1. Is zero a rational number? Can you write it in the form ` \frac{p}{q}\ `, where p and q are integers and q ≠ 0?
solution :
We know that, a number is said to be rational if it can be written in the form ` \frac{p} {q}\ ` , where p and q are integers and q ≠ 0.
Zero can be written in the form `\ 0, \frac{0} {1}, \frac{0} {2}, \frac{0}{3}, \frac{0}{4}, \frac{0} {-1}, \frac{0} {-2}, \frac{0}{-3}, \frac{0}{-4}, ...........\ `
Hence, zero is a rational number.
2. Find six rational numbers between 3 and 4.
solution :
There are infinite rational numbers between 3 and 4.
As we have to find 6 rational numbers between 3 and 4.
we will multiply both the numbers, 3 and 4, with 6+1 = 7.
Example : ` \3×(\frac{7}{7}) = \frac{21}{7}\ ` and ` \4×(\frac{7}{7})= \frac{28}{7} \ `.
The numbers in between ` \frac{21}{7}\ ` and ` \frac{28}{7}\ ` will be rational and will fall between 3 and 4.
Hence, ` \frac{22}{7}, 23/7, 24/7, 25/7, 26/7, 27/7 \ `are the 6 rational numbers between 3 and 4.
3. Find five rational numbers between ` \3/5\ ` and ` \4/5\ `.
solution :
There are infinite rational numbers between ` \3/5\ ` and ` \4/5\ `.
To find out 5 rational numbers between ` \3/5\ ` and ` \4/5\ `.
we will multiply both the numbers ` \3/5\ ` and `\ \4/5\ `
with ` \10/10\ ` (or any number greater than 5)
Example : ` \(3/5) × (10/10) = 30/50\ ` and ` \(4/5) × (10/10) = 40/50\ `.
The numbers in between ` \30/50\ ` and ` \40/50\ ` will be rational and will fall between ` \3/5\ ` and ` \4/5\ `.
Hence, ` \31/50, 32/50, 33/50, 34/50, 35/50\ ` are the 5 rational numbers between ` \3/5\ ` and ` \4/5\ `.
4. State whether the following statements are true
or false. Give reasons for your answers.
(i) Every natural number is a
whole number.
(ii) Every integer
is a whole number.
(iii) Every
rational number is a whole number.
solution :
(i) Every natural number is a whole number.
TRUE
Natural numbers : Numbers starting from 1 to infinity
Example : Natural numbers= 1, 2, 3, 4, 5, 6, 7, ...............,∞.
Whole numbers : Numbers starting from 0 to infinity
Example : Whole numbers= 0, 1, 2, 3, 4, 5, 6, 7,....................., ∞.
Or, we can say that whole numbers have all the elements of natural numbers and zero.
Hence, Every natural number is a whole number.
(ii) Every integer is a whole number.,
FALSE
Integers : Integers are set of numbers that contain positive,negative and 0.
Example : integers are -∞, .........-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, ........, ∞.
Whole numbers : Numbers starting from 0 to infinity.
Example : Whole numbers are 0, 1, 2, 3, 4, 5, .............,∞.
Or, we can say that every integers are not whole numbers.
Hance, Every integer is not a whole number.
(iii) Every rational number is a whole number.
FALSE
Rational numbers : All numbers in the form ` \p/q\ `, where p and q are integers and q ≠ 0.
Example : Rational numbers are ` \0, -1/5 , 2/7, 0/-9, -1/7,..............., ∞\ `.
Whole numbers : Numbers starting from 0 to infinity.
Example : Whole numbers and 0, 1, 2, 3, 4, 5, 6, 7, .............., ∞.
Or, we can say that every Rational numbers not Whole numbers.
Hence, All rational numbers are not whole numbers.
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