LEARN MATH STEP BY STEP THROUGH VERY EASY PROCESS

SOME 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.**