What is the factorial of 0 (0!)?

By mathematical convention, the factorial of 0 is defined as 1. This is because there is exactly one way to arrange zero objects (by doing nothing).

Why can't you calculate the factorial of a negative number?

The factorial function is defined for non-negative integers as the product of all positive integers up to that number. The concept of multiplying a series of decreasing positive integers does not apply to negative numbers, so the factorial is undefined for them.

Why does the calculator have a limit for the input number?

Factorial values grow extremely rapidly. The factorial of 170 is already a very large number (7.257 x 10^306). Numbers larger than this exceed the limits of standard computer data types, resulting in a value of 'Infinity'. Our calculator limits the input to prevent this overflow and ensure accurate results.

Factorial Calculator

Quickly compute the factorial of any non-negative integer.

Calculate n!

Enter a non-negative integer (n)

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About the Factorial Calculator

Understand what a factorial is, how to calculate it, and its applications in mathematics and science.

What is a Factorial?

The factorial of a non-negative integer 'n', denoted by the symbol n!, is the product of all positive integers less than or equal to n. It represents the total number of ways you can arrange 'n' distinct objects in a sequence. For example, the factorial of 5 (written as 5!) is calculated as 5 × 4 × 3 × 2 × 1, which equals 120. This means there are 120 different ways to arrange five distinct items. By special definition, the factorial of 0 (0!) is 1, because there is only one way to arrange zero objects—by doing nothing.

How to Use This Calculator

Our factorial calculator is designed for simplicity and accuracy. To find the factorial of a number, simply enter a non-negative integer into the input box and click the "Calculate Factorial" button. The calculator will instantly compute and display the result. It's important to note that factorial values grow extremely quickly; for this reason, the calculator has a practical limit (around 170) to prevent generating numbers so large that they cause performance issues or result in an 'Infinity' value, which isn't useful for most calculations.

Practical Applications

Factorials are a fundamental concept with significant importance in various branches of mathematics and other scientific fields. They are not just an abstract concept but a practical tool for solving real-world problems.

Permutations and Combinations: This is the most common application. Factorials are the building blocks for calculating the number of ways a set of items can be arranged (permutations) or selected (combinations). This is crucial in statistics, probability theory (e.g., calculating odds in card games), and logistics.
Series Expansions: In calculus, factorials are essential components in the power series expansions of important functions like the exponential function (e^x) and trigonometric functions such as sine and cosine.
Computer Science: Algorithms involving permutations, such as those for solving puzzles like the Rubik's Cube, route optimization problems (like the traveling salesman problem), or scheduling tasks, often rely on factorial-based calculations to explore the solution space.
Probability Theory: Factorials are used to determine the number of possible outcomes in an experiment, forming the basis for calculating probabilities.