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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.


(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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Find six rational numbers between 3 and 4.

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.
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State whether the following statements are true or false. Justify your answers.

(i)    Every irrational number is a real number.
(ii)   Every point on the number line is of the form square root of straight m comma where m is a natural number.
(iii)  Every real number is an irrational number.


(i) True, since collection of real numbers is made up of rational and irrational numbers.
(ii) False, because no negative number can be the square root of any natural number.
(iii) False, for example, 2 is real but not irrational.

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

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.
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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?


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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