Chapter 1 Number Systems Ex 1.5
Classify the following numbers as rational or irrational.
(i) Irrational ∵ 2 is a rational number and √5 is an irrational number.
∴ 2.√5 is an irrational number.
(∵The difference of a rational number and an irrational number is irrational)
(ii) 3 + 23−−√ – 23−−√ = 3 (rational)
(iii) 27√77√ (rational)
(iv) 12√(irrational) ∵ 1 ≠ 0 is a rational number and 2–√≠ 0 is an irrational number.
∴ 12√ is an irrational number. 42
(∵ The quotient of a non-zero rational number with an irrational number is irrational).
(v) 2π (irrational) ∵ 2 is a rational number and π is an irrational number.
∴ 2x is an irrational number. ( ∵The product of a non-zero rational number with an irrational number is an irrational)
Simplify each of the following expressions
Recall, π is defined as the ratio of the circumference (say c) of a circle to its diameter (say d). That is π = cd. This seems to contradict the fact that n is irrational. How will you resolve this contradiction?
Actually cd = 227,which is an approximate value of π.
Represent 9.3−−−√ on the number line.
Firstly we draw AB = 9.3 units. Now, from S, mark a distance of 1 unit. Let this point be C. Let O be the mid-point of AC. Now, draw aemi – circle with centre O and radius OA. Let us draw a line perpendicular to AC passing through point B and intersecting the semi-circle at point D.
∴ The distance BD = 9.3−−−√
Draw an arc with centre B and radius BD, which intersects the number line at point E, then the point E represents 9.3−−−√ .