Thread: 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

If the question is simply can "irrational plus irrational be irrational", then you only need to show an example to answer "yes": . If the question is must "irrational plus irrational be irrational", a counter example shows that the answer is "no": .

To show that is irrational for all n, use a proof "by contradiction": If were rational, we would have for integers a and b. Then showing that is rational, a contradiction.

Since , for n> 2. If x and y are rational, x< y, y- x is a positive number and is positive and less than y- x. Finally, is larger than x but less than y.

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