Introduction
In your earlier classes, we've learnt about the number line and how to represent various types of numbers on it (see Fig. 1.1).
Just imagine you start from zero and go on walking along this number line in the positive direction.
As far as your eyes can see, there are numbers, numbers and numbers
Now suppose you start walking along the number line, and collecting some of the numbers. Get a bag ready to store them.
You might begin with picking up only natural numbers like 1, 2, 3, and so on. You know that this list goes on for ever.
So, now your bag contains infinitely many natural numbers! Recall that we denote this collection by the symbol N.
Now turn and walk all the way back, pick up zero and put it into the bag. You now have the collection of whole numbers which is denoted by the symbol W.
Now, stretching in front of you are many, many negative integers. Put all the negative integers into your bag.
Now, Recall that it is the collection of all integers, and it is denoted by the symbol Z.
Are there some numbers still left on the line? Of course! There are numbers like `color{orange}(1/2,3/4,)` or even `color{orange}((-2005)/(2006))` If you put all such numbers also into the bag, it will now be the collection of rational numbers.
The collection of rational numbers is denoted by Q. ‘Rational’ comes from the word ‘ratio’, and Q comes from the word ‘quotient’.
You may recall the definition of rational numbers:
A number `‘r’` is called a `"rational number,"` if it can be written in the form `p/q` where `p` and `q` are integers and `q ne0`
`"Notice"` that all the numbers now in the bag can be written in the form `p/q` where `p` and `q` are integers and `q ne 0`
For example, `–25` can be written as `(-25)/1` here `p = -25` and `q = 1.`
Therefore, the rational numbers also include the natural numbers, whole numbers and integers.
You also know that the rational numbers do not have a unique representation in the form `p/q` where `p` and `q` are integers and `q ne 0` For example `2 = 2/4 = 10/20=25/50 = 47/94` and so on.
These are equivalent rational numbers (or fractions). However, when we say that `p/q` is rational number, or when we represent `p/q` on the number line, we assume that `q ne 0` and that p and q have no common factors other than `1` (that is, `p` and `q` are co-prime).
So, on the number line, among the infinitely many fractions equivalent to `1/2` we will choose `1/2` to represent all of them.
Just imagine you start from zero and go on walking along this number line in the positive direction.
As far as your eyes can see, there are numbers, numbers and numbers
Now suppose you start walking along the number line, and collecting some of the numbers. Get a bag ready to store them.
You might begin with picking up only natural numbers like 1, 2, 3, and so on. You know that this list goes on for ever.
So, now your bag contains infinitely many natural numbers! Recall that we denote this collection by the symbol N.
Now turn and walk all the way back, pick up zero and put it into the bag. You now have the collection of whole numbers which is denoted by the symbol W.
Now, stretching in front of you are many, many negative integers. Put all the negative integers into your bag.
Now, Recall that it is the collection of all integers, and it is denoted by the symbol Z.
Are there some numbers still left on the line? Of course! There are numbers like `color{orange}(1/2,3/4,)` or even `color{orange}((-2005)/(2006))` If you put all such numbers also into the bag, it will now be the collection of rational numbers.
The collection of rational numbers is denoted by Q. ‘Rational’ comes from the word ‘ratio’, and Q comes from the word ‘quotient’.
You may recall the definition of rational numbers:
A number `‘r’` is called a `"rational number,"` if it can be written in the form `p/q` where `p` and `q` are integers and `q ne0`
`"Notice"` that all the numbers now in the bag can be written in the form `p/q` where `p` and `q` are integers and `q ne 0`
For example, `–25` can be written as `(-25)/1` here `p = -25` and `q = 1.`
Therefore, the rational numbers also include the natural numbers, whole numbers and integers.
You also know that the rational numbers do not have a unique representation in the form `p/q` where `p` and `q` are integers and `q ne 0` For example `2 = 2/4 = 10/20=25/50 = 47/94` and so on.
These are equivalent rational numbers (or fractions). However, when we say that `p/q` is rational number, or when we represent `p/q` on the number line, we assume that `q ne 0` and that p and q have no common factors other than `1` (that is, `p` and `q` are co-prime).
So, on the number line, among the infinitely many fractions equivalent to `1/2` we will choose `1/2` to represent all of them.
