Exploring the Depth of the Binomial Theorem: A Scientist's Perspective

[a1b2c3zzz]

Hello all! This isn't a problem in particular I'm having trouble with, but a much more general question about the binomial theorem.

I'm using Stewart's precal book. The section devoted to the theorem has several problems dealing with proving different aspects of it, mostly having to do with the coefficients. I've been busting my head on these problems today, but the question is, should I be? I always try to intuitively grasp the topics we cover, but as far as I can tell, the binomial theorem is not incredibly deep. It seems like nothing much more than a convenient way to raise numbers to different powers, at least at this level. Am I wasting energy?

Sorry if my question is a little vague!

 

[SteamKing]

The binomial theorem as applied to integer powers and integer coefficients is pretty simple. However, Isaac Newton deduced a way to extend the principle of the binomial theorem with integers to a more general result which allowed non-integer powers and real coefficients, a much more useful theorem:

http://en.wikipedia.org/wiki/Binomial_theorem

The theorem has several different proofs, and it has uses beyond just raising an arbitrary monomial to a given power.

Yes, it is worth studying the basic binomial theorem now, because you will probably encounter it again in more general form.

 

[micromass]

Entire mathematics is just a more convenient thing of doing something complicated!

The binomial theorem gives an expansion for ##(x+y)^n## where ##n## is an integer. Later in calculus, you will extent the theorem for when ##n## is not an integer anymore. That version was the one which Newton found.

The binomial theorem is extremely important and has quite many important consequences, particularly when you also involve calculus.

Whether you choose to spend time on proving parts of it, that's your choice. If you are into mathematics or theoretical physics as a goal, then you should definitely spend time on it. If you're more into practical stuff like engineering or experimental physics, then it's less important.

 

[verty]

Hello all! This isn't a problem in particular I'm having trouble with, but a much more general question about the binomial theorem.

I'm using Stewart's precal book. The section devoted to the theorem has several problems dealing with proving different aspects of it, mostly having to do with the coefficients. I've been busting my head on these problems today, but the question is, should I be? I always try to intuitively grasp the topics we cover, but as far as I can tell, the binomial theorem is not incredibly deep. It seems like nothing much more than a convenient way to raise numbers to different powers, at least at this level. Am I wasting energy?

Sorry if my question is a little vague!

This is how you want to think about it. Math is a language (or a collection of languages) that one uses to describe (or name) mathematical objects. What is the binomial theorem? It is a theorem that names or describes the result of expanding (a + b)^n. It isn't deep because this is a simple thing to do, but it's quicker to use the theorem than to do the expansion manually.

If you are convinced that you understand that operation and you trust the theorem, move on.

 

[a1b2c3zzz]

Great, thanks for the insight guys!