On 10/25/09, ...but here's a question wrote:
> OK, so I define a rational number as a number that can be represented
> as a ratio of two integers.
>
> I also tell my kids that irrational numbers (as decimals) go on
> forever without repeating.
>
> That means any ratio of two integers, converted to decimal, must
> either terminate or repeat.
>
> How do you prove that there is no ratio of two numbers that goes on
> forever without repeating?

That is good deductive reasoning on your part. At this level of math, you
will not be able to prove that there is no ratio of two numbers that goes
on forever without repeating. They will just have to take your world for
it maybe because I think proving it might confuse them. p.s. On a sort of
related note, mathematicians believed for centuries that pi might be
irrational but it took until the 1700's for someone to prove it. I think
a student would probably have to be a math major before they could
understand this proof.

But if you wanted to prove to them that no fractions go on forever without
repeating, I guess you could use decimal expansion like this that I just
copied from wikipedia (but I think it will confuse most of them--you
might want to just tell them that mathematicians have proven it and if
they are really curious about it they can learn about it in advanced math
classes in college):

"The decimal expansion of an irrational number never repeats or
terminates, unlike a rational number. To show this, suppose we divide
integers n by m (where m is nonzero). When long division is applied to
the division of n by m, only m remainders are possible. If 0 appears as a
remainder, the decimal expansion terminates. If 0 never occurs, then the
algorithm can run at most m − 1 steps without using any remainder
more than once. After that, a remainder must recur, and then the decimal
expansion repeats. Conversely, suppose we are faced with a recurring
decimal, we can prove that it is a fraction of two integers. For example:

Here the length of the repitend is 3. We multiply by 103:

Note that since we multiplied by 10 to the power of the length of the
repeating part, we shifted the digits to the left of the decimal point by
exactly that many positions. Therefore, the tail end of 1000A matches the
tail end of A exactly. Here, both 1000A and A have repeating 162 at the
end. Therefore, when we subtract A from both sides, the tail end of 1000A
cancels out of the tail end of A:

Then

(135 is the greatest common divisor of 7155 and 9990). Alternatively,
since 0.5 = 1/2, one can clear fractions by multiplying the numerator and
denominator by 2:

(27 is the greatest common divisor of 1431 and 1998). The bottom line,
53/74 is a quotient of integers and therefore a rational number."

Share This Post | Report This Post

Next Post >>

Posts on this thread, including this one