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High School Students Finding New Proofs of Old Theorems (Part 1): Dividing a line segment with straightedge and compass – Mean Green Math
This is one of my all-time favorite stories to share with students: how a couple of ninth graders in 1995 played with Geometer's Sketchpad and stumbled upon a brand-new way of using only a straightedge and compass to divide a line segment into any number of equal-sized parts. This article was published in 1997 and…
Happy Pythagoras Day! – Mean Green Math
Let me take a one-day break from my current series of posts to wish everyone a Happy Pythagoras Day! Today is 7/24/25 (or 24/7/25 in other parts of the world), and $latex 7^2 = 24^2 + 25^2$. Bonus points if you can figure out (without Googling) when the next Pythagoras Days will be. Hint: It's…
Higher derivatives in ordinary speech – Mean Green Math
Just about every calculus student is taught that the first derivative is useful for finding the slope of a curve and finding velocity from position, and that the second derivative is useful for finding the concavity of a curve and finding acceleration from position. I recently came across a couple of quotes that, taken literally,…
Saturday Night Live: Washington’s Dream – Mean Green Math
Happy Fourth of July. I'm a sucker for G-rated ways of using humor to engage students with concepts in the mathematical curriculum. I never thought that Saturday Night Live would provide a wonderful source of material for this effort. https://www.youtube.com/watch?v=JYqfVE-fykk
Horrible False Analogy – Mean Green Math
I had forgotten the precise assumptions on uniform convergence that guarantees that an infinite series can be differentiated term by term, so that one can safely conclude $latex \displaystyle \frac{d}{dx} \sum_{n=1}^\infty f_n(x) = \sum_{n=1}^\infty f_n'(x)$. This was part of my studies in real analysis as a student, so I remembered there was a theorem but…
Predicate Logic and Popular Culture (Part 277): Kellie Pickler – Mean Green Math
Let $latex T$ be the set of all times, and let $latex G(t)$ measure how good day $latex t$ is. Translate the logical statement $latex \exists t_1 < 0 \exists t_2 < 0 \forall t \in T ( (t \ne t_1 \land t \ne t_2) \Longrightarrow (G(t) < G(t_1) \land G(t) < G(t_2)),$ where time $latex 0$ is today.…
Predicate Logic and Popular Culture (Part 276): Heart – Mean Green Math
Let $latex T$ be the set of all times, and let $latex G(t)$ be the statement “I got by on my own at time $latex t$." Translate the logical statement $latex \forall t \in T ( ((t<0) \longrightarrow G(t) ) \land (t \ge 0) \longrightarrow \sim G(t)),$ where time $latex 0$ is today. This matches the opening line of…
Solving Problems Submitted to MAA Journals (Part 7i) – Mean Green Math
The following problem appeared in Volume 131, Issue 9 (2024) of The American Mathematical Monthly. Let $latex X$ and $latex Y$ be independent normally distributed random variables, each with its own mean and variance. Show that the variance of $latex X$ conditioned on the event $latex X>Y$ is smaller than the variance of $latex X$…
Solving Problems Submitted to MAA Journals (Part 7h) – Mean Green Math
The following problem appeared in Volume 131, Issue 9 (2024) of The American Mathematical Monthly. Let $latex X$ and $latex Y$ be independent normally distributed random variables, each with its own mean and variance. Show that the variance of $latex X$ conditioned on the event $latex X>Y$ is smaller than the variance of $latex X$…
Solving Problems Submitted to MAA Journals (Part 7g) – Mean Green Math
The following problem appeared in Volume 131, Issue 9 (2024) of The American Mathematical Monthly. Let $latex X$ and $latex Y$ be independent normally distributed random variables, each with its own mean and variance. Show that the variance of $latex X$ conditioned on the event $latex X>Y$ is smaller than the variance of $latex X$…
Solving Problems Submitted to MAA Journals (Part 7f) – Mean Green Math
The following problem appeared in Volume 131, Issue 9 (2024) of The American Mathematical Monthly. Let $latex X$ and $latex Y$ be independent normally distributed random variables, each with its own mean and variance. Show that the variance of $latex X$ conditioned on the event $latex X>Y$ is smaller than the variance of $latex X$…
Solving Problems Submitted to MAA Journals (Part 7e) – Mean Green Math
The following problem appeared in Volume 131, Issue 9 (2024) of The American Mathematical Monthly. Let $latex X$ and $latex Y$ be independent normally distributed random variables, each with its own mean and variance. Show that the variance of $latex X$ conditioned on the event $latex X>Y$ is smaller than the variance of $latex X$…
Solving Problems Submitted to MAA Journals (Part 7d) – Mean Green Math
The following problem appeared in Volume 131, Issue 9 (2024) of The American Mathematical Monthly. Let $latex X$ and $latex Y$ be independent normally distributed random variables, each with its own mean and variance. Show that the variance of $latex X$ conditioned on the event $latex X>Y$ is smaller than the variance of $latex X$…
Solving Problems Submitted to MAA Journals (Part 7c) – Mean Green Math
The following problem appeared in Volume 131, Issue 9 (2024) of The American Mathematical Monthly. Let $latex X$ and $latex Y$ be independent normally distributed random variables, each with its own mean and variance. Show that the variance of $latex X$ conditioned on the event $latex X>Y$ is smaller than the variance of $latex X$…
Solving Problems Submitted to MAA Journals (Part 7b) – Mean Green Math
The following problem appeared in Volume 131, Issue 9 (2024) of The American Mathematical Monthly. Let $latex X$ and $latex Y$ be independent normally distributed random variables, each with its own mean and variance. Show that the variance of $latex X$ conditioned on the event $latex X>Y$ is smaller than the variance of $latex X$…
Solving Problems Submitted to MAA Journals (Part 7a) – Mean Green Math
The following problem appeared in Volume 131, Issue 9 (2024) of The American Mathematical Monthly. Let $latex X$ and $latex Y$ be independent normally distributed random variables, each with its own mean and variance. Show that the variance of $latex X$ conditioned on the event $latex X>Y$ is smaller than the variance of $latex X$…
Solving Problems Submitted to MAA Journals (Part 6e) – Mean Green Math
The following problem appeared in Volume 97, Issue 3 (2024) of Mathematics Magazine. Two points $latex P$ and $latex Q$ are chosen at random (uniformly) from the interior of a unit circle. What is the probability that the circle whose diameter is segment $latex overline{PQ}$ lies entirely in the interior of the unit circle? Let…
Solving Problems Submitted to MAA Journals (Part 6d) – Mean Green Math
The following problem appeared in Volume 97, Issue 3 (2024) of Mathematics Magazine. Two points $latex P$ and $latex Q$ are chosen at random (uniformly) from the interior of a unit circle. What is the probability that the circle whose diameter is segment $latex overline{PQ}$ lies entirely in the interior of the unit circle? As…
Solving Problems Submitted to MAA Journals (Part 6c) – Mean Green Math
The following problem appeared in Volume 97, Issue 3 (2024) of Mathematics Magazine. Two points $latex P$ and $latex Q$ are chosen at random (uniformly) from the interior of a unit circle. What is the probability that the circle whose diameter is segment $latex \overline{PQ}$ lies entirely in the interior of the unit circle? As…
Solving Problems Submitted to MAA Journals (Part 6b) – Mean Green Math
The following problem appeared in Volume 97, Issue 3 (2024) of Mathematics Magazine. Two points $latex P$ and $latex Q$ are chosen at random (uniformly) from the interior of a unit circle. What is the probability that the circle whose diameter is segment $latex \overline{PQ}$ lies entirely in the interior of the unit circle? As…
Solving Problems Submitted to MAA Journals (Part 6a) – Mean Green Math
The following problem appeared in Volume 97, Issue 3 (2024) of Mathematics Magazine. Two points $latex P$ and $latex Q$ are chosen at random (uniformly) from the interior of a unit circle. What is the probability that the circle whose diameter is segment $latex \overline{PQ}$ lies entirely in the interior of the unit circle? It…
Solving Problems Submitted to MAA Journals (Part 5d) – Mean Green Math
The following problem appeared in Volume 96, Issue 3 (2023) of Mathematics Magazine. Evaluate the following sums in closed form: $latex f(x) = \displaystyle \sum_{n=0}^\infty \left( \cos x - 1 + \frac{x^2}{2!} - \frac{x^4}{4!} \dots + (-1)^{n-1} \frac{x^{2n}}{(2n)!} \right)$ and $latex g(x) = \displaystyle \sum_{n=0}^\infty \left( \sin x - x + \frac{x^3}{3!} - \frac{x^5}{5!} \dots…
Solving Problems Submitted to MAA Journals (Part 5e) – Mean Green Math
The following problem appeared in Volume 96, Issue 3 (2023) of Mathematics Magazine. Evaluate the following sums in closed form: $latex f(x) = \displaystyle \sum_{n=0}^\infty \left( \cos x - 1 + \frac{x^2}{2!} - \frac{x^4}{4!} \dots + (-1)^{n-1} \frac{x^{2n}}{(2n)!} \right)$ and $latex g(x) = \displaystyle \sum_{n=0}^\infty \left( \sin x - x + \frac{x^3}{3!} - \frac{x^5}{5!} \dots…
Solving Problems Submitted to MAA Journals (Part 5c) – Mean Green Math
The following problem appeared in Volume 96, Issue 3 (2023) of Mathematics Magazine. Evaluate the following sums in closed form: $latex f(x) = \displaystyle \sum_{n=0}^\infty \left( \cos x - 1 + \frac{x^2}{2!} - \frac{x^4}{4!} \dots + (-1)^{n-1} \frac{x^{2n}}{(2n)!} \right)$ and $latex g(x) = \displaystyle \sum_{n=0}^\infty \left( \sin x - x + \frac{x^3}{3!} - \frac{x^5}{5!} \dots…
Solving Problems Submitted to MAA Journals (Part 5b) – Mean Green Math
The following problem appeared in Volume 96, Issue 3 (2023) of Mathematics Magazine. Evaluate the following sums in closed form: $latex f(x) = \displaystyle \sum_{n=0}^\infty \left( \cos x - 1 + \frac{x^2}{2!} - \frac{x^4}{4!} \dots + (-1)^{n-1} \frac{x^{2n}}{(2n)!} \right)$ and $latex g(x) = \displaystyle \sum_{n=0}^\infty \left( \sin x - x + \frac{x^3}{3!} - \frac{x^5}{5!} \dots…
Solving Problems Submitted to MAA Journals (Part 5a) – Mean Green Math
The following problem appeared in Volume 96, Issue 3 (2023) of Mathematics Magazine. Evaluate the following sums in closed form: $latex \displaystyle \sum_{n=0}^\infty \left( \cos x - 1 + \frac{x^2}{2!} - \frac{x^4}{4!} \dots + (-1)^{n-1} \frac{x^{2n}}{(2n)!} \right)$ and $latex \displaystyle \sum_{n=0}^\infty \left( \sin x - x + \frac{x^3}{3!} - \frac{x^5}{5!} \dots + (-1)^{n-1} \frac{x^{2n+1}}{(2n+1)!} \right)$.…
Solving Problems Submitted to MAA Journals (Part 4) – Mean Green Math
The following problem appeared in Volume 96, Issue 3 (2023) of Mathematics Magazine. Let $latex A_1, \dots, A_n$ be arbitrary events in a probability field. Denote by $latex B_k$ the event that at least $latex k$ of $latex A_1, \dots A_n$ occur. Prove that $latex \displaystyle \sum_{k=1}^n P(B_k) = \sum_{k=1}^n P(A_k)$. I'll admit when I…
Solving Problems Submitted to MAA Journals (Part 3) – Mean Green Math
The following problem appeared in Volume 53, Issue 4 (2022) of The College Mathematics Journal. Define, for every non-negative integer $latex n$, the $latex n$th Catalan number by $latex C_n := \displaystyle \frac{1}{n+1} {2n \choose n}$. Consider the sequence of complex polynomials in $latex z$ defined by $latex z_k := z_{k-1}^2 + z$ for every…
Solving Problems Submitted to MAA Journals (Part 2b) – Mean Green Math
The following problem appeared in Volume 53, Issue 4 (2022) of The College Mathematics Journal. This was the second-half of a two-part problem. Suppose that $latex X$ and $latex Y$ are independent, uniform random variables over $latex [0,1]$. Define $latex U_X$, $latex V_X$, $latex B_X$, and $latex W_X$ as follows: $latex U_X$ is uniform over…
Solving Problems Submitted to MAA Journals (Part 2a) – Mean Green Math
The following problem appeared in Volume 53, Issue 4 (2022) of The College Mathematics Journal. This was the first problem that was I able to solve in over 30 years of subscribing to MAA journals. Suppose that $latex X$ and $latex Y$ are independent, uniform random variables over $latex [0,1]$. Now define the random variable…
Solving Problems Submitted to MAA Journals (Part 1) – Mean Green Math
I first became a member of the Mathematical Association of America in 1988. My mentor in high school gave me a one-year membership as a high school graduation present, and I've maintained my membership ever since. Most years, I've been a subscriber to three journals: The American Mathematical Monthly, Mathematics Magazine, and College Mathematics Journal.…
500,000 page views – Mean Green Math
I'm taking a break from my usual posts on mathematics and mathematics education to note a symbolic milestone: meangreenmath.com has had more than 500,000 total page views since its inception in June 2013. Many thanks to the followers of this blog, and I hope that you'll continue to find this blog to be a useful…
My Favorite One-Liners: Index – Mean Green Math
I'm doing something that I should have done a long time ago: collecting a series of posts into one single post. The links below show my series on my favorite one-liners. Mathematical Wisecracks for Almost Any Occasion: Part 2, Part 7, Part 8, Part 12, Part 21, Part 28, Part 29, Part 41, Part 46,…
Confirming Einstein’s General Theory of Relativity with Calculus: Index – Mean Green Math
I'm doing something that I should have done a long time ago: collecting a series of posts into one single post. The links below show my series on general relativity and the precession of Mercury's orbit. Part 1: Introduction Part 1a: Introduction Part 1b: Observed precession of Mercury Part 1c: Outline of argument Part 2:…
My Mathematical Magic Show: Index – Mean Green Math
I'm doing something that I should have done a long time ago: collecting a series of posts into one single post. The links below show the mathematical magic show that I'll perform from time to time. Part 1: Introduction. Part 2a, Part 2b, and Part 2c: The 1089 trick. Part 3a, Part 3b, and Part…
Lagrange Points and Polynomial Equations: Part 5 – Mean Green Math
This series was motivated by a terrific article that I read in the American Mathematical Monthly about Lagrange points, which are (from Wikipedia) "points of equilibrium for small-mass objects under the gravitational influence of two massive orbiting bodies." There are five such points in the Sun-Earth system, called $latex L_1$, $latex L_2$, $latex L_3$, $latex L_4$, and…
Lagrange Points and Polynomial Equations: Part 4 – Mean Green Math
From Wikipedia, Lagrange points are points of equilibrium for small-mass objects under the gravitational influence of two massive orbiting bodies. There are five such points in the Sun-Earth system, called $latex L_1$, $latex L_2$, $latex L_3$, $latex L_4$, and $latex L_5$. The stable equilibrium points $latex L_4$ and $latex L_5$ are easiest to explain: they are the…