Useless Numerology for 2016: Part 3

The following entertaining (but useless) facts about the number 2,016 appeared in a recent Facebook post (and subsequent comments) by the American Mathematical Monthly.

$2016 = 1+2+3 + \dots + 62 + 63$

$2016 = 2^{11} - 2^5$

In this post, we’ll explore why these two expressions have to be equal.

The sum $1 + 2 + 3 + \dots + 62 + 63$ is an arithmetic series. The first term is $1$ , the last term is $63$ , and there are $63$ terms in the series. Using the formula for an arithmetic series, we find

$1 + 2 + 3 + \dots + 62 + 63 = \displaystyle \frac{(63)(1 + 63)}{2}$

$= \displaystyle \frac{63 \times 64}{2}$

$= 63 \times 32$

$= (64-1) \times 32$

$= (2^6 - 1) \times 2^5$

$= 2^{11} - 2^5$ .

Useless Numerology for 2016: Part 2

The following entertaining (but useless) facts about the number 2,016 appeared in a recent Facebook post (and subsequent comments) by the American Mathematical Monthly.

$2016 = 2^{10} + 2^9 + 2^8 + 2^7 + 2^6 + 2^5$

$2016 = 2^{11} - 2^5$

Not surprisingly, there’s a natural reason why these two expressions are equal. (However, there isn’t a natural reason why the answer happens to match the current year other than coincidence.)

To begin, $2^5 + 2^6 + 2^7 + 2^8 + 2^9 + 2^{10}$ is a finite geometric series. The first term is $2^5 = 32$ , the common ratio is $2$ , and there are $6$ terms in the series. Using the formula for a finite geometric series,

$2^5 + 2^6 + 2^7 + 2^8 + 2^9 + 2^{10} = \displaystyle \frac{2^5 (1-2^6)}{1-2} = \frac{2^5-2^{11}}{-1} = 2^{11} - 2^5$ ,

thus establishing that these two expressions are equal.

A Mathematician’s New Year’s Resolutions

Math with Bad Drawings

Mathematician New Years 0 Mathematician New Years 1

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Predicate Logic and Popular Culture (Part 20): Mariah Carey

Let $W(x)$ be the proposition “I want $x$ for Christmas.” Translate the logical statement

$W(\hbox{you}) \land \forall x (x \ne \hbox{you} \Longrightarrow \lnot W(x))$ ,

where the domain is all things.

The clunky way of translating this into English is,”I want you for Christmas, and if something isn’t you, then I don’t want that for Christmas.”

Context: This semester, I taught discrete mathematics for the first time. Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.

Voiceless Video Syllabus

I’m a couple months late with this… after all, school started in August… but nevertheless I recently stumbled on this voiceless video syllabus by Joshua Katz, a mathematics teacher in Florida.

I really enjoyed this.

Well done, sir. Your students are very lucky to have you as a teacher.

How to Call Out Mistakes without Trampling the Mistaken

Math With Bad Drawings had a nice post about strategies teachers can use to address a student who volunteers an incorrect answer. This is a natural dilemma for teachers who want to encourage class participation but who don’t want to deflate students who make mistakes. I recommend this essay highly: http://mathwithbaddrawings.com/2014/11/05/wrong-but-not-stupid-or-how-to-call-out-mistakes-without-trampling-the-mistaken/

One recent strategy that I’ve employed is quoting the trolls from Frozen: https://meangreenmath.com/2014/06/14/a-bit-of-a-fixer-upper/

50,000 page views

I’m taking a one-day break from my usual posts on mathematics and mathematics education to note a symbolic milestone: meangreenmath.com has had more than 50,000 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 resource to you.

Some other (probably useless) statistics: this blog has been viewed by readers from 167 different countries. Top viewership: the United States, India, the Philippines, Canada, the United Kingdom, Australia, Brazil, the European Union, United Arab Emirates, Germany, Taiwan, and Pakistan.

Twelve most viewed posts or series (written by me):

Twelve most viewed posts (guest presenters):

Top twelve search engine terms that landed people on my blog:

systems of equations project / system of equations project ideas / system of linear equations project / system of equations projects / solving systems of equations projects / real life system of linear equations / solving systems of equations project / linear systems project / etc.
log table / how to find square root using log book / how to find square root in log book / how to find square root using log / logarithms with square roots / log tables / how to find square root of a number using log table / how to use log table for square roots / etc.
geometry tricks
law of exponents foldable / exponent rules foldable / foldable exponent rules
mean green math / green math / meangreenmath (hey, it works!)
cavalieri’s principle / cavalieri’s principle proof
tables / square root table / mathematical table / math tables
law of exponents fun activity / law of exponents activity
5e lesson plan math
congruent segments in real life
student art work with circles and parabolas
finger trick for multiplying by 9

Twelve other search engine terms that caught my attention:

deercrossing sign phone call
engaging students in reading lesson
examples of inductive and deductive reasoning in the declaration of independence
grape and triangle pun
math and 2048
common core and math number line
meetkunde trucjes
dominoes falling over
slide rule advertisement
the stereotypes about math that hold americans back
youtube/danceofchanceprobabilityfromanormalcurve
honey i shrunk the kids dilation project

If I’m still here at that time, I’ll make a summary post like this again when this blog has over 100,000 page views.

Your Humble Servant, Isaac Newton

From the YouTube description:

Almost fifty years ago, Cambridge University Press published the correspondence of Isaac Newton, a seven-volume, 3000-page collection of letters that provides insight into this great, if difficult, genius. William Dunham shares his favorite examples of Newton as correspondent. He ends with Newton’s most-quoted line about standing on the shoulders of giants and how his search for its place of origin led him, improbably, to a library in Philadelphia.

Circumference

Source: http://www.xkcd.com/1184/

Further comments, from Nicholas Vanserg, “Mathmanship,” The American Scientist, Vol. 46, No. 3 (1958):

In an article published a few years ago, the writer intimated with befitting subtlety that since most concepts of science are relatively simple (once you understand them), any ambitious scientist must, in self-preservation, prevent his colleagues from discovering that his ideas are simple too…

The object of… Mathmanship is to place unsuspected obstacles in the way of the pursuer until he is obliged, by a series of delays and frustrations, to give up the chase and concede his mental inferiority to the author…

[U]se a superscript as a key to a real footnote. The knowledge seeker reads that $S$ is $-36.7^{14}$ calories and thinks, “Gee what a whale of a lot of calories,” until he reads to the bottom of the page, finds footnote 14 and says, “oh.”

Collaborative Mathematics: Challenge 11

I’m a couple months late with this, but my colleague Jason Ermer at Collaborative Mathematics has published Challenge 11 on his website: http://www.collaborativemathematics.org/