Number systemsModule - I : Number systemsModule - I Concepts covered :
1. Various systems of numbers
2. Rational numbers
3. Finding rational numbers between
two given rational numbers.
Slide 2 : Systems of numbers Natural numbers:
The numbers we use for counting are called counting numbers or natural numbers.
The collection of natural numbers is represented by the letter N.
There is no end for counting. Hence the collection of natural numbers is infinite.
Slide 3 : Whole numbers:
When zero is included in the collection of natural numbers then we get the collection of whole numbers.
The collection of whole numbers is represented by the letter W.
The collection of whole numbers is infinite.
Slide 4 : Integers:
If we include the negative integers to the collection of whole numbers we get the collection of integers.
The collection of integers is denoted by the
letter Z.
Therefore, the collection of integers consists of negative integers, zero and positive integers.
The collection of integers is infinite.
Rational numbers : Rational numbers If we include numbers of the form a/b to the collection of integers we get the set of rational numbers.
The collection of rational numbers is denoted by the letter Q.
Rational comes from the word ‘ratio’ and Q from the word ‘quotient’.
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
Slide 6 : A doubt may arise if 34 is a rational number or not. 34 is a rational number as it can be written as 34/1, where
p = 34 and q = 1.
Therefore, the rational numbers also include the natural numbers, whole numbers and integers.
Rational numbers do not have a unique representation in the form p/q, where p and q are integers and .
Slide 7 : Let us consider the given case:
and so on.
These are equivalent rational numbers
(or fractions).
Hence, we say that p/q is a rational number, or when we represent p/q on the number line, we assume that
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 , we choose to represent all of them.
Example : Example Are the following statements true or false? Give reasons for your answers.
Every whole number is a natural number.
Sol: False, because zero is a whole number but not a
natural number.
(ii) Every integer is a rational number.
Sol: True, because every integer m can be expressed in
the form m/1, and so it is a rational number.
(iii) Every rational number is an integer.
Sol: False, because is not an integer.
Rational numbers between two given rational numbers : Rational numbers between two given rational numbers There are infinitely many rational numbers between any two given rational numbers.
To find a rational number between r and s, you can add r and s and divide the sum by 2, that is (r + s)/2 lies
between r and s.
Example : Example Find five rational numbers between 1 and 2.
There are two ways of solving this problem.
We can find a rational number between r and s, by adding r and s and then dividing the sum by 2.
Rational number between 1 and 2
Rational number between 1 and 3/2
Rational number between 3/2 and 2
Slide 11 : Rational number between 3/2 and 5/4
Rational number between 3/2 and 7/4
Therefore, 5/4, 11/8, 3/2, 13/8, 7/4 are the five
rational numbers lying between 1 and 2.
Slide 12 : ii) We can find all the five rational numbers in one
step.
As we want five numbers, we write 1 and 2 as rational numbers with denominator 5 + 1, that is 1 = 6/6
and 2 = 12/6.
We will see that 7/6, 8/6, 9/6, 10/6 and 11/6 are all rational numbers between 1 and 2.
So, the five numbers 7/6, 4/3, 3/2, 5/3 and 11/6 lie between 1 and 2.
Exercise 1.1 : Exercise 1.1 Is zero a rational number? Can you write it in the form p/q, where p and q are integers and q 0.
Sol: Yes zero is a rational number. 0 = 0/1.
Find six rational numbers between 3 and 4.
Sol: We need to insert 6 rational numbers between 3 and 4, we
write 3 and 4 as rational numbers with denominator
6 + 1, or 3 = 3 ? 7/7 = 21/7 and 4 = 4 ? 7/7 = 28/7.
So, 22/7, 23/7, 24/7, 25/7, 26/7 and 27/7 lie between
3 and 4.
Therefore, the required rational numbers between 3
and 4 are 22/7, 23/7, 24/7,25/7, 26/7 and 27/7.
Slide 14 : 3. Find five rational numbers between 3/5 and 4/5.
Sol: We need to find five rational numbers between 3/5
and 4/5, so we multiply and divide the given
rational numbers with 6.
So, 3/5 = 3/5 ? 6/6 = 18/30 and
4/5 = 4/5 ? 6/6 = 24/30.
So, 19/30, 20/30, 21/30, 22/30 and 23/30 lie
between 3/5 and 4/5.
Therefore the required rational numbers between
3/5 and 4/5 are 19/30, 2/3, 7/10, 11/15 and 23/30.
Slide 15 : 4.State whether the following statements are true
or false. Give reasons for your answers.
Every natural number is a whole number.
Sol: True. The collection of whole numbers contains
all the natural numbers.
(ii) Every integer is a whole number.
Sol: False. For example is not a whole number.
(iii) Every rational number is a whole number.
Sol: False. For example 3/5 is a rational number but
not a whole number.