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An Interesting Identity…
For the long time readers of mine, you might recall what is called Euler’s Totient function. I recommend you take a look back at that previous article if you are not familar with some of ϕ\phi‘s properties: What I would like to show…
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Addicted to Primes (P-Adic)
Let’s begin with an interesting problem. (I guess you can judge whether or not it’s interesting!) Find three squares, each with rational side lengths, whose areas add to a bigger square such that the side length of the second square equals the area…
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Integrability of Popcorn
Note that it’s assumed that the reader is familiar with Riemann Darboux integrals. This includes the next theorem: Theorem (Integrals with Epsilon): Let be a bounded function on Then, is Riemann integrable if and only if for all there exists a partition of…
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There are more Real Numbers than Counting Numbers!
There are different sizes of infinity. I know, that is a surprising (and amazing) fact that needs to be justified (or proved)! As you may know, there are an infinite number of natural numbers and there are an infinite number of real numbers…
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Fantastic Fibonacci: Newbie at Number Theory and Let’s Get Real… Analysis Team Up
We spent a long time in our Newbie as Number Theory series, as well as some time in our current Let’s Get Real… Analysis series. Today, we will use tools from number theory and analysis to learn a little about the incredible Fibonacci…
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The Pythagorean Theorem
The Pythagorean Theorem isn’t only one of the most useful theorems in mathematics, it’s also one of the most beautiful! There are also more than 350 different proofs out there for you to enjoy! Today, I wanted to cover one of the most…
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Let’s Get Real… Analysis (Part 10): Cool Stuff with Cauchy Sequences
What does it mean for a sequence to be a Cauchy sequence? Why are Cauchy sequences important? Great questions! Hopefully by the end of this article you can answer these for yourself! The Need for More Recall back in Let’s Get Real… Analysis…
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Let’s Get Real… Analysis (Part 9): limsup’s, liminf’s, and the Bolzano-Weierstrass Theorem
We’ve asked and answered the question, “Can we ever prove that a sequence must converge without needing to know or find its limit beforehand?” in Let’s Get Real… Analysis (Part 5): The Monotone Convergence Theorem. Today, we will be proving something that has…
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Let’s Get Real… Analysis (Part 8): Superb Subsequence
Sometimes we have sequences that don’t converge, and yet, it’s possible to take elements from the divergent sequence to form a convergent sequence. We refer to such sequences as subsequences, which will be the topic of interest today. But you knew that, you…
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Let’s Get Real… Analysis (Part 7): Limit Laws for Sequences
We’re continuing in our Let’s Get Real… Analysis series mission to gather more tools for analyzing sequences. Today, we will learn what I call, our limit laws. These will likely be intuitive, but they give us an opportunity to practice writing proofs, which…
What Comes Next: 1, 2, 4, 8, 16, … hint it’s not 32
Hello,
Welcome to A Kick in the Discovery! If you’re curious where this phrase came from, that’s good! It came from the great physicist Richard Feynman. He was curious about anything and everything and he loved to explain science to others. I want to communicate the same enthusiasm about things that I find amazing! More than that, I want to practice teaching and explaining more and more complex ideas. Communication is very important and very challenging, and I hope this blog helps me learn this skill. With all this said, please bear with me while I practice my writing skills! It will be a process.
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