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Premise 1: All straight lines have the value of length equal to the numerical value of the end point, provided the starting point of the line is assigned the numerical value zero.

Premise 2: No point can be assigned the value y.xxxxxx... or y.abcdef....(Example: 8.9999.... or 8.39465..). We can either have the point with the assigned numerical value to be either y.xxx, y.xx...x, y.abcdef..g, etc, but not y.xxxxx.... or y.abcdef....

Premise 3: All lines have starting and ending point, thus they have the value of length equal to the numerical value of the ending point. From the premise 2, the value must be either y.xxx, y.xx...x, or y.abcd...g, etc.

Premise 4: Diagonal of the square whose side is the unit of length, has got starting and ending point. Therefore, the length of the diagonal should be a value which can be expressed as the fraction with terminating decimal form.

By this argument, length of the diagonal ($\sqrt2$) seems to have fractional form with a terminating decimal form, which (I think) is not true, then what is going wrong in the argument? Or else is it that the diagonal (in this case) has no starting and ending point?

Sensebe
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    Your premise 2 is wrong. And the proof is exactly the irrationality of $\sqrt{2}$. – Emilio Novati Mar 19 '15 at 13:51
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    Premise 2 is basically saying that all lengths are rational, which is known to be false. – Wojowu Mar 19 '15 at 13:53
  • This problem got into me while reading these Dedekind's lines from his book "Essays on the theory of numbers": the ancient Greeks already knew and had demonstrated that there are lengths incommensurable with a given unit of length, e. g., the diagonal of the square whose side is the unit of length. If we layoff such a length from the point 0 upon the line we obtain an end-point which corresponds to no rational number. – Sensebe Mar 19 '15 at 13:57
  • @EmilioNovati: I have added the word "thus" in the premise 2 now. If the premise is still wrong, please comment on it. Thank you. – Sensebe Mar 19 '15 at 14:04
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    the wrong statement in 2) is ''which can be expressed as fraction that has terminating decimal form''. – Emilio Novati Mar 19 '15 at 14:17
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    This reads to me like one of those mathematical proofs of the existence of God. – TonyK Mar 19 '15 at 14:19
  • Why do you think premises 2 and 3 are/should be true? – Wojowu Mar 19 '15 at 14:28
  • @EmilioNovati: Please consider processing with the newly added premise. Thank you for the time. [Please others take a note that the previous premise 2 in now premise 3] – Sensebe Mar 19 '15 at 14:31
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    Under your new premise 2 we already have problems with intervals of length $\frac{1}{3}$, because it'd have to be 0.333... – Wojowu Mar 19 '15 at 14:41
  • @Wojowu: Yes you can never ever divide line of length 1 (using the measurement as defined in the question) into 3 equal parts. – Sensebe Mar 19 '15 at 14:48
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    But in general line segments can be divided into 3 equal pieces, which makes your premises invalid. – Wojowu Mar 19 '15 at 15:01
  • @Wojowu: I think we can't, even though we feel, we can. Can you describe how we can divide the line segment of length 1 into three equal pieces? – Sensebe Mar 19 '15 at 15:56
  • I thank everyone for the involvement. I would like to know related books on this matter. Can you all suggest any? – Sensebe Mar 19 '15 at 16:26
  • We can use compass and straightedge construction to find a point which is 1/3 of the way between two points. So we can divide the line segment into three equal parts by putting two points. – Wojowu Mar 19 '15 at 17:04
  • @Wojowu: I think straightedge don't give us correct answers. It gives us approximate value, which is not helpful here. I don't think there is any way to divide into 3 equal pieces. – Sensebe Mar 20 '15 at 00:32

1 Answers1

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Although I'm a little confused by your language, I would interpret this in one of two ways:

  1. "Premises" 1 through 3 (they're actually axioms) redefine lines and points in $\Bbb R^2$ in a way other than the one we're typically familiar with. Under these rules, I would say that Premise 4 fails because the diagonal of a square is not a line, since it's end point violates Premise 1.

  2. You're operating under a different metric. In other words, distances aren't measured in the the traditional way in your defined space. For instance, if you work in a 2-D metric space using the Taxi Cab metric

$d((x1 , y1), (x2 , y2)) = |x1 - x2| + |y1 - y2|$

then Premise 4 works out fine, although the length of the diagonal is actually 2 in this case, not an irrational value. Indeed, the length of the diagonal of a square under this metric (provided the square is parallel to the axes) is just the sum of the height and width of the square, so it's rational.

TL;DR

Either the length of the diagonal in this space is irrational, in which case the diagonal isn't a "line", or the metric is such that the length of the diagonal of a square with rational sides are never irrational.

Zimul8r
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