00:01
We want to prove that every rational number r can be written as r equal p over q where p and q are integer numbers, q is positive, and p and q are relatively prime.
00:16
Moreover, with integers p and q are uniquely determined by r.
00:26
So let's see the case r equal zero separately.
00:30
In that case, we can write 0 as 0 divided by 1.
00:53
And then we have 0, 1 are integers, of course.
00:59
The denominator 1 is positive.
01:04
And 0 and 1 are relatively prime.
01:15
Because the only divisors of 1 are 1 and negative 1, which are also divisors.
01:24
Alts of zero.
01:25
So they are relatively prime.
01:28
And this is the only way to write the fraction zero in this way because if you put another denominator non -equal to zero, that is two, three, etc.
01:43
We will have that that number has a divisor, which is the number itself, which is different from one and negative one.
01:53
And it won't be relatively prime with zero because any number is a divisor of zero and because of that, putting the denominator order than one or negative one will have not non -relatively prime numerator and denominator.
02:10
The negative one is also a possibility and we will have all these properties.
02:16
So let's say we have two ways in this case of expressing the rational number with the characteristic set, but the denominator.
02:28
Equal to negative 1 will be negative not positive.
02:31
So this is the only way we can write the number 0 as a fraction of integer numbers in such a way that denominator is positive, and numerator and denominator are relatively prime.
02:48
So we approve the statement for r equals 0.
02:53
So let's say now that r is not 0.
02:59
That is the fraction representing the rational number r has a numerator different from zero.
03:08
So if r is the rational number, we know that it is, in fact, a fraction m over n with m and n into the numbers and n different from zero.
03:25
That's all we know.
03:28
And having that, we can now talk about the sign of n.
03:39
We can say that you can always consider n positive.
03:58
Because if we have several possibilities, if f, if m is positive and n is positive, we have it already.
04:07
Then the denominator is positive already.
04:29
If both are negative, in which case the fraction is positive, as in the first case, then we can write r, which is m over n, as negative m over negative n.
04:45
The fraction is the same because negative 1 over negative 1 is 1.
04:52
So the fraction is exactly the same.
04:53
But in this, written this way, the denominator is the same.
04:57
Positive because n is negative.
05:06
If m is positive and n is negative in which case the rational number is negative then r which is n over n we will have a negative denominator we can write this again this way it's exactly the same fraction but now the denominator is positive and finally the other possibilities is the fourth possibility m is negative and is positive.
05:45
Then the denominator is already positive.
05:54
So in all the four only possibilities we have, we can rewrite in these two cases, the fraction by changing signs of both the numerator and denominator, and with that we have the same fraction, the same rational number, but with positive denominator.
06:14
So we can always consider the denominator positive.
06:19
That's the first thing we do.
06:23
And now let's see how we redefine r to have the properties of the numerator and denominator being relatively prime.
06:39
So let's define l as the greatest common divisor of m and n.
06:51
We know that that exists because that exists for any integer numbers.
06:57
The greatest common divisor is positive, and by definition, the greatest common divisor of m and n, it's a divisor of m and n simultaneously, that is, l divides m and l divides m.
07:23
This means that we can write m as l times an integer, let's say p, and n, and n is equal to.
07:46
To l times another integer q.
07:50
So where p and q are into the numbers.
08:05
With that, then r, which is the ratio number, which is equal to m over n, will be equal to m is l times p over l times q.
08:20
L cancel out.
08:22
L is positive, so it's not zero.
08:24
And then we get p over q.
08:29
That is the fraction.
08:30
Or the rational number is exactly equal to p over q or p and q are the deers which made the divisibility of l 2m and n.
08:49
Now we have found that.
08:53
Let's prove that p and q are relatively prime, but first let's see the sign of q.
09:02
Well, we said n can always be chosen to be positive.
09:07
And l is also positive and for that reason the sign of m and the sign of m is the same sign of p because m equals p this is positive so the sign of the product is decided by the sign of p so the sign of m will be the sign of p they have the same sign another way said and same thing for n and q because l is positive the signs of n and q are the same.
09:45
So q is positive since n can always be chosen to be positive.
10:01
And l is positive.
10:08
So q is positive.
10:10
And we will prove that p and q are relatively prime.
10:28
Let's see that.
10:29
For that we show or we define any common divisor of p and q, that e be a common divisor of p and q.
10:56
Then, that is, e divides p and e divides q.
11:03
Now because e divides p, p is equal to e times, let's say, q.
11:10
K1 and b e divides q that is q is e times another integer k2 so k1 k2 are integer numbers and we have that if we multiply this by l we get lp remember l times p is so here up is m so we get m which is l times p is e k1 and doing the same thing here we multiply by l both sides of the equation we get lq which is remember equal n we have this it's n but l times q multiplying this both sides by l we get l e k2 so m is l e times k1 and n is l e times k2 so this means that l times e divides m and n...