Why is 170 the highest factorial

170 factorial (170!) is commonly cited as the highest factorial exactly representable using standard double-precision floating-point format (the 64-bit float format used by most programming languages) because 171! exceeds approximately 1.798 × 10^308, which is the maximum value this format can represent—anything larger returns as "infinity" in computational systems.

This limitation arises from how computers store floating-point numbers using the IEEE 754 standard, which allocates specific bit ranges for the number's mantissa (significant digits) and exponent. While 170! equals approximately 7.257 × 10^306 (just within range), 171! jumps to roughly 1.241 × 10^309, exceeding the format's capacity. However, this is only a limitation for standard floating-point representation—using arbitrary-precision arithmetic libraries, you can calculate factorials far beyond 170! with complete accuracy. Software implementations like Python's integer type, Java's BigInteger, or specialized mathematical software like Mathematica and Maple routinely compute factorials in the thousands or millions without overflow. The "170 limit" matters primarily for software using native floating-point types for performance reasons or for applications where approximate values suffice. Understanding these computational boundaries helps programmers choose appropriate data types when working with large numbers and explains why factorial calculations sometimes behave unexpectedly when approaching these limits.