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H. C. Chan, in terms of : Some Recent Developments, Proceedings of the Twelfth International Conference in Fibonacci Numbers, (2010): 17-25. Read Pi in terms of Phi (Fib Conf 2006).
H. C. Chan, in terms of , Fibonacci Quart. 44 (2006): 141–144. Read Pi in terms of phi.
H. C. Chan, More Formulas for , Amer. Math. Monthly 113: 452-455. Read More formulas for Pi.
H. C. Chan, Machin-type formulas expressing in terms of , Fibonacci Quart. 46/47 (2008/2009): 32–37 Read Pi via Machin.
I wrote this note in 2008 to introduce complex numbers. It is posted for the Math Teachers at Play blog carnival.
What is the ?
In the set of Real Numbers, , the solution is undefined. We must consider the set of Complex Numbers, . Complex numbers are numbers of the form of , where and are real numbers: and is the imaginary unit. Note is defined by or equivalently .
Thus, we have:
We now have . So we must ask "What is the ?". To evaluate we will use Euler’s formula, So by substituting , giving
Note we have isolated and now arrived at the following equality: . Taking the square root of both sides gives
Thus, we have now have and following through application of Euler’s formula to , gives
So going back to the original problem and substituting , gives
Hence, the solution, , is a complex number!
In college, we often learn of many infinite series that give the value of pi including one called the Leibniz series, named after Gottfried Leibniz. It is also called the Gregory–Leibniz series, recognizing the work of James Gregory. This unnecessarily attributes the discovery to the west, however, the formula was first discovered in India by Madhava of Sangamagrama and so is also called the Madhava–Leibniz series. Indian mathematicians made vast and fundamental contributions to our modern mathematics.
“Why aren’t we giving our students a chance to even hear about these things, let alone giving them an opportunity to actually do some mathematics, and to come up with their own ideas, opinions, and reactions? What other subject is routinely taught without any mention of its history, philosophy, thematic development, aesthetic criteria, and current status? What other subject shuns its primary sources— beautiful works of art by some of the most creative minds in history— in favor of third-rate textbook bastardizations?”
“In 2009, mathematician Timothy Gowers posed this question to the blogosphere: “Is massively collaborative mathematics possible?” He described an unsolved math problem and asked for help figuring it out. Over the next few hours and days, commenters began to pick at the problem together. They brought up incomplete ideas, which were expanded and incorporated into other peoples’ ideas, until Gowers posted 37 days later that the problem had (probably) been solved.”