# Thread: x and y are irrational, can x+y be irrational

1. ## x and y are irrational, can x+y be irrational

the answer is yes, but i need to show an example, would i just do it like pi+ square root 2 or would i need to do a proof by contradiciton, could i please have a worked solution please.

also if n is any positive integer then sqrt(2)/n is irrational. notice that this means we can find irrational numbers as small as we like simply by increasing n.
use the fact that sqrt(2)/n can be made as small as we desire to show that between every pair of distinct rational numbers there is an irrational number. that is suppose x Є Q and y Є Q and that x < y show that there is an irrational number z with x < z < y

2. If the question is simply can "irrational plus irrational be irrational", then you only need to show an example to answer "yes": $\displaystyle \sqrt{2}+ \sqrt{2}= 2\sqrt{2}$. If the question is must "irrational plus irrational be irrational", a counter example shows that the answer is "no": $\displaystyle \sqrt{2}+ (1-\sqrt{2})= 1$.

To show that $\displaystyle \sqrt{2}/n$ is irrational for all n, use a proof "by contradiction": If $\displaystyle \sqrt{2}/n$ were rational, we would have $\displaystyle \sqrt{2}/n= a/b$ for integers a and b. Then $\displaystyle \sqrt{2}= (an)/b$ showing that $\displaystyle \sqrt{2}$ is rational, a contradiction.

Since $\displaystyle 1< \sqrt{2}< 2$, $\displaystyle \sqrt{2}/n< 1$ for n> 2. If x and y are rational, x< y, y- x is a positive number and $\displaystyle (y- x)\sqrt{2}/n$ is positive and less than y- x. Finally, $\displaystyle x+ (y-x)\sqrt{2}/n$ is larger than x but less than y.

3. Originally Posted by b0mb3rz
the answer is yes, but i need to show an example, would i just do it like pi+ square root 2 or would i need to do a proof by contradiciton, could i please have a worked solution please.

also if n is any positive integer then sqrt(2)/n is irrational. notice that this means we can find irrational numbers as small as we like simply by increasing n.
use the fact that sqrt(2)/n can be made as small as we desire to show that between every pair of distinct rational numbers there is an irrational number. that is suppose x Є Q and y Є Q and that x < y show that there is an irrational number z with x < z < y

A simpler example if you want ( It has nothing to do with square root (2)
or pi
x=0.01001000100001.....

This is not recurring & has infinite digits in a sequence

y=0.02002000200002....

This is also not recurring & is same

x+y= 0.03003000300003....
neither of these is rational