On Incomplete Truth

After finding fallacies in constructed algebraic statements, I have conjectured upon statements that are true under limited conditions.

Incomplete (Conditional) Truth Conjecture: When a statement is a conditional truth, the solution will result in one or more fallacies.

Demonstration:
Statements
(23)^2=529 True
When a=2, b=3, (10a+b)^2=529 True
(10a+b)^2=100a^2+20ab+b^2 True
When a=2, b=3, 200a+10ab+20b+b^2=529 True
100a^2+20ab+b^2=200a+10ab+20b+b^2 Not universally true, condition: when a=2, b=3
100a^2+10ab=200a+20b Not universally true, condition: when a=2, b=3 100a^2-200a=20b-10ab True when a=2, b=3
100a(a-2)=10b(2-a)
10a(a-2)=b(2-a) True when a=2
-(a-2)=(2-a)
-10a=b
When a=2, b=3
-20=3 (Fallacy)

When b= 10a(a-2)/(2-a) is evaluated with the variable a being 2 and the variable b being 3, another fallacy is identified: 3=undefined.

History of the Pascal's Triangle before Pascal

The Pascal's triangle is composed of the coefficients of binomial expansions . The coefficients contain a plethora of properties: combinatorial computation, Fibonacci numbers, and powers of the number eleven. When was this triangle created and by whom? The concept of investigating binomial coefficients beyond the second power originated in India around the 10th century AD. The aim was to search for a new method of counting (combinatorics). However, the development of binomial coefficients in India never drove the Indians to drawing the Pascal's triangle. It was not until the upper half 11th century, when a Chinese mathematician Jia Xian (fl. 11th century) worked on binomial coefficients and constructed the Jia Xian's triangle (later known as the Yang Hui's triangle). The Persian mathematician Omar Khayyam (1048-1131) was also credited for working on binomial coefficients. One thing is for sure, the Pacal's triangle, Fibonacci series and binomial coefficients were products of Asian mathematics.

Notes
1) "The Staircase to Mount Meru" was mathematical representation of an Indian Myth, which is coincidentally the Fibonacci number series, stacked into the shape of a mountain. When the numbers are shifted to form a right angle staricase, the staircase resembles the Pascal's triangle.

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