# Where Did the Binomial Theorem Originate?

• quantizedzeus
In summary, the binomial theorem is a result of mathematicians trying to solve problems in algebra. It comes from a proof by Euclid and was extended by Newton.
quantizedzeus
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...!...

quantizedzeus said:
Where did the binomial theorem come from...?

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.

http://ualr.edu/lasmoller/Newton.ht...r404&utm_content=click&utm_campaign=custom404

And here is a really interesting page about Al-Karaji, born in Baghdad in the year 953.

http://www-groups.dcs.st-andrews.ac.uk./~history/Biographies/Al-Karaji.html

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.$(a+b)^2=a^2+2ab+b^2$ 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.

[PLAIN]http://www.mathgym.com.au/history/pythagoras/prop8.gif

This comes down to the binomial theorem...

Last edited by a moderator:
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...

## 1. What is the Binomial Theorem?

The Binomial Theorem is a mathematical formula used to expand a binomial expression raised to a power. It allows us to find the coefficients of each term in the expansion, making it easier to solve complex equations.

## 2. Who discovered the Binomial Theorem?

The Binomial Theorem was first discovered by Chinese mathematician Yang Hui in the 13th century. However, it was later rediscovered and popularized by European mathematicians such as Isaac Newton and Blaise Pascal in the 17th century.

## 3. How is the Binomial Theorem used in real life?

The Binomial Theorem has various applications in the fields of physics, engineering, and statistics. It is used to model and solve problems involving binomial distributions, such as in genetics, finance, and biology. It is also used in the development of algorithms and computer programs.

## 4. What is the importance of the Binomial Theorem in mathematics?

The Binomial Theorem is an essential tool in algebra and calculus. It allows us to simplify and solve complex equations, making it easier to understand and manipulate mathematical concepts. It also serves as the basis for other mathematical theories and formulas.

## 5. Can the Binomial Theorem be generalized to other types of expressions?

Yes, the Binomial Theorem can be generalized to other types of expressions, such as polynomials and trigonometric functions. This is known as the Multinomial Theorem, which expands a multinomial expression raised to a power and determines the coefficients of each term in the expansion.

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