Question

which number produces a rational number when added to 0.5

Answer 1

Let, the number be x

Therefore, x +0.5= p/q where q is non zero and (p, q) co-prime.

Then, x= -0.5 + (p/q)

now you can substitute any value of p and q to get x.

Answer 2

Answer:

Any rational number.

Step-by-step explanation:

∴ $x+0.5$ is rational if and only if x is rational.

To proveL:  we proceed as follows:

Suppose that  $x+0.5$ is rational.

Then there are some integers p , q with $q\neq 0$ such that

$x+\frac{1}{2}=\frac{p}{q}$

Subtract both sides by $\frac{1}{2}$, we get

$x=\frac{p}{q}-\frac{1}{2}=\frac{2p-q}{2q}$ which is rational.

Conversely, if x is rational, then there are integers a,b with b>0 such that    $x=\frac{a}{b}$  then we have,  

$x+\frac{1}{2}=\frac{a}{b}+\frac{1}{2}=\frac{2a+b}{2b}$ which is also a rational.