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ioda
3 years ago
14

Definition: A fraction a/b is reduced if a and b are relatively prime.

Mathematics
1 answer:
Mariulka [41]3 years ago
3 0

Answer:

it is true

Step-by-step explanation:

We can demonstrate this by contradiction.

First choosing a random rational and assuming that exist two ways to represent this rational. Call that rational x, a, b and a', b ' the couples of relatively prime to express x.

Then we have

x=x\\\frac{a}{b}=\frac{a'}{b'}

Isolating a:

a=\frac{a'b}{b'}

a is an integer and a' is a relatively prime with b', for this reason b has to be a factor of b'. Suppose this factorization b'=nb, replacing it:

a=\frac{a'nb'}{b'}=a'n

But now we have that n is a factor of a and is a factor of b, it means that a and b are not relatively prime, that is a contradiction with our premises. The sentence is true.

And referring to the positive denominator. If we want to express a positive rational the denominator and numerator will be both positive and if is a negative one we choose a positive denominator and a negative numerator.

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