Original Problem
Given a set $ N=\{a_1,…,a_{n}\}$ with $ n$ positive numbers and $ \sum_i a_i=1$ , find a subset whose sum is $ x_*$ , where $ x_*$ is a known irrational number and $ x_*\approx 0.52$ .
I proved its hardness by the following arguments.
Instance
Given a set $ N=\{a_1,…,a_{n+2}\}$ with $ n+2$ numbers where
 $ a_1,…,a_n$ are positive and rational
 $ \sum_{i=1}^n a_i = .02$
 $ a_{n+1}=x_*0.01$
 $ a_{n+2}=0.99x_*$
determine whether we can find a subset of $ N$ , such that the sum of the subset is $ x_*$ . .
NPcomplete

Since 𝑥∗ is irrational, the desired subset cannot contain both of the last two numbers.

Since the sum of any subset not containing the n+1st element is smaller than 𝑥∗, the desired subset must contain the n+1st element.

The remaining question is to find a subset of the first n numbers whose sum is .01
So the original problem is NPcomplete.
My problem
Someone argued that since $ x_*$ is irrational, I can’t store irrational numbers in a machine properly and my proof is not correct. How to address it?