Homework SourseProve that there are irrational numbers x and y such that x
Prove that there are irrational numbers x and y such that x + y is rational.
Solution
To give one answer, let k=(r^0.5+1)/2. Notice that both 1/k and 1/k^2 are irrational, but their sum is 1. That is
1/k^2 + 1/k = 1
1+k=k^2
Strictly speaking, however, for any irrational x and y such that x+y is rational, there exists a rational a1,a2 and irrational b such that x=a1+band y=a2b meaning that we could write, for rational p:
x+y=p
(a1+b)+(a2b)=p
| To give one answer, let k=(r^0.5+1)/2. Notice that both 1/k and 1/k^2 are irrational, but their sum is 1. That is 1/k^2 + 1/k = 1 This stems immediately from the fact that k satisfies (and very often stems from) the equation:1+k=k^2 and if we divide through by k^2, we get the first equation.Strictly speaking, however, for any irrational x and y such that x+y is rational, there exists a rational a1,a2 and irrational b such that x=a1+band y=a2b meaning that we could write, for rational p: x+y=p (a1+b)+(a2b)=p where the irrational terms obviously cancel. So, in some sense, we can\'t avoid subtraction here - no matter what example we choose, we can always express it in a way that makes the cancellation obvious. |
