Why must we distinguish between rational and irrational numbers?

Ayanna Goodman

Ayanna Goodman

Answered question

2022-11-12

Why must we distinguish between rational and irrational numbers?

Answer & Explanation

brulotfao

brulotfao

Beginner2022-11-13Added 16 answers

Here's one example of where the difference between rational numbers and irrational numbers matters. Consider a circle of circumference 1 (in any units you choose), and suppose we have an ant (of infinitesimal size, of course) on the circle that moves forward by f f instantaneously once per second. Then the ant will return to its starting point if and only if f is a rational number.
Maybe that was a little contrived. How about this instead? Consider an infinite square lattice with a chosen point O. Choose another point P and draw the line segment O P. Pick an angle θ and draw a line L starting from O so that the angle between L and O P is θ. Then, the line L passes through a lattice point other than O if and only if tan θ is rational.
In general the difference between rational and irrational becomes most apparent when you have some kind of periodicity in space or time, as in the examples above.
kituoti126

kituoti126

Beginner2022-11-14Added 8 answers

One thing is in how you construct them. Starting from the natural numbers (and 0) you construct the integers by saying that Z is the smallest set that contains the naturals and is a group under addition. Similarly, the rationals Q is the smallest set containing Z that forms a group under multiplication (when 0 is taken out). The reals R can then be constructed by defining it to be the smallest set containing Q in which every bounded set has a least upper bound.

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