Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫  π  -¹ ² ³ °

You are not logged in.

## #1 2012-10-18 11:41:59

ShivamS
Member
Registered: 2011-02-07
Posts: 3,648

### Rational Numbers

For those of you who face those relatively hard rational number questions, here is a short answer to most of them. Usually, you see "Find n rational numbers between x and y". In the case that y<x, you can use the formula: d = (y-x)/(n+1). I used this formula long ago and it saved me from a lot of tests.

Last edited by ShivamS (2014-12-30 02:48:34)

Offline

## #2 2012-10-18 19:41:35

bob bundy
Registered: 2010-06-20
Posts: 8,591

### Re: Rational Numbers

hi Shivamcoder3013

A gremlin had crept into yout post so that was rendered as an emoticon on my computer.  This seems to happen with some browsers.  If you put a space = ( then it avoids this problem.

Now to the post itself.  I'm a bit confused about how you use this formula.  Please would you give an example.  Thanks.  [intended emoticon]

Bob

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob Bundy

Offline

## #3 2012-10-19 06:14:25

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,039

### Re: Rational Numbers

bob bundy wrote:

hi Shivamcoder3013

A gremlin had crept into yout post so that was rendered as an emoticon on my computer.  This seems to happen with some browsers.  If you put a space = ( then it avoids this problem.

Now to the post itself.  I'm a bit confused about how you use this formula.  Please would you give an example.  Thanks.  [intended emoticon]

Bob

The emoticons can also be avoided by checking the "never show smilies" options in the "Post reply" screen.

I think he is just dividing the interval (x,y) into n+1 pieces. Adding a number of these to x will give us n rational numbers.

Here lies the reader who will never open this book. He is forever dead.
Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

Offline

## #4 2012-10-20 06:27:39

ShivamS
Member
Registered: 2011-02-07
Posts: 3,648

### Re: Rational Numbers

Yes, I should have clarified. After you use the specified formula, you get the Rational # in between by: x+d, x+2d, x+3d...x+nd. To give an example, lets say we have to find 3 rational numbers between 2 and 3. Therefore: x=2 y=3 n=3. Thus, d= (3-2)/(3+1)=1/4. So the n numbers are: 2 1/4, 2  2/4 and 2 3/4. This is fairly obvious, I just wanted to give the generalized formula.

Offline

## #5 2012-10-20 15:17:44

noelevans
Member
Registered: 2012-07-20
Posts: 236

### Re: Rational Numbers

Hi Y'all!

Indeed a nice way to get the n numbers in between! However the following question arises.

Are we talking about n fractions between two fractions or n rational numbers between two rational
numbers? (or fractions between two rational numbers, or perhaps rational numbers between two
fractions?)  It seems that the question is for fractions.  Given two rational numbers 2.5 and 3.0 we
can get any number of rational numbers between these by just starting with 2.5 and tack on any
digits we wish beyond the 5 in 2.5.  For example 2.53, 2.56, 2.568, 2.5809 are four rational
numbers between 2.5 and 3.0.  And there are many other options starting with 2.6, 2.7, 2.8 or 2.9.

For four fractions between 5/2 and 3/1 we can use  11/4, 22/8, 33/12, and 44/16.  These are
different fractions between 5/2 and 3/1, but all correspond to only one rational number 2.75
between 2.5 and 3.

We tend to use for each rational number the reduced fraction that corresponds to it.  .5 and 1/2
are used more or less interchangeably although they are not technically speaking the same
thing.  But for .285714... (repeating infinitely this 6 digit block) we tend to prefer the fraction 2/7.

.75 represents the total amount represented by each fraction in {3/4, 6/8, 9/12, 12/16...} but
each of these fractions indicate how that amount is divided up (For example, 3 fourths  pie,
6 eights pie, 9 twelfths pie, 12 sixteenths pie, ) We especially pick on 75/100 when reading .75 but
have you ever seen a pie cut into 100 pieces? A nice pecan pie with whole pecans  would become
a pecan pudding with diced pecans!

Picky, picky, picky, eh?

Just had to get my 2/100 dollar in (or is it 4/200 or 6/300 etc. dollar?).

Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).
LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

Offline

## #6 2012-10-20 19:50:44

bob bundy
Registered: 2010-06-20
Posts: 8,591

### Re: Rational Numbers

hi Shivamcoder3013

Thanks for the clarification.

In the case that y<x

x<y maybe ?

Bob

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob Bundy

Offline

## #7 2012-10-21 08:49:04

ShivamS
Member
Registered: 2011-02-07
Posts: 3,648

### Re: Rational Numbers

Yes Bob, my mistake.

Offline

## #8 2012-10-22 06:52:40

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,039

### Re: Rational Numbers

noelevans wrote:

Hi Y'all!

Indeed a nice way to get the n numbers in between! However the following question arises.

Are we talking about n fractions between two fractions or n rational numbers between two rational
numbers? (or fractions between two rational numbers, or perhaps rational numbers between two
fractions?)  It seems that the question is for fractions.  Given two rational numbers 2.5 and 3.0 we
can get any number of rational numbers between these by just starting with 2.5 and tack on any
digits we wish beyond the 5 in 2.5.  For example 2.53, 2.56, 2.568, 2.5809 are four rational
numbers between 2.5 and 3.0.  And there are many other options starting with 2.6, 2.7, 2.8 or 2.9.

For four fractions between 5/2 and 3/1 we can use  11/4, 22/8, 33/12, and 44/16.  These are
different fractions between 5/2 and 3/1, but all correspond to only one rational number 2.75
between 2.5 and 3.

We tend to use for each rational number the reduced fraction that corresponds to it.  .5 and 1/2
are used more or less interchangeably although they are not technically speaking the same
thing.  But for .285714... (repeating infinitely this 6 digit block) we tend to prefer the fraction 2/7.

.75 represents the total amount represented by each fraction in {3/4, 6/8, 9/12, 12/16...} but
each of these fractions indicate how that amount is divided up (For example, 3 fourths  pie,
6 eights pie, 9 twelfths pie, 12 sixteenths pie, ) We especially pick on 75/100 when reading .75 but
have you ever seen a pie cut into 100 pieces? A nice pecan pie with whole pecans  would become
a pecan pudding with diced pecans!

Picky, picky, picky, eh?

Just had to get my 2/100 dollar in (or is it 4/200 or 6/300 etc. dollar?).

Rational numbers are the same as fractional numbers. 11/4, 22/8, 33/12, and 44/16 are not different fractions (i.e. fractional numbers), they are different representations of the same number.

Here lies the reader who will never open this book. He is forever dead.
Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

Offline

## #9 2012-10-22 11:04:38

noelevans
Member
Registered: 2012-07-20
Posts: 236

### Re: Rational Numbers

Hi stefy!

Perhaps we were taught differently.  I was taught that using the set W={0,1,2,3,...} of whole numbers, the set of fractions derived thereby is F = {a/b | a and b are whole numbers, b not 0}.
Then the rational number .75 would be the class {3/4, 6/8, 9/12,...} of equivalent fractions.

Having differing viewpoints often leads to interesting conversations, and occasionally to new ideas
in mathematics.  You apparently have a quite varied and extensive background in mathematics.
More power to you!

Interesting side note:  Most people say you can't have zero in the denominator of a fraction
because division by zero is not defined.  I see the reason for not having zero in the denominator
as being the definition precludes it from the very beginning.  Just writing something like 2/0 is
violating the definition.

Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).
LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

Offline

## #10 2014-12-29 22:17:16

DeanPemberton
Member
From: USA
Registered: 2014-10-30
Posts: 20
Website

### Re: Rational Numbers

ShivamS wrote:

Yes, I should have clarified. After you use the specified formula, you get the Rational # in between by: x+d, x+2d, x+3d...x+nd. To give an example, lets say we have to find 3 rational numbers between 2 and 3. Therefore: x=2 y=3 n=3. Thus, d= (3-2)/(3+1)=1/4. So the n numbers are: 2 1/4, 2  2/4 and 2 3/4. This is fairly obvious, I just wanted to give the generalized formula.

Yes I agree with you I also use this formula of rational number and I got the actual answer.

Offline