Theorem: All real numbers can be written in decimal form. Decimal representation
ID: 3030779 • Letter: T
Question
Theorem: All real numbers can be written in decimal form.
Decimal representations either (I) terminate in infinite strings of 0's, or don't. If they don't terminate, the infinite tail end either (II) has a repeating pattern or (III) doesn't.
Prove this theorem for numbers between 0 and 1 using pigeonhole principle. Assume x is rational.
x = a/b, a and b are positive integers, a < b. Divide, by the usual algorithm, a by b. Assume that the division never terminates. At each step there is a remainder, r, a positive integer, with 0 < r < b. There are b - 1 such possible values for r. Since the division never terminates, we will soon have at least b such remainders, r. Since b - 1 < b, Pigeonhole Principle proves that 2 of the remainders are the same. Furthermore we are bringing down 0 each time. Why?, so the divisions necessarily repeat themselves, so x is of type (II).
Explanation / Answer
We are bringing down 0 each time as the remainder is in 0 < r < b. For determining the next digit in division pocess we partition the remainder into next tength part.Since in division of a decimal number when remainder is not zero, the process is repeated for the next or remaining fractional part.
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