# Prove that the rationals are dense

• PcumP_Ravenclaw
In summary, there are infinitely many rational numbers between a/b and c/d, where m is any positive integer.
PcumP_Ravenclaw
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Show that there are infinitely many rational numbers ## \frac{a +m*c}{b + m*d} ## between the two rational numbers ## a/b## ##c/d ##. m is any positive integer.

My attempt:

first make common denominator

##
\frac{a*d}{b*d}##
##\frac{c*b}{d*b}
##
all numbers going from ##a*d## to ##c*b## divided by ##b*d## is in this interval. but these are finite so we have to increase the resolution. say multiply numerator and denominator by 1000 so now ## a*d*1000 ## to ## c*b*1000 ## divided by ## b*d*1000 ## .

It can be 100000 or anything very large say multiplied to infinity. how do I write this mathematically?

I am not sure I understand what you are trying to argue. Is ad<cb? How are you going to show the form stated in the problem?
Try looking at values for m from zero to infinity. You should be able to conclude that they are all in the interval [a/b,c/d).

PcumP_Ravenclaw said:
Show that there are infinitely many rational numbers ## \frac{a +m*c}{b + m*d} ## between the two rational numbers ## a/b## ##c/d ##. m is any positive integer.

My attempt:

first make common denominator

##
\frac{a*d}{b*d}##
##\frac{c*b}{d*b}
##
all numbers going from ##a*d## to ##c*b## divided by ##b*d## is in this interval. but these are finite so we have to increase the resolution. say multiply numerator and denominator by 1000 so now ## a*d*1000 ## to ## c*b*1000 ## divided by ## b*d*1000 ## .

It can be 100000 or anything very large say multiplied to infinity. how do I write this mathematically?
I would do this in steps.
1. Show that ##\frac{a + mc}{b + md}## is less than c/d, for any choice of a positive integer m.
2. Show that ##\frac{a + mc}{b + md}## is greater than a/b, for any choice of a positive integer.
I am assuming that a/b < c/d.
If you can prove the above propositions, you're done, since m can be any integer in the set {1, 2, 3, ...}

PcumP_Ravenclaw
RUber said:
I am not sure I understand what you are trying to argue. Is ad<cb? How are you going to show the form stated in the problem?
Try looking at values for m from zero to infinity. You should be able to conclude that they are all in the interval [a/b,c/d).
I want to know how to derive the form ## \frac{a +m*c}{b+m*d} ##

PcumP_Ravenclaw said:
I want to know how to derive the form ## \frac{a +m*c}{b+m*d} ##
I don't believe that you need to derive it, just show that there are an infinite number of rationals of this form between a/b and c/d.

I think that how it works is that given a/b and c/d, where a/b < c/d, the rational number with m = 1, ## \frac{a +1*c}{b+1*d} ##, is halfway between a/b and c/d. The same fraction with m = 2 gives a number that is 2/3 of the way between a/b and c/d. When m = 3, the fraction gives a number that is 3/4 of the way between a/b and c/d, and so on. These are the results I get when I choose a/b = 1/4 and c/d = 1/2, for a few selected values of m.

Is not halfway between ## a/b ## and ## c/d ## equal to ## \frac{a*d+c*b}{2*b*d} ##?

PcumP_Ravenclaw said:
Is not halfway between ## a/b ## and ## c/d ## equal to ## \frac{a*d+c*b}{2*b*d} ##?
Yes it is.

Furthermore, the result for ##\displaystyle \ \frac{a+m\cdot c}{b+m\cdot d} \ ## depends not only on the two rational numbers, a/b and c/d but depends on the particular values used for each of a, b, c, and d .

For Mark's example with a/b = 1/4 and c/d = 1/2:
Using a = 1, b = 4, c = 2, d = 4 the result with m = 1 is: ##\displaystyle \ \frac{1+1\cdot 2}{4+1\cdot 4}=\frac38 \ .##

Using a = 1, b = 4, c = 1, d = 2 the result with m = 1 is: ##\displaystyle \ \frac{1+1\cdot 1}{4+1\cdot 2}=\frac13 \ .##​

Each of these results is between 1/4 and 1/2. Only the first is halfway between .

PcumP_Ravenclaw
Can I say that my way to find rational numbers inbetween ## a/b ## and ## c/d ## is
## \frac{a*d+c*b}{m*b*d} ## where m is an integer from 1 to infinity. I realize that this will miss certain rational numbers but there is no way to encompass all the rational numbers using any formula involving integers like m? because between m = 1 and m = 2 there are infinitely many fractions? so the formula ##
\frac{a+m\cdot c}{b+m\cdot d}
## is also similar to mine missing some rational numbers so whose formula has higher resolution?

Thank you!

PcumP_Ravenclaw said:
Can I say that my way to find rational numbers inbetween ## a/b ## and ## c/d ## is
## \frac{a*d+c*b}{m*b*d} ## where m is an integer from 1 to infinity.
that only works for m=2. You need some more references to m in the expression.
Anyway, the way I read the question, you are required to show specifically that there are infinitely many of the form ##\frac{a +m*c}{b + m*d}##

It seems pretty straightforward that if m = 0, you have a/b and the limit as m goes to infinity will surely be c/d. From my vantage point, the only challenge would be to show explicitly that ##\frac ab \leq \frac{a+mc}{b+md} < \frac cd ## for all natural m.
This might be easier if you look at this as some sort of average.
Perhaps a is a total number of something done by b people, and c is the total number done by d people, the ##\frac{a+c}{b+d}## is the average done by all the people.
If every additional group of d people can do c things, than you can continue to average these events out by taking the total number done (a+mc) and dividing by the total number of people (b+md).
From this perspective, it should be clear that your mean can never exceed your largest value (c/d) and it can never be less than your smallest value (a/b).

PcumP_Ravenclaw
haruspex said:
that only works for m=2. You need some more references to m in the expression.
Anyway, the way I read the question, you are required to show specifically that there are infinitely many of the form ##\frac{a +m*c}{b + m*d}##
I agree with haruspex here. (I usually do.)

If you can show that ##\displaystyle \ \displaystyle \ \frac{a+c}{b+d} \ ## is between ##\displaystyle \ \frac ab \ ## and ##\displaystyle \ \frac cd \,,\ ## then that will show that ##\displaystyle \ \frac{a+m\cdot c}{b+m\cdot d} \ ## is between ##\displaystyle \ \frac ab \ ## and ##\displaystyle \ \frac cd \ ## in general. Can you see why?

PcumP_Ravenclaw

## 1. What does it mean for the rationals to be dense?

Being dense means that there are no "gaps" in the set of rational numbers. This means that between any two rational numbers, there is always another rational number. In other words, the rational numbers are closely packed with no missing values.

## 2. How can you prove that the rationals are dense?

One way to prove the density of the rationals is by using the Archimedean property. This property states that for any two real numbers, there exists a natural number that lies between them. Since the rational numbers are a subset of the real numbers, this property can be used to show that there are no "gaps" in the set of rational numbers.

## 3. Can you provide an example of the density of the rationals?

Yes, an example of the density of the rationals is the number line between 1 and 2. We know that there are infinitely many rational numbers between 1 and 2, such as 1.5, 1.25, 1.1, etc. This shows that there are no "gaps" in the set of rational numbers between 1 and 2.

## 4. How does the density of the rationals relate to the concept of limits?

The density of the rationals is closely related to the concept of limits. In calculus, limits are used to describe the behavior of a function as the input approaches a certain value. The density of the rationals ensures that there are always rational numbers that can get arbitrarily close to the limiting value, allowing us to accurately describe the behavior of a function.

## 5. Why is proving the density of the rationals important in mathematics?

The density of the rationals is an important concept in mathematics because it allows us to accurately measure and describe the behavior of real-world phenomena. It also serves as the foundation for many mathematical concepts, such as limits, continuity, and differentiability, which are essential in fields such as physics, engineering, and economics.

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