👉 Breach math, also known as the Bretschneider bound, is a mathematical result that provides an upper limit on the size of the set of real numbers that can be expressed as the difference of two rational numbers. In simpler terms, it states that for any real number \( x \), there exists a rational number \( \frac{p}{q} \) (where \( p \) and \( q \) are integers and \( q \neq 0 \)) such that \( |x - \frac{p}{q}| < \frac{1}{q^{1+\epsilon}} \) for any positive \( \epsilon > 0 \). This means that no matter how small you want the difference between a real number and its rational approximation to be, there will always be some rational number that can make this difference smaller than any arbitrarily small positive value. This theorem is significant because it shows that the set of real numbers that can be represented exactly by rational differences is "dense" in the real numbers, but not all real numbers can be approximated by rationals to any desired degree of precision.
