The Most Internationally Recognized Math Equation Through History

The search for the universally recognized math equation that captivates the attention of people around the globe may seem like a daunting task. However, the question itself is multifaceted, as recognition can vary between mathematicians and the general public. While the general public might be more likely to recognize Emc2, mathematicians might point towards equations such as the Fibonacci Identity.

A Historical Perspective

From ancient Greece to modern Europe, one of the most profoundly recognized mathematical identities is the Fibonacci’s Identity. This identity, first mentioned in the The Book of Squares by the 13th century mathematician Fibonacci, has its roots in a puzzle that was originally solved by the Greek mathematician Diophantus around 250 AD. Diophantus used this identity to create four distinct Pythagorean Triples with the same hypotenuse. Although he didn't state the identity in its full algebraic form due to the lack of algebraic notation of the time, his examples clearly indicate he was aware of this rule.

Mathematical Significance

The Fibonacci Identity, known formally as (a^2 b^2 c^2 d^2 a c b^2 d^2 - a d c^2 b^2), is deeply rooted in the realm of quadratic equations and complex number theory. It can be expressed as the multiplication of complex numbers:

[ (a - i b)(c i d) (a c - b d) i (a d b c) ]

When we take the squared magnitudes of these complex numbers, we obtain Fibonacci’s Identity:

[ a^2 b^2 c^2 d^2 (a c - b d)^2 (a d b c)^2 ]

Brahmagupta’s Extension

Further generalization of the Fibonacci Identity was done by the Indian mathematician Brahmagupta in his work Brāhmasphu?asiddhānta around 628 AD. His identity is known as Brahmagupta’s Identity, which extends the Fibonacci Identity to encompass quadratic surds:

[ (a b sqrt{n})(c d sqrt{n}) (a c n - b d) (a d b - c n) sqrt{n} ]

This identity leads to the following theorem and proof, demonstrating the multiplication of surds and their magnitudes:

Theorem: Q(ab√n)Q(cd√n) Q(acn - bd) (a d b - c n)√n

The proof, which is provided below, reinforces the mathematical significance of these identities.

The Mathematical Proof

Proof:

Start with the multiplication of magnitudes: Show that (Q(absqrt{n})Q(cdsqrt{n}) a^2-nb^2c^2-nd^2) Prove that (Q(absqrt{n})Q(cdsqrt{n}) Q(acnbd - adbcsqrt{n})) Finally, demonstrate that (acnbd - adbcsqrt{n} Q(acnbd) - (adbcsqrt{n}))

The Brahmagupta identity is further simplified by negating (n) and subtituting (n-n), leading to:

[a^2nb^2c^2nd^2 acnbd - adbc^2]

Negating (b) gives a different form:

[a^2nb^2c^2nd^2 acnbd - ad - b^2c^2]

Conclusion

In conclusion, while the recognition of math equations can vary, the Fibonacci Identity stands as a remarkable equation that has crossed multiple civilizations and time periods. Its significance lies in the foundation it provides for complex number theory, quadratic equations, and number theory, making it one of the most historically and internationally recognized mathematical identities.