CLASS-9SOME MORE RESULT IN IRRATIONAL NUMBER

Some More Results on Irrational Numbers

Results.1) Show that the negative of an irrational number is irrational.

Solution :- Let ‘x’ be irrational and if possible, let (-x) be rational.

Then, (-x) is rational

=>  - (-x) is rational  [ negative of rational is rational ]

=>  x is rational

This contradicts the fact that, x is irrational

The contradiction arises by assuming that, -x is rational.

Thus whenever ‘x’ is irrational, then –x is irrational.

Result.2)- Show that the sum of a rational and an irrational is irrational

Solution:- Let ‘x’ is rational and ‘y’ is irrational

Then, we have to show that (x + y) is irrational.

If, possible let, (x + y) be rational, then,

(x + y) is rational and ‘x’ is rational

=> [(x + y) – x] is rational        [difference of two rational is rational]

=> ‘y’ is rational

This contradicts the fact that ‘y’ is irrational

The contradiction arises by assuming that (x + y) is rational

Thus, whenever ‘x’ is rational ‘y’ is irrational, then (x + y) is irrational.

i.e., (2 + √5), (1/3 + √2), (-3/7 + √5) are all irrational.

Result.3) Show that the product of a non-zero rational with an irrational is irrational.

Solution.) Let, ‘x’ be a non-zero rational and ‘y’ be irrational

Then, we have to show that ‘xy’ is irrational.

If possible, let ‘xy’ be not irrational

Then, ‘xy’ is rational

Now, xy is rational and ‘x’ is non-zero rational

=> (xy) ÷ x is rational       [quotient of two rationals is rational]

=>  (xy)/x is rational

=>  ‘y’ is rational.

This contradicts the fact that ‘y’ is irrational.

Since, the contradiction arises by assuming that ‘xy’ is rational, so xy is irrational. Thus, 6√7, 4√3/7, -5√7/11, etc, are all irrational.