Simplify each of the following expressions:
(i) (3 3 2 2 + + ) ( ) (ii) (3 3 3 3 + − ) ( )
(iii) ( )
2
5 2 + (iv) ( 5 2 5 2 − + ) ( )
3. Recall, π is defined as the ratio of the circumference (say c) of a circle to its diameter
(say d). That is, π =
c
d
⋅ This seems to contradict the fact that π is irrational. How will
you resolve this contradiction?
Answers
Given : π is defined as the ratio of the circumference (say c) of a circle to its diameter. Does it contradict the fact that π is irrational
To find : resolve this contradiction
Solution:
π is defined as the ratio of the circumference (say c) of a circle to its diameter
π = Circumference of circle / Diameter of circle
c = Circumference of circle
d = Diameter of circle
=> π = c/d
any number which can be written in the form of p/q
is rational number
where p , q ∈ Z and q ≠ 0
Hence π would be rational number if
c , d ∈ Z and d ≠ 0
but c & d both are not integer together
hence π is not rational
so π is irrational
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