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Polynomials

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Question
CBSEENMA9001434
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Is zero a rational number? Can you write it in the form straight p over straight q commawhere p and q are integers and q ≠ 0?

Solution

Yes! zero is a rational number. We can write zero in the form straight p over straight q comma where p and q are integers and q ≠ 0 as follows:  
0 equals 0 over 1 equals 0 over 2 equals 0 over 3 space e t c..
Note. Denominator q can also be taken as negative integer.

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Question
CBSEENMA9001435
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Find six rational numbers between 3 and 4.

Solution

There can be infinitely many rational numbers between 3 and 4.
fraction numerator 3 plus 4 over denominator 2 end fraction equals 7 over 2 fraction numerator 3 plus begin display style 7 over 2 end style over denominator 2 end fraction equals 13 over 4 fraction numerator 3 plus begin display style 13 over 4 end style over denominator 2 end fraction equals 25 over 8 fraction numerator 3 plus begin display style 25 over 8 end style over denominator 2 end fraction equals 49 over 16 fraction numerator 3 plus begin display style 49 over 16 end style over denominator 2 end fraction equals 97 over 32 fraction numerator 3 plus begin display style 97 over 32 end style over denominator 2 end fraction equals 193 over 64
Thus, six ratonal numbers between 3 and 4 are   7 over 2 comma space 13 over 4 comma space 25 over 8 comma space 49 over 16 comma space 97 over 32 space and space 193 over 64.
Alter 
open table attributes columnalign right end attributes row cell 3 equals 3 over 1 equals fraction numerator 3 cross times 7 over denominator 3 cross times 7 end fraction equals 21 over 7 end cell row cell 4 equals 4 over 1 equals fraction numerator 4 cross times 7 over denominator 4 cross times 7 end fraction equals 28 over 7 end cell end table close curly brackets space 6 plus 1 equals 7
Thus, six rational numbers between 3 and 4 can be taken as  
22 over 7 comma space 23 over 7 comma space 24 over 7 comma space 25 over 7 comma space fraction numerator 26 over denominator space 7 end fraction space and space 27 over 7.
Note. This is known as the method of finding rational numbers in one step.

Question
CBSEENMA9001436
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Find five rational numbers between  3 over 5 space and space space 4 over 5.

Solution

3 over 5 equals fraction numerator 3 cross times 10 over denominator 5 cross times 10 end fraction equals 30 over 50 4 over 5 equals fraction numerator 4 cross times 10 over denominator 5 cross times 10 end fraction equals 40 over 50
Therefore, five rational numbers between 3 over 4 space a n d space 4 over 5 can be taken as  31 over 50 comma space 32 over 50 comma space 33 over 50 comma space 34 over 50 comma space 35 over 50.

Question
CBSEENMA9001437
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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) True, since the collection of whole numbers contains all natural numbers.
(ii) False, for example, - 2 is an integer but not a whole number.
(iii) False, for example, 1 half is a rational number but not a whole number.

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