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Name: Ken M.
Status: student	
Age:  N/A
Location: N/A
Country: N/A
Date: 8/26/2005


Question:
In every textbook of mathematics or work on the history of mathematics, Sir Isaac Newton's expression for the binomial theorem in negative powers is merely stated, not derived. How did Newton derive the binomial theorem for negative exponents? I have not found the answer in any reference work.



Replies:
The binomial expansion works for positive, negative, fractional, and even complex exponents although the formulas get a bit long. Two on-line references will guide you through the process for the non-positive cases. The first: http://hyperphysics.phy-astr.gsu.edu/hbase/alg3.html#bef gives a short explanation so that you can convince yourself "it works". The second: http://www.escape.com/~paulg53/math/pi/newton/ gives the derivation in all its messiness. If you still need some more cases to fill in between these two extremes, I suggest you do a "Google" search on the search term: "binomial theorem negative exponents". You will find many "hits" with all levels of generality and messiness. You are correct in your comment that the derivation for other than positive exponents is left as "One can show..." or "It is left as an exercise for the student..." that mathematicians are famous for!

Vince Calder



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