploration and great to hear about students exploring ideas!

Cheers,
Stephen La Rocque.

Dawn,

Claude

Hi Dawn,

I suspect that this is partly about "language". Am I correct in guessing that the trouble arises from the fact that the line stops but the decimal expansion of the number doesn't? In that case, do lengths like 1 1/3 cm (1.33333.... cms) cause trouble? Perhaps less so because we have the alternate notation 1 1/3? Or the length of the diagonal of a square of side 1? We'd normally say it is the square root of 2 rather than 1.41.... What about what you'd get by cutting a circle of diameter 1 at some point and unravelling it into a straight line? Again, we have the alternate notation pi.

I would say that the line has a length, and the length is a number. Irrational numbers are numbers (as the name suggests), so it is possible that these can measure length. Being irrational is a property that a number may or may not have. For further development, I suggest looking into what implications x being irrational might have for constructing a line segment of length x using a ruler and compass. It is always possible to do it using similar right triangles when x is rational.

Does this help?
Victoria

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