Teaching irrational numbers

I was asking students in my discrete mathematics lecture, "What is an irrational number?"

The answers I got were along the lines of, "It is a number whose representation after the decimal point never repeats or terminates." (Yeah, they did not put it quite that formally.)

OK, but... I really think that understanding is inadequate, because it begs the question of how the heck we know that the decimal representation of, say, π, never repeats or terminates! Certainly, no one has ever calculated that representation out to infinity and found no repetition or termination. (Yes, I know, the very phrasing of what I am saying in the previous sentence is ridiculous: but that's my point!)

To really understand why some number is irrational, one has to grasp why trying to represent that number as a ratio sends one onto a never-ending spiral of closer and closer fractional approximations of the irrational number. And once students get that, they can understand that an irrational number is precisely the limit of the process by which we attempt to express that number as a ratio, as our number of attempts goes to infinity.

Comments

  1. But Gene, all real numbers, rational and irrational, can be represented as the limit of a neverending sequence of fractional approximations. 3/10, 33/100, 333/1000, etc. approaches but never equals 1/3.

    And we don’t need to calculate a decimal out to infinity to know how the decimal will behave. We can predict it in advance. To take a simple example, .1011011101111011111... is a perfectly predictable decimal which we know will never repeat, and so we can tell right off the bat that it’s irrational.

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    Replies
    1. "But Gene, all real numbers, rational and irrational, can be represented as the limit of a neverending sequence of fractional approximations. 3/10, 33/100, 333/1000, etc. approaches but never equals 1/3. "

      Yes, but I'm pretty sure 1/3 *does* equal 1/3.

      " To take a simple example, .1011011101111011111... is a perfectly predictable decimal which we know will never repeat, and so we can tell right off the bat that it’s irrational."

      Well, yeah, if you specify a number by a rule about its digits, *then* we know what the digits are. But that's not how we know the square root of two is irrational.

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    2. You might try this out:
      https://youtu.be/yk6wbvNPZW0

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    3. Keshav, this one probably gets at what I am saying even better: https://www.youtube.com/watch?v=f1yDExNAEMg
      Or you might think about what a Dedekind cut is.

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  2. You once said the first Pythagorean who discovered irrationality committed suicide. Had they known about uncomputable numbers, they might've gone on a killing spree.

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  3. I used to teach HS discrete math. I always used the joke "Discrete math is between consenting adults behind closed doors."

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