Importance of Binomial Theorem

[22990atinesh]

I know Binomial Theorem is a quick way of expanding a Binomial Expression that has been raised to some power i.e $(a+b)^n$. But why is it so important to expand $(a+b)^n$. What is the practical use of this in Science and Engineering.

 

[UltrafastPED]

Here is some background:
http://ualr.edu/lasmoller/Newton.html

Newton showed how to use fractional exponents; this leads to infinite series which converge if the exponent is between -1 and +1. In many cases the first few terms are used in approximations, especially if the expression is of the form (1+x)^n.

 

[22990atinesh]

Here is some background:
http://ualr.edu/lasmoller/Newton.html

Newton showed how to use fractional exponents; this leads to infinite series which converge if the exponent is between -1 and +1. In many cases the first few terms are used in approximations, especially if the expression is of the form (1+x)^n.

Hello UltrafastPED,
The link that you gave didn't answered my question. My question isn't whether $(a+b)^n$ is important or $(1+x)^n$. My question is Why there is so much need to solve equations like $(a+b)^n$ or $(1+x)^n$, So that we have devised Binomial Theorem to solve such equations. When in Engineering and Science we come across such equations.

 

[SteamKing]

Whenever we need to expand (a+b)^{n}, application of the binomial theorem means we don't have to multiply a bunch of binomial expressions together. Kids nowadays take for granted having a symbolic algebra program like Mathematica or Maple, but in the olden days, the B.T. could save a lot of time doing algebra (and be more accurate to boot, avoiding a lot of mistakes which might otherwise go undetected). The binomial coefficients are also the same as the entries in Pascal's triangle, and there is a simple algorithm which allows one to calculate these entries by using the first two rows of the triangle.

http://en.wikipedia.org/wiki/Pascal's_triangle

 

[22990atinesh]

Whenever we need to expand (a+b)^{n}, application of the binomial theorem means we don't have to multiply a bunch of binomial expressions together. Kids nowadays take for granted having a symbolic algebra program like Mathematica or Maple, but in the olden days, the B.T. could save a lot of time doing algebra (and be more accurate to boot, avoiding a lot of mistakes which might otherwise go undetected). The binomial coefficients are also the same as the entries in Pascal's triangle, and there is a simple algorithm which allows one to calculate these entries by using the first two rows of the triangle.

http://en.wikipedia.org/wiki/Pascal's_triangle

You didn't understand my question. Please read my above comment on "UltrafastPED"

 

[SteamKing]

We need it when we need it. It's like asking where in Science and Engineering we need the quadratic equation.

I think you have a fundamental misunderstanding about mathematics. The binomial theorem wasn't devised because people were so overwhelmed with multiplying monomials together that they needed a better way to do them. The binomial theorem was devised because someone noticed that multiplying a series of identical monomials together gave certain coefficients to the various terms in the product. It was later discovered that these coefficients bore a certain relationship with the number of combinations one could have when selecting two or more objects. Later on, other properties were discovered when the binomial theorem was extended to non negative integer exponents. There are lots of things in mathematics which have no or very little practical application, but it's nice to have them sitting on the shelf when an application or use arises.

 

[homeomorphic]

If you understand it, it becomes kind of obvious, so you can see that it's just a cute observation, rather than some humongous theory you have to spend 20 years trying to understand, and then the practical relevance is...well, sort of irrelevant.

 

[crownedbishop]

The binomial theorem can be of great help in analysis if you are trying get some estimates, and you have something in the form of (a+b)ⁿ. I know it's used several times for this reason in baby rudin. Practically, it can be a lot faster to evaluate the coefficients of each term then by just multiplying each term out, like you mentioned.