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- Thread starter abbeyofthelema
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- #2

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I can't prove it, but its true.

- #3

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well thank you, is there any particular reason that you believe it to be true?

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Galileo

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Simply use the definition of a rational number.abbeyofthelema said:

A rational number can be written in the form:

[tex]\frac{a}{b}[/tex],

with a and b integers and b not equal to zero.

Suppose you have two rational numbers. Compute their product (and their sum while you're at it). Is the result again of the above form?

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The fact that rationals happen to be the set of all reals with periodic limiting behaviour in their decimal representations is a derived property. The definition of a rational number is just that it can be represented as a quotient of integers.

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that's great. so then the product of two rational numbers must always be rational :)

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[tex]\frac{a}{b} ~ , ~ ~ a, b \in \mathbb{N} [/tex]

It's easy to see that a product of two natrual number must be natrual, thus

[tex] \frac{a}{b} ~ * \frac {c}{d} ~ = ~ \frac{ac}{bd}, ~ ~ a, b, c, d \in \mathbb{N}[/tex]

Setting ac to a and bd to b in the def. finish the proof.