Irrational numbers cannot be represented exactly in binary for the same reason they cannot be represented exactly in decimal: their digits continue infinitely without repeating. Binary, like decimal, is a positional number system but uses only two digits (0 and 1). When an irrational number such as √2 or π is converted to binary, the sequence of digits never terminates or forms a repeating pattern. Since computers have finite memory, they cannot store an infinite binary sequence. As a result, irrational numbers are approximated to a limited number of bits using floating-point representation. This approximation leads to a small loss of accuracy known as rounding error. In programming, this can impact calculations that rely on exact values, such as equality comparisons or cumulative operations over large datasets. Developers must therefore use techniques like error bounds or tolerance levels when comparing irrational values and be cautious in applications where precision is critical.

Despite their non-repeating nature, researchers have found some interesting properties in the decimal expansions of irrational numbers such as π and e, but no true pattern has ever been proven. These numbers are believed to be normal, meaning that their digits are uniformly distributed — each digit from 0 to 9 should appear roughly the same number of times in the long run, and all digit sequences of a given length should occur with equal frequency. However, this has not been mathematically proven for π or e. Extensive statistical analyses have shown that their decimal expansions behave as if they are normal, and this randomness has intrigued both mathematicians and computer scientists. Some digits and sequences appear in interesting places (e.g. a long string of 9s in π is known as the “Feynman point”), but these occurrences are considered coincidental rather than indicative of structure. Overall, the unpredictability of their digits makes them suitable for randomness testing and other computational applications.

Irrational numbers are not typically used directly as keys or identifiers in computing systems because of their approximate nature and infinite decimal expansions. Keys or identifiers, such as those used in databases, cryptographic systems, or memory addresses, require precise and reproducible values. Since irrational numbers must be approximated to a finite precision, small differences in rounding or representation across platforms could result in inconsistencies, making them unsuitable for unique identification. However, in some niche contexts, irrational numbers are used indirectly. For example, their digits can seed pseudo-random number generators (PRNGs) due to their statistical randomness. In fractal generation or procedural graphics, certain irrational constants like the golden ratio (approximately 1.618…) are used to avoid artefacts or visible repetition. But even then, only a fixed-precision approximation is used. For all practical purposes, irrational numbers are avoided in critical systems requiring unique or exact identifiers, as their approximated form could lead to ambiguous or duplicate references.

Scientific applications rely heavily on irrational numbers, but since these numbers cannot be represented exactly, accuracy is maintained through careful approximation and numerical techniques. Most systems use double-precision floating-point (64 bits), which offers around 15–17 significant decimal digits of accuracy — usually enough for physical simulations, engineering models, or statistical calculations. For more demanding tasks, arbitrary-precision arithmetic libraries (e.g. GMP or MPFR) allow irrational numbers to be approximated to thousands of digits, significantly reducing rounding errors. Additionally, algorithms are often designed to minimise the impact of rounding, such as through error correction methods, interval arithmetic, or symbolic computation where expressions are manipulated algebraically until the final step. In environments like scientific computing or computer algebra systems (e.g. MATLAB, Mathematica), irrational numbers may be carried symbolically as π or √2 until a numerical result is explicitly required. Ultimately, while irrational numbers are never stored exactly, their approximations are managed to be accurate enough for the specific precision required by the application.

No, a computer cannot, in general, determine whether a number is irrational unless it is given that information explicitly through mathematical context or symbolic representation. Computers operate using finite precision arithmetic, meaning that every number entered into a system is effectively a rational approximation. For example, entering 3.1415926535 into a program will always be interpreted as a rational number, even if it was intended as an approximation of π. If a number is input symbolically (like sqrt(2) or pi), some systems can recognise and treat it as irrational internally. However, if the number is given only as a decimal or binary approximation, there is no way for the computer to prove whether the number is irrational or just a very long rational decimal. The distinction between rational and irrational is a mathematical property, not something that can be directly inferred from a truncated numerical value. Therefore, irrationality must either be defined mathematically in code or handled through symbolic methods rather than numerical inspection.