Proving Irrationality of Sums and Products of Irrational Numbers

[hashimcom]

hi
i m hashim i want to solve a qquestion
1.if x is rational & y is irrational proof x+y is irrational?
2. if x not equal to zero...y irrational proof x\y is irrational??
3.if x,y is irrational ..dose it implise to x+y is irrational or x*y is irrational

thanks
please
hashim

 

[d_leet]

1. is easy, what have you tried so far?
2. Do you mean x/y? And is x supposed to be rational? If not then 2 is false.

 

[sutupidmath]

1. since x rational we can write it as p/q, where they cannot be simplifyed anymore.
suppose now that x+y is rational,
so it also can be written like r/s, where r,s are integers
so x+y=r/s,

p/q +y=r/s , y=r/s -p/q, so we come to a contradiction, since the right hand of the equation is also a rational, but it contradicts the fact that y is irrational, so our first assumtion that x+y is rational is wrong.

 

[sutupidmath]

3. for the it looks like trivial.

 

[hashimcom]

3. for the it looks like trivial.

what it mean?
please

& where num 2 proof
please

hashim
thanks

 

[sutupidmath]

well, i am not going to show u the whole proof for the last one. But try to reason the same way i did on problem 1.

 

[hashimcom]

mr sutupidmath ...
czn u say my proof for 2
pf:
since x non zerc, so x either rational or irrational
i. if x rational & x non zero...
x =p\q wher p,q is intger...
y is irrational.
now suppose x\y is rastional
x\y=r\s...r,s is integer.
x\y=(p\q)\y=r\s
y=(s\r)*(p\q)...which is rational...contradict
ii. x is irratinal, x non zero...y irrational
suppose x\y = r\s
x = r\s*y...how to continue now?

please help me?
thanks
hashim

 

[HallsofIvy]

The way you have stated it, "2. if x not equal to zero...y irrational proof x\y is irrational", it's NOT true! For example, \pi/\pi= 1. The ratio of two irrational numbers certainly can be rational. You probably meant: If x is a rational number, not equal to 0, and y is irrational, then x/y is irrational. For all of these, you don't need to go back to the definition of rational numbers as m/n. Use the fact that the rational numberse are closed under the operations of addition, subtraction, multiplication, and division (with divisor not 0).

 

[sutupidmath]

The way you have stated it, "2. if x not equal to zero...y irrational proof x\y is irrational", it's NOT true! For example, \pi/\pi= 1. QUOTE]

what if we put it this way. x, y both irrational and x is not equal to y, then what could we say for x/y??
can we go this way, since x irrational it cannot be written as p/q, for the same reason y cannot be written as r/s, where r,s,p,q are all integers. then is it safe to reason this way

x/y cannot be equal to (p/q)/(r/s)= ps/qr whic is obviously rational, so we can conclude that x/y under these conditions is irrational right??

 

[d_leet]

The way you have stated it, "2. if x not equal to zero...y irrational proof x\y is irrational", it's NOT true! For example, \pi/\pi= 1. QUOTE]

what if we put it this way. x, y both irrational and x is not equal to y, then what could we say for x/y??
can we go this way, since x irrational it cannot be written as p/q, for the same reason y cannot be written as r/s, where r,s,p,q are all integers. then is it safe to reason this way

x/y cannot be equal to (p/q)/(r/s)= ps/qr whic is obviously rational, so we can conclude that x/y under these conditions is irrational right??

No, because you presupposed that line with x and y not being equal to (p/q) and (r/s) respectively, so obviouls they can't equal ps/qr. For a counter example consider 5\pi/6\pi= 5/6

 

[sutupidmath]

ok then, what could we say as a conclusion at this case. Because obviously we cannot conclude that at any case when x and y are irrational x/y is rational. FOr example, sqr3/sqr2 is irrational, or sqr6/sqr3, is obviously irrational.

 

[d_leet]

ok then, what could we say as a conclusion at this case. Because obviously we cannot conclude that at any case when x and y are irrational x/y is rational. FOr example, sqr3/sqr2 is irrational, or sqr6/sqr3, is obviously irrational.

You can't say anything, sometimes it's rational, sometimes it isn't.