Lesson plan
Binomial approximation
The binomial approximation is useful for approximately calculating powers of sums of 1 and a small number x. It states that : (1 + x)^\alpha \approx 1 + \alpha x. It is valid when |x| and |\alpha x| \ll 1 where x and \alpha may be real or complex numbers. The benefit of this approximation is that \alpha is converted from an exponent to a multiplicative factor. This can greatly simplify mathematical expressions (as in the example below) and is a common tool in physics. The approximation can be proven several ways, and is closely related to the binomial theorem. By Bernoulli's inequality, the left-hand side of the approximation is greater than or equal to the right-hand side whenever x>-1 and \alpha \geq 1 .
Article summary
The binomial approximation is useful for approximately calculating powers of sums of 1 and a small number x. It states that : (1 + x)^\alpha \approx 1 + \alpha x. It is valid when |x| and |\alpha x| \ll 1 where x and \alpha may be real or complex numbers. The benefit of this approximation is that \alpha is converted from an exponent to a multiplicative factor. This can greatly simplify mathematical expressions (as in the example below) and is a common tool in physics. The approximation can be proven several ways, and is closely related to the binomial theorem. By Bernoulli's inequality, the left-hand side of the approximation is greater than or equal to the right-hand side whenever x>-1 and \alpha \geq 1 .
Key claims
No claims recorded.
Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
Bibliography
No sources recorded.
Key claims and bibliography are generated from the same tracked claims and sources as the article page.