The Product of Rational Numbers

  • #1
I've been grinding my brain at this problem because im trying to figure out if the product of two rational numbers is always, never, or sometimes rational. a rational number would either have to terminate, or be infinitely periodic, so i would say that the product of two rational numbers is always rational, but i cant say this for sure :(
 

Answers and Replies

  • #2
2,209
1
I can't prove it, but its true.
 
  • #3
coolio

well thank you, is there any particular reason that you believe it to be true?
 
  • #4
Galileo
Science Advisor
Homework Helper
1,989
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abbeyofthelema said:
I've been grinding my brain at this problem because im trying to figure out if the product of two rational numbers is always, never, or sometimes rational. a rational number would either have to terminate, or be infinitely periodic, so i would say that the product of two rational numbers is always rational, but i cant say this for sure :(
Simply use the definition of a rational number.

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?
 
  • #5
998
0
Yes. If you want to try to prove something, then 100% of the time your best bet for a first step is to write down the definitions of the things you're working with :wink:

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.
 
  • #6
definitely

that's great. so then the product of two rational numbers must always be rational :)
 
  • #7
180
0
Def:
[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.
 

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