Saturday 29 March 2008

The History of Casting Out Nines

In my youth, back in the dark ages before calculators, one of the common mathematical tools we were taught was called casting out nines. It seems it is not common anymore, perhaps the errors students make with calculators no longer submit to the simple check of this ancient method. If you happen to be of the generation who have been kissed on the forehead by the Gods of Electronics and actually don't know the method, you can find some notes on my page, and also here is a brief video from YouTube.

I have a note on my MathWords page on the subject from a respected math historian (Albrecht Heefer) that tells me, "Casting out nines is believed to be of Indian origin, but it does not occur before 950. Maximus Planudes called it 'Arithmetic after the Indian method". Along the way I seem to have a note from him telling me that I can find more confirmation on the web site of David Singmaster, the famous historian of mathematical recreation; but while searching there, I seem to have a note that claims the first mention of casting out nines was by the Latin writer Iamblichus in 325 BC... But he was talking about Nichomachus a Pythagorean who lived around 100 AD.

Now the common thought, or at least as I thought I understood it, was that the inventors of the hindu-arabic numerals had developed casting out nines and it sort of made its way into the west with the introduction of the Arabic numbers. Leonardo of Pisa, the famous Fibonacci whose bunny sequence you remember from school (of course you do, 1, 1, 2, 3, 5, 8, 13, 21...... That sequence) was a major influence in bringing both to the west with his famous book, the Liber Abaci, (the book of calculating) around 1202.

But the fact is that the general public held on to their Roman numerals for several centuries, and legal documents had to have them in some areas up into the 15th century.. Now the problem, at least for me, is that it seems much less likely that someone would develop casting out nines using Roman numerals.. see if you are using Arabic numerals, you take a number and add up the digits... 2534 would give 2+5+3+4 = 14 and then adding 1+4 = 5 so we know that if you divide 2534 by nine, you get a remainder of 5. Now in Roman numerals we write 2534 as MMDXXXIIII ... It just looks less likely to jump out.. Ok, maybe if they never used the D for 500 and V for five, I can see it becoming obvious, so maybe that is how it came about. If you write the Roman numbers with only unit (that's how math types say ONE) multipliers, like M for 1000 or X for 10 or C for 100, then all you would have to do is count the number of digits (not add them up). For example MMCCXII has seven digits, so the number 2212 should have a digit root of 7, which it does. And for really long numbers, you could throw away groups of nine in the same way we do with casting out nines..... MAYBE... but I wonder..

Anyway, I'm still looking for that Rogue Scholar out there who happens to have the original of Nicomachus' "Introduction to Arithmetic" laying around on his bookshelf and would like to translate for me to explain where he says it came from (if indeed he did).

Friday 28 March 2008

Geometry in the News

It's spring time in Possum Trot, and my morning walks are joined by daffodils parading alone the roadside. Reminds me of the beautiful geometry of them that I wrote of some years ago in England.

I’ve been thinking about geometry a lot lately. Partly that is due to the fact that I’m going through Trig and Vectors. Partly it is probably because it has popped up in science stories I have been reading lately. On the same day I wrote about the Daffodils in the Snow, I read a note from a researcher on why they respond differently to wind than other similar flowers. The short answer is geometry.

When you watch Tulips, for instance, they will lean away from the wind. Daffodils, on the other hand, remain almost completely erect, but turn their tilted heads away from the wind. William Wordsworth must have had this in mind when he wrote,

“ Ten thousand saw I at a glance
Tossing their heads in sprightly dance.”

The reason the daffodil twists like a weather vane and the tulip bends more in the wind is the geometry of the cross-section of the stem. A tulip stem is nearly round, and so it can bend, but not twist. The same effect causes your garden hose to crimp up and cut off the flow of water when it can’t twist. They are very good at bending, very poor at twisting. The engineering types call the twisting motion torsion, and it is related to the cumulative sum of the fourth power of the distances from the center to the edge of the stem. Circular things are far away in all directions. But a Daffodil has a cross section that is more elliptical. If you pick it up you can see the difference in the long and short axis easily with the naked eye. Since the sum of the fourth powers of the distances is lower, it is more able to turn away from the wind. Scientists studied one type of daffodil and reported that up to about 22 mph, the stems stayed essentially erect, and the trumpets all turned away from the wind. After that, the flowers both turned and bent some, but they cannot bend as low as a tulip in any wind… all geometry.

The geometry of sharks popped up too. It seems that shark geometry may be to thank for the engineering advances that will result in a few more swimming records falling at the Olympics in China this year, although I assume the swimmers will want some of the credit. It seems a shark has dimples on its scales that breaks up the flow of water, reducing the drag so it uses less energy. Speedo reckons it can make better swimsuits making its swimmers go faster using something similar. The new body suits are already in the pools and having an impact.

Another bit of research shows how, with such a huge blue ocean to wander around, how exactly do marine predators like sharks find their next meal? Yeah…. You guessed it, geometry. It has been known that many land animals search for food much like a shopper in the super market searches for a particular item. The math term is called a Levy walk (actually a Levy flight), a fractal type structure from geometry where the small parts are self-similar. Ok, the actual rules are a little technical but in essence it means that the animal undergoes lots of short-distance journeys interspersed with fewer longer-distance journeys. Just as you go to an area of the store where you think the item is located, then circle around in that area looking for it. If you don’t find it, you go off to another area and begin a close search there.

Geometry, making your life better… See, because if two sides and the included angle of one triangle….

Thursday 27 March 2008

I Told You So. People Don't Believe Statistics

A while back I wrote to suggest that, contrary to the popular quip, "You Can't Prove Anything with Statistics." Now there is an official UK government survey to demonstrate that, in the main, I was right. The Office for National Statistics in England took a poll of more than 1,000 adults and found "the majority thought figures were manipulated for political purposes."

NO! Gasp! Could that be true?

The poll shows that only 36% of people questioned believed official statistics were accurate. Worse, 84% said they thought the Government twisted the figures when talking politics.

The 36% figure shows a drop from 37% a couple of years ago (2005).

The press release also reportedly found television was the most powerful influence on opinion, with the internet growing rapidly.

At least there is some good news for those of us who love statistics. You can play this hot rap-stats hit from the folks at Johns Hopkins Bio-stats... As my grandson Xander so often exclaims...."ROCK---EN"

Sunday 23 March 2008

Snow and Daffodils

I’m approaching the end of my seventh year living in the East Anglia region of the UK, and we just had our second real snow of that period, Easter morning, 2008. I’m not a big fan of snow. Ok, I don’t mind snow; I just don’t like the cold it seems to be regularly associated with. That may seem strange since my permanent home is in the northern part of the lower peninsula of Michigan, but I think that is one of the things that keeps me working late in life. If I retire I have to go back to Michigan and shovel snow… and they have had lots more days of lots more snow than here in East Anglia.
I love the Daffodils pushing up through the snow. There is probably a life lesson in there, but the trouble with life lessons is figuring out what they mean. Next week the snow will most likely be distant memory, and the daffodils will still be standing tall. The daffodils are more constant, and so I notice the snow. The temporary discomfort draws my attention, but it makes me appreciate the daffodils. Would I have noticed them this morning if the snow had not fallen? Maybe life’s little tribulations are just there to make us see the blessings that are more constant. They are building an elevator in my school next to my room, and for some reason it could NOT be done in the summer. Some days the ear-splitting drilling and hammering noise is almost continuous, and on occasion I have had to retreat to the cafeteria for a make-shift class. Through all this, I am inspired by the way my young charges continue to stay on task. We laugh at the crazy spectacle of trying to learn calculus and analysis in the cacophony, but through it all, my little daffodils stick their heads above the snow and persevere.

Thursday 20 March 2008

The Pain is All in Your Head (no brain=no pain?)

I’ve been reading several interesting articles lately about the brain, and the strange way it seems to work (and NOT work at times). Some of it ties in with the incredible effect of placebos that I have mentioned in recent posts.

Jack Tsao, A US Navy officer/neurologist at Bethesda Maryland has found some interesting treatments for the phantom pain that often tortures folks with amputated limbs. He recruited twenty-two amputees with one missing leg and randomly assigned three treatments. One group were instructed to sit in front of a mirror and move the non-amputated leg as they watched the movement in the mirror and imagined moving the missing leg. A second group did the same, but with the mirror covered, and a third group was told just to imagine moving the missing leg. And the results? After four weeks of the therapy the second group had over half the patients report increased pain over the four weeks of treatment. For the third group, who only imagined moving the missing limb, two-thirds reported increased pain. For the mirror group, 100% (that’s not a typo).. All of them reported a reduction in pain.

A suggested explanation is that after the amputation the mind notices the lack of sensation from the missing leg, and sort of turns up the sensory volume in the nerve chain for that region of the body, producing increased pain sensation. The therapy seems to fool the mind in regions of so-called “mirror neurons” in the brain which respond when we move or watch someone else move.

As a teacher I get to watch kids behavior during tests, and one of the things you notice is the “looking for the answer on the ceiling” gazes that have nothing to do with cheating. Students will gaze to the left or right or at the ceiling as they ponder the solution to a math question. In the 1970’s there was lots of research about this tendency to gaze off to one side or the other when we are trying to think of the answer to a question. For a while it seemed that there might be an association between the direction of the gaze, left or right, and the type of question but that explanation seems to be too simplistic. There is still definitely an effect, and it appears that if you are prevented from this “lateral eye movement” which actually has an acronym, LEM, that you are less able to answer the question. (See Glenberg, Arthur M., Jennifer L. Schroeder, and David A. Robertson (1998). Averting the gaze disengages the environment and facilitates remembering. Memory & Cognition 26/4: 651-8.)

You can sort of test the idea that harder questions are more likely to induce the LEM by asking a series of questions on this link, that progressively get harder. According to the researchers, only a third of the sample of college students could answer 20 or more of the 33 questions. Ask them of a friend, and watch the eyes. Record the questions that require LEM; according to the research they should occur more on the later, harder questions. Here are a couple of them to whet your appetite, with the answers below.

Six easiest ones are
1. What’s the name of the comic strip character who eats spinach to increase his strength?
2. What’s the last name of the brothers who flew the first airplane at Kitty Hawk, North Carolina?
3. What’s the name of the crime of purposely setting a building or property on fire?
4. What’s the name of Dorothy’s dog in "The Wizard of Oz"?
5. What’s the name of the man who rode horseback in 1775 to warn that the British were coming?
6. What’s the last name of the famous magician and escape artist who died of appendicitis?

And the final six (supposedly very hard) are
28. What’s the name of the mountain range separating Asia from Europe?
29. What’s the name of the first person to run the mile in under four minutes?
30. What was the name of the Cuban leader overthrown by Castro?
31. What was the last name of the artist who painted "American Gothic"?
32. What was the name of the town through which Lady Godiva supposedly made her famous naked ride?
33. What’s the name of the highest mountain in South America?

If you need the answers, here they are for these questions, with the percentage of the college sample who answered them correctly…

1. Popeye (94%)
2. Wright (92%)
3. Arson (88%)
4. Toto (84%)
5. Paul Revere (82%)
6. Houdini (80%)

28. Ural (9%)
29. Bannister (7%)
30. (4%)
31. Wood (3%)
32. Coventry (1%)
33. Aconcagua (0%)

So how did you do… Do you notice yourself looking right or left more on the hard ones??? (HEY, over here!!!)

Wednesday 12 March 2008

PI Day is COMING, PI Day is Coming!!!!

Yeah! Finally a math Holiday, and at my school we actually give the kids the day off (well, some may think that is for course scheduling, but actually it is to celebrate Pi Day. March 14 (3.14....get it?) has been unofficially designated as PI day. (Who would have the authority to make it official?)

Many of you who may feel a little embarrassed that you haven't memorized Pi beyond two or three digits may want to run this video of a parody of the Don McLean's classic "American Pie." It has the first 29 digits (child's play) of the mathematical Pi. Enjoy!

For a little about the history of Pi, you can read my entry at Mathwords, and even learn a little about William Jones, who was the first person to tie the Greek Letter Pi to the circumference-diameter relation of a circle (and you thought the Greeks always used Pi for that didn't you?).

Anne Buckner sent me a link to a Pi movie on BrainPOP... You have to register and they send you emails, but mostly harmless. Today is also Albert Einstein's birthdate, so they have a quick movie about Big Al as well.

And just in case you didn't know where else to find it.... Pi, more than you ever wanted to know...

3. 14159265358979323846264338327950288419716939937510 58209749445923078164062862089986280348253421170679 82148086513282306647093844609550582231725359408128 ....

The Math of a Traffic Jam

Ok, every one has been in one. The traffic seems backed up for miles and you poke along, and then, suddenly you are moving again. No dead bodies on the side of the road, no carnage, no accident at all, and yet.... the traffic almost at a standstill.
For years mathematicians and computer geeks have simulated these "schockwave" traffic jams, but now a team of Japenese researchers have created a physical example. They put a bunch of cars, 230 to be exact, on a circular track, and told them to drive at a steady 30 mph. At first it all seemed to go well, but then, those tiny variations in speed caused some grouping up... somebody touched a brake and then... well just watch the video....

Monday 10 March 2008


No, Really, I have proof. If you pay me more as a teacher, your kid will learn more... well, at least you will feel he is learning more, and isn't that just about as good.

The research comes from the medical folks, and is one more extension of the many studies lately about the "placebo effect". It seems that, as I have mentioned earlier, there is more and more evidence that the ideas you have about the treatment you are getting has a significant effect on how well it works.

So what is new? Well along comes a study from Duke University that shows that people who think their drug costs more, get more pain relief.....the high priced spread really tastes better. They took 82 people and told them they were testing a new pain reliever. They pre-tested the people with a standardized wrist schock test, and then introduced the mock pain killer, and retested the folks.... OH, but along the way, they mention to one group that the pills cost $2.50 each.... but to the other group, they mention that they got them for the marked down price of 10 cents each... and can you guess which group had lower pain effects....NOPE... both of them... !! but the group who paid more, got MORE relief from their pseudo drug. While 61% of the dime-a-pill group reported a reduction in pain on the second trial, 85% of the big-buck drugs group had a reduction in pain.

So that's my idea.... I figure if you paid your kids math teacher about $250,000, you gotta' be thinking, "Wow! He must be a great teacher." and that makes you think your kid is learning and you are happy and the kids is happy and I'm driving around in my hot new car feeling WAY happy cause I can now afford the really good $2.50 pain killers...

Scientific Logic... It makes perfect sense... Dollars and Sense. Send those checks directly to my new home in the south of France.

Saturday 1 March 2008

Close order Drill and the square root of -1

I’ve been thinking about groups lately, both the mathematical kind and the social kind. The thoughts were prompted by another trip with some of my kids to the Center for Mathematical Sciences in Cambridge to hear Professor Marcus de Sautoy speak on his new book, Finding Moonshine, which is soon to be released [the American title seems to be different] and is about symmetry and mathematical group theory. When mathematicians talk about symmetry, we talk about groups.

The social group that joins me on my trips seems diverse at first glance. There are a few seniors, a few juniors, and a smattering of sophomores. There are about the same number of boys and girls, and they are as different as high school boys and girls can be. The girls will giggle at themselves, amazed that they can be excited about going to a math lecture, while the boys grow more quiet than usual when they are out of their natural turf. We stand outside one of the lecture rooms and study a board covered with symbols that neither they nor I can decipher. They try to recognize some of the symbols, a “!” factorial symbol here, “and” and “or” logic symbols there. “Is that Sigma for summation?” Some one suggests its some kind of probability problem. “What’s that one, With the Pi in the parenthesis?” Grateful that I actually know one they don’t, I explain, “That’s a capital Gamma, a function like factorial that works for all real values, not just integers.” They say “Ahh” as if they understand. We’ll expand on that on another trip perhaps, a day when an expert can peal back another level of the mathematical mystery just a bit.

What makes some kids, some people, open their minds to the complexities of math, drawn to the cryptic symbols they don’t understand? It must be the same drive that led Champollion to decipher the Hieroglyphs. How is it that one kid can be blinded to the relationship of imaginary numbers by the simple hurdle of its name, while another wants to visualize a snowflake that exists in a universe with 196,883 dimensions.

All these thoughts wandered through my mind as I watched a group of ROTC students passing beneath my window and stop at attention in the open courtyard below. As they practices the simple stationary turns common to such formations, “A’ Ten Hut”, “Right Face”, About Face”, “Left Face”, I realized that I was watching them perform a physical demonstration of the same relations they would swear were too complex to understand in their algebra classes.

The mathematical term Isomorphism is from the Greek roots for “same body”. The parade ground moves represented an order four group that was isomorphic to the multiplicative relationships between the imaginary quantities they found so impossible to comprehend. The four activities on the drill pad could each be paired with the four primitive mathematical quantities, 1, -1, I, and –I, so that they each would produce the same result.

One is the identity, Like “Attention” it keeps the position fixed. About face is the same as -1. If we think of right face as i, the square root of -1, then left face would be –i, the opposite of i. Any two actions on the pad operated like multiplication of its counterparts in the abstract number set. About face followed by about face was like Attention, not turning at all, just as multiplying -1 by -1 returned us to the mathematical identity. Right face followed by about face produced left face, and mathematically -1 x I gives –i. What about that mysterious i x i that so confused them in the algebra class. Right face followed by right face was just about face, the drill pad’s symbol for -1. No student in the small squad in front of me would have thought it difficult to imagine what his position would be if I asked him how he would be standing if he made back to back right faces.

The same abstractions that made math so powerful, that allowed us to represent the drill team, and the multiplication of complex numbers with the same symbols was a beauty they could not see. But some will twist their faces and squint against the dimensional curse of being born in three-space, hoping to get a glimpse of a four-space hypercube, one step closer to the Monster. Only 196,879 more dimensions to climb. But on the way home, they start with baby steps. Someone asks, “Can there be a group with just two elements?’ I was going to answer, when one conjectures, “Yeah, I think just one and negative one would be a group under multiplication.” They discuss it for awhile, ignoring the old man driving the car; after all, they learned tonight that “a mathematician seldom produces great work over the age of forty”, and if they don’t know exactly how old I am, they know that my age is much greater than forty; or as they would say, my age = .