numb3rth30ry:
A beautiful geometric visualization of positive-valued binomial expansions, the algebraic expressions produced by raising sums of variables (a, b) to natural-valued powers n. The algebraic structures and procedures of binomial expansion are described by the binomial theorem–which, in turn, is proved by the above figures.
Given (a+b)^n, there will be (n+1)-many terms, c(a^(n-m))(b^m), where c is a constant, and a and b are variables.
By tradition, these terms are arranged in descending order by powers of the leading term, a. Accordingly, the binomial expansion of (a+b)^n begins with the term having the highest power of a, which is a^n for all n.
The remaining n-many terms are ordered such that the exponent of each successive a term (n-m) decreases by one. Correspondingly, the exponent of the b term (m) increases by one, such that (n-m)+m=n.
The binomial coefficients c for each successive term c(a^(n-m))(b^m) are described by Pascal’s triangle. Given (a+b)^n, the nth row of Pascal’s triangle contains (n+1)-many numbers, which are the coefficients c, listed in the order described above.
Note that the exponent n is the dimension of the figures pictured above. This is no coincidence; the term (a+b)^n can be depicted geometrically by a figure whose measure (length, area, volume, hypervolume, etc.) is the quantity produced by multiplying (a+b), n-many times.
Additionally, observe that the coefficients c give the quantity of n-dimensional figures required to fill in the missing bits of area, as illustrated by the figures. For example, in order to fill the volume (a+b)^3, we use two cubes of volume a^3 and b^3, and the remainder is filled by 3 “plate-like” figures and 3 “tube-like” figures (each of whose dimensions are (a^2)b and a(b^2), respectively).
Mathematics is beautiful. <3
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