Big Factorials - Numberphile

Summary

This video explores the concept of factorials and introduces Stirling's approximation, a mathematical shortcut for calculating large factorials. The presenter explains how factorials are the product of integers up to a given number and how Stirling’s approximation simplifies calculations for large numbers. Though it may not always give exact results, the approximation's ratio approaches 1 as the number increases, making it useful for large values of N. Examples are given to highlight the practical use of this approximation in fields like mathematics and physics, showing how it saves time and simplifies complex calculations.

Takeaways

• 😀 Factorials are the product of all positive integers from 1 to N, e.g., 3! = 1 × 2 × 3 = 6 and 5! = 1 × 2 × 3 × 4 × 5 = 120.
• 😀 Factorials grow extremely fast, making their direct computation difficult for large numbers like 100! or 2,000!.
• 😀 Stirling's approximation provides a way to estimate factorials for large N with a formula that simplifies the process: N! ≈ (N^N / e^N) × √(2πN).
• 😀 Stirling's approximation is not exact, but it becomes increasingly accurate as N gets larger, with the ratio of exact factorial to approximation approaching 1.
• 😀 For small numbers like 5!, Stirling's approximation gives a result close but not exact, e.g., 118.019 instead of 120, with a ratio of 1.01678.
• 😀 For large numbers like 100!, Stirling’s approximation captures the magnitude correctly, though the actual difference between the factorial and approximation is large.
• 😀 The ratio of exact factorial to approximation becomes increasingly close to 1 as N increases. For example, 100! has a ratio of 1.00083, and 2,000! has a ratio of 1.00004.
• 😀 Despite large absolute differences for very large factorials, Stirling’s approximation gives a highly accurate relative size, making it useful for large N calculations.
• 😀 Stirling's approximation is widely used in mathematical fields like combinatorics, statistical physics, and any area involving large numbers, simplifying complex calculations.
• 😀 The formula for Stirling’s approximation involves constants like e (Euler's constant) and pi, which arise from integral representations and advanced calculus methods.

Q & A

What is a factorial?

A factorial of a number is the product of all positive integers from 1 up to that number. For example, 3 factorial (3!) is 1 * 2 * 3 = 6.

How is Stirling's approximation different from the exact factorial?

Stirling's approximation is a formula used to estimate factorials, especially for large numbers. While it doesn't provide the exact value, it gives a result that becomes more accurate as the number grows larger.

What is the formula for Stirling's approximation?

Stirling's approximation for a large number N is: N! ≈ N^N * e^(-N) * √(2πN). This approximation becomes more accurate as N increases.

Why is Stirling's approximation considered useful despite not being exact?

Although Stirling's approximation isn't exact, it's highly useful for large numbers because it simplifies calculations, reducing the computational effort required compared to directly calculating the factorial.

What does the symbol '≈' mean in the context of Stirling's approximation?

'≈' means 'approximately equal to'. In Stirling's approximation, it indicates that the factorial N! is approximately equal to the expression on the right-hand side, but not exactly equal.

How does the ratio of the actual factorial to the approximation behave as N increases?

As N increases, the ratio of the actual factorial to Stirling's approximation gets closer and closer to 1, even though the absolute difference between the two values grows larger.

Can Stirling's approximation be used for small numbers?

Stirling's approximation is not very useful for small numbers because it doesn't provide accurate results for small values of N. It is most effective for large numbers.

Why does the ratio of N! to the approximation approach 1 as N grows large?

As N grows larger, the relative error between N! and its approximation becomes negligible. The ratio approaches 1, meaning the approximation accurately represents the magnitude of the factorial.

What role does Euler's constant (e) play in Stirling's approximation?

Euler's constant (e) appears in Stirling's approximation as part of the formula, contributing to the accuracy of the approximation for large values of N. It arises from the mathematical properties of exponential growth.

Who commonly uses Stirling's approximation, and why?

Stirling's approximation is widely used in fields such as mathematics, combinatorics, and statistical physics. It helps simplify complex calculations involving large numbers, such as in the study of large systems or particle behavior.