**Rational Numbers**

- A rational Number is something that can be written as p divided by q where pa nd q both are integers.
- THe number on the top is called numerator So, p divided by q; p is called the numerator and q is called the denominator.
- Q stands for rational numbers as it is special so, this Q with these fat boundaries denote the rational numbers.
- In rational number the same thing can be written in many ways.
- It is useful to add, substract and compare rational numbers.
- If we have two fractions which have different denominations, there is no way to directly compare them.
- The only way them is to somehow convert them into equivalent fractions such that they have the same denominator.
- The usual way is just to find a number such that both the denominators multiply into that number are factors of that number.
- It is not really important that the denominator is the smallest common multiple of the two denominators but it must be some common multiple so that we can bring it all to a common number that you can then compare.
- The representation is not unique for rational numbers.
- The reduced form of a rational number is one where there are no common multiples between the common factors between the top and the bottom.
- So, p by q is of the form, where we can not find any factor f such that f divides p and q.
- The greatest common divisor state that the largest number which divides both top and the bottom.

**Property Of Rational Number**

- We know that there is something which is the next integer and the previous integer.
- So, for every integer m, the next one is m plus 1 and the previous one is m minus 1 and it does not matter if this is positive or negative.
- The property of this next and previous is that there is nothing in between right.
- So, there is no integer between m and m plus 1, there is no integer between m and m minus 1.
- So, that is what next means, it is not some bigger integer or smaller integer.

**Is it possible to talk about next or previous rational?**

- It is not possible to talk about them.
- It is because, between any two rational, we can find another one because we can always take the average of two numbers.
- Rember that if we take the average of any 2 numbers, then it must be between those 2 numbers as because it is the sum of the numbers divided by 2.
- So, the average can not be smaller than both or can not be bigger than both.
- So, if the 2 numbers are not the same, then it must lie strictly between them.
- If the numbers are same then there average is the same.
- Taking the average of any two rational numbers we can find another rational number.
- In other word rational number are dense. Here dense means that they are closely packed together.
- As we can not find any gaps in the rational numbers because between any two rational number we can find another rational number.
- Moreover it is not true for the integers as because we saw that in the number line, there is a gap between m and m plus 1, there is no integer there right.
- The rational numbers are dense and conversely, we say that the integers and natural number are discrete.
- So, a discrete set has this kind of next property and a dense set has no next property between any 2 numbers, will find another number.

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