Posts tagged ‘Tom Lehrer’
A digital computer is one who adds on his fingers, and this definition reminds me of a quote by Tom Lehrer.
Base eight is just like base ten, really… if you’re missing two fingers!
Many students know the trick for multiplying by 9 on their fingers, which I mentioned in yesterday’s post. However, most folks don’t know the following finger trick for multiplying two numbers that are greater than 5 but less than or equal to 10.
- On each hand, raise a number of fingers to indicate the difference between each factor and 5. For instance, 8 – 5 = 3 and 7 – 5 = 2, so to multiply 8 × 7, raise three fingers on the left hand and two fingers on the right hand, as shown below.
- Count the total number of fingers that are raised on both hands, and append a 0. As shown above, there are 3 + 2 = 5 fingers raised, so this gives an intermediate result of 50.
- Count the number of fingers that are folded on each hand, and find their product. Again using the example above, there are two fingers folded on the left and three fingers folded on the right, and 3 × 2 = 6.
- Add the results of Steps 2 and 3. For our example, 50 + 6 = 56, so 8 × 7 = 56.
We can use algebra to understand why this trick works.
Let L = the number of fingers raised on the left hand, and let R = the number of fingers raised on the right hand. Then the numbers we are multiplying are (5 + L) and (5 + R).
The total number of fingers raised is L + R. Consequently, the result from Step 2 is equal to 10(L + R).
The number of fingers not raised on the left hand is 5 – L, and the number of fingers not raised on the left hand is 5 – R. Their product, which is required in Step 3, is (5 – L)(5 – R) = 25 – 5(L + R) + LR.
Finally, combining them in Step 4 gives
10(L + R) + 25 – 5(L + R) + LR = 25 + 5(L + R) +LR = (5 + L)(5 + R),
which is the product we initially wanted to compute.
I recently learned one other method of multiplication. It does not require fingers, and I’m not even sure it’s useful, but it is interesting.
To find the product of two numbers, a and b, do the following:
Plot the points A(0, a), B(b, 0), and C(0, 1).
Draw segment CB, and then construct segment AD parallel to CB.
As a result, point D(d, 0) is a point on the horizontal axis such that a × b = d.
For instance, to compute 6 × 3, plot A(0, 6), B(3, 0), and C(0, 1). Then draw a line through A that is parallel to CB. This line will intersect the horizontal axis at the point D(18, 0), so 6 × 3 = 18.
On cold days, I look for creative ways for my sons to burn energy indoors. If I were forced to give it a title, today’s game would be called, “Up and Down the Stairs with a Song.” Eli ran up the stairs to the second floor while Alex ran down the stairs to the basement; then they both returned to the main level where they sang a song; then each boy ran up or down the other set of stairs; and, finally, they returned to the main level and rang the “dinger,” a bell included with one of their toddler games.
During the game, the song that Eli sang was mathematical:
1, 2, 3,
4, 5, 6, …
That’s how the numbers go.
7, 8, 9,
10, 11, 12, …
That’s all you really need to know.
13, 14, 15, 16, …
On and on they go.
17, 18, 19, 20, …
Gotta line ’em up just so!
The song is from “1-2-3, Count With Me,” a Sesame Street video starring Ernie (sans Bert). My favorite song from the video is Martian Beauty, but sadly, it just doesn’t hold the same appeal for Eli and Alex.
There have been lots of math songs through the ages, and the number has risen exponentially with YouTube. Generally, math songs are humorous. (Maybe because no one would listen to a math song that wasn’t funny?) Below are my top
6. 7 8 9 – Barenaked Ladies
Thanks to Joshua Zucker for reminding me of this gem! How could I have forgotten? One of the moldiest of all oldie math jokes turned into a song. Shame on me for not including this in the original “Top 5 Math Songs” list.
5. That’s How the Numbers Go – Ernie (Sesame Street)
This is the song from the Sesame Street video “1-2-3, Count With Me.” I worried that it wasn’t sophisticated enough for readers of MJ4MF, but if Steven Strogatz can reference the video in a column for the New York Times, well, that’s credibility enough for me.
4. Lateralus – Tool
Even if you don’t like the genre, Lateralus by Tool gets big props for its intricate use of the Fibonacci sequence. The time signatures of the chorus change from 9/8 to 8/8 to 7/8. Drummer Danny Carey said the song “was originally titled 9-8-7 for the time signatures. Then it turned out that 987 was the 17th number of the Fibonacci sequence. So that was cool.” They exploited this relationship in several ways.
- The number of syllables in the verses follow the pattern 1, 1, 2, 3, 5, 8, 5, 3, 2, 1, 1, 2, 3, 5, 8, 13, 8, 5, 3, which rise and fall with the Fibonacci sequence.
- The song mentions spirals several times.
- The first word is sung at 1:37 into the song, and 97 seconds ≈ 1.618 minutes, which just happens to be the golden ratio, a number strongly associated with the Fibonacci sequence.
As it turns out, Tool has several other mathy songs, including Parabola, Forty Six and 2, and Cesaro Summability.
3. What You Know About Math? – Ethan Gilbert and Aaron Flack
Ethan Gilbert and Aaron Flack created a numerical sensation with What You Know About Math. With almost 4 million visits on YouTube, the song deserves a place on this list.
2. New Math – Tom Lehrer
This list could have easily been composed entirely of Tom Lehrer songs, since his titles include New Math, The Derivative Song, and Lobachevsky, but that seems unfair to the other artists who have contributed so much to the math music genre. So I tried to pick just one song that represents his body of work. (Personally, I think his best song is The Elements, but it’s not very mathy. And my apologies for linking to that particular video… I beg your forgiveness for including a link to a video that spells Lehrer’s name wrong and mistates the song title, but I chose it because it nicely displays the lyrics while the song plays.)
1. Finte Simple Group of Order Two – The Klein Four
The lyrics of Finite Simple Group of Order Two contain enough bad math puns to keep an undergraduate math major chuckling an entire semester. Doesn’t hurt that these guys are pretty good singers, too.
Ever notice that cartoon hands only have four fingers? There’s a very simple reason for that — it’s easier to draw a hand with four fingers than with five fingers. But have you ever wondered what it would be like to have only four fingers on each hand? For one thing, you’d count in base eight, not base ten. For another, you’d never be able to give someone a high five.
Base eight and base ten are relevant to a math joke that’s appropriate for today:
Why do mathematicians sometimes confuse Halloween and Christmas?
Because Oct 31 = Dec 25.
It’s a wonderful coincidence that October 31 and December 25 both happen to be days with significance, which is why that joke works. It’s no coincidence, however, that the abbreviations for October (Oct) and December (Dec) could also be the abbreviations for the octal (base 8) and decimal (base 10) number systems. In the old Roman calendar, October was the 8th month, and December was the 10th month. When the Julian calendar was created, July and August were added in the middle of the year, pushing October and December to the 10th and 12th slots, respectively.