Where did the binomial theorem come from.....???
... God? Nature? The Platonic universe of ideal forms? Logic? The human mind? Or do you mean who was the mathematician who discovered/concocted it?
I just wanted to know how was the binomial theorem discovered.....which problems or consequences led mathematicians towards its discoverery.....!!!....
Pascal gets credit for the basic version, but Isaac Newton extended it to negative and real numbers. That was pretty insightful and clever.
In looking this up I ran across a guy named al-Karaji who worked out (a+b)^5 in the year 1029. It amazing to think of someone that far away from us in time and space, sitting there by himself, multiplying polynomials before anyone else even knew what that meant.
Here's a nice little short article about Newton and the binomial theorem in general.
And here is a really interesting page about Al-Karaji, born in Baghdad in the year 953.
For whatever reason, the history of math always fascinates me. People thinking about these things so long ago, leaving their thoughts to us so that we can go farther.
I think the first statement of the binomial theorem for n=2, i.e.[itex](a+b)^2=a^2+2ab+b^2[/itex] can be found in Euclid. Indeed, in book II of the elements we find
Proposition 4. If a straight line be cut at random, the square on the whole is equal to the squares on the segments and twice the rectangle contained by the segments.
This figure can be helpful:
This comes down to the binomial theorem...
I like to think of Pascal's triangle as the sequence of sequences of "triangular" numbers in different dimensions - 0 dimensions is the line which is all "1"s, 1 dimension is the line of counting numbers, 2 dimensions is the regular triangular numbers (1,3,6,10...), the next line is the tetrahedral numbers (1,4,10,20...), after that each line is a higher dimensional sequence of tetrahedral numbers.
It is possible to set up a numerical place value system analogous to the unit - square- cube... sequence of traditional place value systems but instead using unit-triangle-tetrahedron... . Numbers have more than one representation in this system, which might have some use, though I haven't been able to think of one.
More on-topic for this forum, the binomial theorem has a deep relationship to the number of elements of a given grade ("blades" of a given grade) in a Clifford or geometric algebra- a 0-dimensional algebra has 1 grade, the scalar numbers. A 1-dimensional algebra has the scalars and 1 vector blade, representing directed intervals. A 2-D algebra has 1 scalar, 2 orthogonal vectors and 1 area element. 3-D has 1 scalar, 3 vectors, 3 areas (planes of rotation) and 1 volume element. 4-D has 1 scalar, 4 vectors, 6 areas, 4 volumes and 1 4-D volume. Higher dimension algebras get very big, e.g. an 8-D algebra has 256 blades with binomial[8,n] blades of dimension n.
Thanks everyone....i've got it..btw can anyone tell me where i can find a lot of mathematical problems for practice....specially permutation and combination problems.......
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