In an effort to develop better number sense (and to create my own journey from fish to infinity), I have been going through Khan Academy’s math material from the very beginning. So far, I have been able to develop strong intuitions and metaphors for most elementary mathematical ideas. I now see that arithmetic, division, negative numbers, fractions, decimals, factors, multiples, etc. all have clear grounding in reality; I have a much stronger grasp of the material.
So far, the only place my intuition has broken down is irrational numbers. It was surprising to me because it happened so suddenly, like the ground giving way. It was as if we saw this gaping hole in the ground, gingerly walked around it, and pretended it wasn’t there. This bothers me.
Is there any straightforward metaphor for irrational numbers? Could you explain it to a child?
Here is a physical metaphor:
Draw a circle, and prepare two sticks:
- One with the length of the circle’s diameter (2r)
- One with the length of the circle’s circumference (2πr)
You cannot cut both sticks into pieces of the same length, no matter what length you choose.
In other words, you cannot measure both sticks using the same measurement-unit.
You can try meters, feet, inches, miles, or even invent your own measurement-unit.
You will never be able to accomplish this, because the ratio between the sticks is irrational.