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Multiplication:

I've much enjoyed the discussion about the multiplication posts, and I agree that calculating in one's head is a good skill to have. Even if there's a calculator handy, people are often reluctant to use it (perhaps overestimating the effort it would take, or not wanting to look ignorant or skeptical), and if one can do even approximate calculations in one's head, one can often spot errors that one would have otherwise missed. Also, my sense is that learning arithmetic is good mental training for other thought processes, but I may be mistaken on that.

Nonetheless, my point was different: The important thing is not knowing how to multiply, but what, when, and why to multiply. Once you understand what calculation to do to check something, you can do it easily enough on a calculator. But the calculator won't explain to you how to structure the calculation, or whether a calculation ought to be done.

That, incidentally, is one reason that I'm so frustrated when people say "I never minded math" (usually referring to arithmetic or at most algebra) "in school, but I just never liked the word problems." All of life is word problems! A problem in the real world never comes to you as "figure out 123 x 456 + 789." Understanding how to translate the real-world condition into numbers is the important skill, not the calculation. And while most school-level word problems are usually only a very first step in that direction, if people can't master that first step, their knowledge of raw arithmetic or algebra won't help them actually deal with the world.

Joel B. (mail):
Spot on!
2.6.2006 1:06pm
JLR (mail):
Prof Volokh, you are right on point regarding the importance of knowing how, when, and why to use arithmetic in real world situations.

You write something that I find intriguing:

"My sense is that learning arithmetic is good mental training for other thought processes, but I may be mistaken on that."

I wonder if there are any neuroscientific studies that might substantiate that hypothesis and identify specific mental tasks that are aided through mastery of mental arithmetic.
2.6.2006 1:30pm
BU2L (mail):
I definitely support the statement, intuitive as it may be, that "arithmetic is good mental training for other thought processes."

Having a more mathematical mindset, I feel, comes in handy in situations that require risk or efficiency assessment, even if you are not doing any actual calculation. i can't quite articulate it, but it definitely feels like arithmetic is a helpful part of the big picture.
2.6.2006 1:34pm
WB:
The more you reveal this "arithmetic" power to the public, particularly its application to the real world, the greater the risk becomes that it could fall into the hands of the terrorists, to deadly effect.
2.6.2006 1:43pm
John Lynch:
The "as a percent of what" riposte is or should be a standard part of any critical thinking. While it is certainly math, it is more than math but an essential of critical evaluation.
2.6.2006 1:57pm
Oris (mail) (www):
Word problems are the answer to what has got to be the most annoying question a teacher ever hears: "But when are we going to have to use this stuff in real life?" As such, I could never really fathom why people didn't like them. I felt like they made the problem both more concrete and more interesting.
2.6.2006 2:06pm
Bruce:
Interesting -- I've never heard the word problem complaint. My difficulties with math stem from not having had *enough* context. I still remember sine, cosine and tangent functions by picturing the unit circle, for example. But I've mostly forgotten quadratic equations, and most of calculus, because I didn't have a visual framework to plug it into. Rote memorization has a really short half-life.
2.6.2006 2:07pm
nyejm (mail) (www):
123 x 456 + 789 = 56877!
2.6.2006 2:29pm
Mary Katherine Day-Petrano (mail):
Mr. Volokh, I agre with everything you have to say. however, some people with certain disabilities (autistics who are not numbers oriented) just do not have the capability.

Ahh, but not all is lost -- some of us more mathematically challenged do have the abilities to leabr formal Logic and apply THAT to the real world. Maybe the next best thing, but a workable alternative nevertheless.

Now, for the next scientific breakthough -- can anyone figure out how to inject the missing genetic mathematical material in to the brains of non-number oriented autistics? And, while you're at it, how about the missing emotion-reading brain cells as well?
2.6.2006 2:35pm
JGR (mail):
"Understanding how to translate the real-world condition into numbers is the important skill, not the calculation."
This reminded me of an interesting fact: Negative numbers were invented by accountants, not mathematicians. (This is mentioned in Jacques Barzun's 'From Dawn To Decadence, 1500 To the Present').
Since many of the readers of these posts appear to have a strong interest in math, let me throw out that the best general book on mathematics that I ever read is John D Barrow's 'Pi In The Sky'. This is not so much a book for the pure math lovers, but a wonderful and even literary exploration into all the philosophical problems surrounding math.
Back on topic, maybe the reason so many of us went off on a rant about doing math in our heads is that we JUST DID NOT WANT TO ACCEPT that people who graduate high school can't figure out how to calculate how many minutes are in a year and whatnot. "Where ignorance is bliss, 'tis folly to be wise".
2.6.2006 2:36pm
Mary Katherine Day-Petrano (mail):
Apologize for the typos

agre=agree

leabr=learn
2.6.2006 2:42pm
JGR (mail):
"Now, for the next scientific breakthough -- can anyone figure out how to inject the missing genetic mathematical material in to the brains of non-number oriented autistics?"
Mary, What luck! I just discovered this in my lab and will mail you the patent for only $10,000 !!! (Limited time offer)(Patent comes with proof of purchase)
2.6.2006 2:44pm
Duncan Frissell (mail):
I always preferred language to maths but ever since 1976 when I played with Visicalc (at an open sales presentation by Apple at the Hyatt Regency San Francisco) I haven't minded number crunching.

I found that with spreadsheets I didn't have to know how to structure problems before calculating. I just construct a formula and test the answer and if it's wrong, I keep changing the formula untill I get it right. Since you can easily estimate and change things with spreadsheets and perform "double entry bookkeeping" on your formulas, it generally works out.
2.6.2006 3:21pm
tefta (mail):
Estimating calculations has saved me from making errors like the one you uncovered in this post. It's a fairly easy skill to master and cultivate.

Solving problems without paper and pencil is a gift. It's been a marvel to me that my father who was educated only to possibly third or fourth grade in a tiny village in Europe, instinctively understood mathematics and could come up with correct answers directly out of his head while I had to slog through formulas to come up with the solution. He didn't have the words to explain the process -- the best he could do was to say the answer was there in his head.

You're right that even grade school number learning is referred to as mathematics. Arithmetic has disappeared.. Third graders learning the multiplication tables are told they are doing math, or even worse, their maths!
2.6.2006 3:40pm
GregC (mail):
To follow up on JGR's post, I can also recommend Peter Bernstein's "Capital Ideas" and "Against the Gods". Bernstein is a financial writer/reporter, so both are geared toward understanding the math behind financial instruments. But both provide a wonderfully written history and explaination of mathematics that even the numeraphobic can love.
2.6.2006 3:41pm
Deoxy (mail):
You should go back and read some of the horrid word problems in schools these days (or at least, 10 years ago).

The real world I can handle (quite well, if I do say so myself). Math course "word problems" are a psychosis all of their own.
2.6.2006 3:54pm
Frank Drackmann (mail):
It doesn't matter if you run or walk, excepting extremes of weight, a mile burns about 100 calories, of course, sleeping burns 60 calories hour just to keep your body funtioning. So if you run 6 miles in an hour, you're only 540 calories ahead of the guy who slept the hour.
2.6.2006 3:59pm
Houston Lawyer:
My mother has been teaching piano for almost 50 years. She is of the school of thought that certain physical exercises improve your math and logic abilities. I have no idea whether this actually works.
2.6.2006 4:20pm
K Bennight (mail):
I don't doubt your point about math, but I think there's credulity at work here, too. Reporters don't think skeptically of things they are sympathetic with.
2.6.2006 4:23pm
Bob (mail):
People often refuse to use even the simplest addition in their everyday life. I am always amazed when my girlfriend goes shopping and does not keep even a rough estimate in her head of how much her purchase should cost. Doing this has saved me money countless times due to double scanning or mislabeling. It just seems like common sense to me.

And by the way at what point does the "math" taught in schools become something that should properly be called math.
2.6.2006 4:44pm
tefta (mail):
Bob, Rule of thumb. If you can do it by counting on your fingers, it's arithmetic, if not, it's mathematics. Sorry, I couldn't resist.

From dictionary.com

Arithmetic: The mathematics of integers, rational numbers, real numbers, or complex numbers under addition, subtraction, multiplication, and division.

Mathematics: a.The study of the measurement, properties, and relationships of quantities and sets, using numbers and symbols; b. A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement.
2.6.2006 4:55pm
Silicon Valley Jim:
I'm one of those who can do reasonably intricate arithmetic in my head; this may well be, at least in part, a consequence of having obtained nearly all my schooling before calculators priced less than half a typical head-of-family wage were available. I agree 100%, however, that the important thing is not knowing how to multiply, but what, when, and why to multiply (or, of course, add, subtract, etc.)
2.6.2006 4:56pm
CEB:
This will probably sound obvious and dumb to the more mathematically-inclined, but I just learned the other day how multiplication in binary "looks" exactly the same as multiplication in decimal. E.g., 10 X 11 = 110 in decimal, and 10 (2) X 11 (3) = 110 (6) in binary. Neat!
2.6.2006 5:00pm
max (mail):
The important thing is not knowing how to multiply, but what, when, and why to multiply.

Amen.

It's almost a cliche to state that students often have to re-learn mathematics to succeed at more advanced mathematical applications. I'd gone through this process myself. Much of this re-education concerns the acquisition of skills involving precisely the understanding E.V. implies. We, in the U.S., don't seem to provide an effective bridge between basic and 'advanced' maths.

I suspect that this is partly due to an exaggerated presumption of the innateness of mathematical ability - that such ability is a discrete heritable trait similar to the acuity of visual recall or proprioception. You'll notice this presumption by remarks on the 'irony' of prominent mathematicians and scientists relating that they were frustrated by rote mathematics but excel at abstract mathematical logic. Similarly the often hyper-pedantic mode of math instruction presented in college level sequences seems justifiable only if you assume that these courses are intended to speak to 'natural' mathematicians.

And so, by assuming that all good mathematicians 'just get it', we neglect to provide a thorough education in the foundational concepts upon which advanced maths are built - thus the bias confirms the result.
2.6.2006 5:04pm
alkali (mail) (www):
I have in my head a bunch of one-significant-digit estimators that I use for spot checking things I see in the paper:

World population: 6 billion (I should use 7)
US population: 300 million
US GDP: 10 billion (a bit low)
Federal budget: 3 billion (a bit high; pre-2006, I used 2 billion)
Days in a year: 400
Workdays in a year: 200 (sometimes I'll use 250)

Does anyone else have any of these?
2.6.2006 5:07pm
Traveler:
Alkali: I think you have a typo, and mean "trillion" for the US budget and GDP.
2.6.2006 5:11pm
Ellen:
One estimate I've used is 25,000 days as a typical lifespan. I recall an incident when "I wonder how many people die on a typical day in the United States" came up in casual conversation. I divided 250 million (population estimate) by 25,000 and came up with 10,000, then suggested 8000 as my estimate (to reflect the fact that the population is biased toward younger ages.) It turns out that, when we looked it up, the actual number is closer to 6000. But the others were amazed that I could make an estimate that was anywhere near correct. They simply had no idea of what inputs to use or how to estimate.
2.6.2006 5:25pm
Silicon Valley Jim:
Alkali:

I have a few:

One employee-year costs about $150,000, including benefits, a share of management, equipment, etc.

Area of the United States: 3 million square miles without Alaska, 3.5 with.

As Frank Drackmann notes, one mile on foot burns 100 calories. One mile on a bicycle burns about 35 calories. One (non-light) beer (or one non-diet soft drink) contains about 150 calories (a little low, but it's helpful to remember that 2 beers=3 miles on foot or 9 to 10 on a bicycle.

Not strictly mathematical, but helpful (came in useful recently in correcting the alumni magazine of a major private university in Silicon Valley): one acre = one football field (the NFL kind, not a soccer or rugby pitch) without the end zones.

One other thing, that doesn't quite match the others: if your total doesn't match by a factor of nine (or ninety, or 900), it's likely that you transposed two numbers. If, for example, the difference is 18, you transposed two numbers with values apart by two, e.g., a 3 and a 5. If it's 45, you transposed two numbers with values apart by 5, e.g., a 7 and a 2.
2.6.2006 5:25pm
tefta (mail):
alkali, yes, doesn't everyone?
2.6.2006 5:26pm
Freddy Hill (mail):
The important thing is not knowing how to multiply, but what, when, and why to multiply

I think it is also important to know with whom to multiply.

Sorry.
2.6.2006 5:30pm
Tony (mail):
I might also comment that when it comes to calculators, some calculators are better than others. RPN calculators are vastly superior for almost any kind of calculation, and encourage clearer thinking about the "structure" of those calculations. For those of you who aren't familiar with them, the buttons you'd press on a regular calculator to add 2+2 are:

2 + 2 =

whereas on RPN they are

2 enter 2 enter +

...which seems pointless until you try to calculate something more complicated, at which point it becomes immediately indispensible.

What boggles my mind is that the Windows "calculator" emulates this ridiculous interface. For a really great desktop calculator, download R from http://www.r-project.org/ . It's a very complicated and technical tool at its deeper levels, but for doing everyday calculations it's also super duper. Wish I had a version for my Treo!
2.6.2006 5:32pm
Don Miller (mail):
I had a math teacher in high school who insisted on teaching us how to solve real life problems using algebra and arithmetic.

I still use his technique for multiplying 2 digit numbers in my head. (hint: same technique as solving (ax+b)(cx+d)).

I am always scribbling down simple little algebra problems and solving for X to figure out the formula I need. Maybe it's doublechecking the correct sales tax rate, or maybe it is refiguring out the conversion of celsius to farenheit. Recently I took a Paramedic class, and it was converting from Lbs to Kilogram to calculate dose rates for various medications.

Life is full of the power of numbers. I have loved your commentary on it.
2.6.2006 5:38pm
Silicon Valley Jim:
Thought of some more that I've used. I've used some of them quite frequently at times:

One work year in the United States: 2,000 hours (insert joke here about the country of your choice)

One year: 9,000 hours (a bit high) or half a million minutes, or 30 million seconds

One day: 1,500 minutes or 90,000 seconds


Tony:

I agree entirely about RPN. I think that I would agree entirely even if two of my grandparents hadn't come to the United States from Poland.
2.6.2006 5:39pm
Chuck Jackson (mail):
alkali:

Rule of thumb Farenheit = 2*Celsius+30.

Correct: Farenheit = (9/5)*Celsius+32.

Rule of thumb is easier to calculate and is reasonably accurate from about -10 C to 30 C. (error is less than 4 degrees F)
2.6.2006 5:50pm
Mary Katherine Day-Petrano (mail):
JGR --

""Now, for the next scientific breakthough -- can anyone figure out how to inject the missing genetic mathematical material in to the brains of non-number oriented autistics?"
Mary, What luck! I just discovered this in my lab and will mail you the patent for only $10,000 !!! (Limited time offer)(Patent comes with proof of purchase)"


Deal. After I get my bar admission and win my first big case (and get that $10,000), I'll take the cure and being a prodigious mathematics career (with my newly created brain cells). Sorta opposite the career trajectory of EV.
2.6.2006 5:54pm
Mary Katherine Day-Petrano (mail):
ooops, typo: being=begin
2.6.2006 5:55pm
Mary Katherine Day-Petrano (mail):
JGR, if the patent comes with the proof of purchase, does that mean I get the royalties from all the other autistics after I complete the purchase? (I'll start looking for my new horse farm already).
2.6.2006 5:57pm
M.E.Butler (mail):
Was there anybody else who saw the movie Proof and had to confirm (in his/her head, while watching the movie) that 1729 really is the sum of two different numbers cubed? (10^3 + 9^3 and 12^3 +1^3)

Cubing 12 in one's head isn't too difficult if there's a break in the dialogue--the one sticky part being the addition of 1440 and 288.
2.6.2006 6:03pm
Bob M - Chicago (mail):
A friend of mine made an interesting observation once.

People are very willing to tell you that they are not very good or even terrible with math.

Yet, those same people would be horrified if they found out a friend or
colleague couldn't read.

Why should the stigma be any less - or absent completely - for innumeracy than for illiteracy?
2.6.2006 6:04pm
Silicon Valley Jim:
Cubing 12 in one's head isn't too difficult if there's a break in the dialogue--the one sticky part being the addition of 1440 and 288.

It's also one of those numbers that you remember if you attended Catholic schools earlier than, say, 1965. It's the volume in square inches of a cubic foot.
2.6.2006 6:05pm
Duncan Frissell (mail):
"A pint's a pound the world around"

Actually a gallon of water weighs about 8.3 pounds (@STP) but the above is close enough for government work.
2.6.2006 6:07pm
Oris (mail) (www):
Silicon Valley Jim, I had never heard the factor of nine trick. That's neat. It reminds me of how delighted I was as a child learning that you can multiply by nine on your fingers (and that all multiples of nine add up to nine). For anyone who has never encountered this one:
  • Hold up all ten fingers.
  • Counting from left to right, put down the finger of the number you want to multiply by nine.
    (E.g., if you're multiplying by three, put down your third finger.)
  • The resulting two groups of fingers represent your tens (on the left of the finger that is down) and ones (on the right) places.

Thus, 3 x 9 = 2 fingers on the left (3rd is down) 7 fingers on the right = 27

To this day I look at my hands if my brain isn't coming up with the multiple of nine fast enough for me.
2.6.2006 6:09pm
Math Graduate (mail):
My post graduate work is in mathematics.

I always recommend that school kids learn simple algebra along with grammar, for they are both grammars and have similiarities. The other element missing in education and practice is the concept of unit arithmetic, which simplifies the formulation.

Example: Miles/hour * hour = miles The hour unit cancels because I know my simple algebra, so the correct forumla is pointed to me. What are the errors we always see in the media? Wrong units, billion, instead of trillion; hour instead of minute, etc.
2.6.2006 6:17pm
nyejm (mail) (www):
To go along with the shortcut for multiplying nines, there is a trick for multiplying 11 as well. But it only works with two-digit multipliers. First, you "split" the multiplier. That is, just take the number from the tens column of the multiplier and put it in the hundreds column of the answer, and leave the number from the ones column in the multiplier in the ones column in the answer. Then, add up the number from the tens column of the multiplier and ones column of the multiplier, and put that in the tens digit in your answer (don't forget to carry over to the hundreds if necessary). And there's your answer.

For example, 11 x 36. Split the 3 and the 6 to get 3_6. Add 3 + 6 and put it in the tens column to get 396. 11 x 36 = 396. Or 11 x 87. Split them up to get 8_7. Add 8+7 to get 15 and add that to the tens column. Carrying the one makes your new number 957. 11 x 87 = 957. Piece of cake.
2.6.2006 6:22pm
Telford Work (mail) (www):
This will probably sound obvious and dumb to the more mathematically-inclined, but I just learned the other day how multiplication in binary "looks" exactly the same as multiplication in decimal. E.g., 10 X 11 = 110 in decimal, and 10 (2) X 11 (3) = 110 (6) in binary. Neat!

11 x 11 = 121 in decimal; 11 x 11 = 1001 in binary. Needing to "carry the 1" makes the difference. That example is just a coincidence, like 2 + 2 equalling 2 x 2.

Was there anybody else who saw the movie Proof and had to confirm (in his/her head, while watching the movie) that 1729 really is the sum of two different numbers cubed? (10^3 + 9^3 and 12^3 +1^3) Cubing 12 in one's head isn't too difficult if there's a break in the dialogue--the one sticky part being the addition of 1440 and 288.

12 ^ 3 = 1728; 1 ^ 3 = 1
10 ^ 3 = 1000; 9 ^ 3 = 729

The irony!

BTW, I may be somewhat mathematically inclined but that doesn't stop me from making lots of mathematical mistakes of my own....
2.6.2006 6:38pm
Splunge (mail):
Our ancestors did most of their math in their heads, perforce, and this helps illustrate why some of our weights and measures seem strangely "illogical" to the modern world with its heavy emphasis on decimals and calculators.

The obvious example is the hour of 60 minutes, which can be exactly divided into halves, thirds, quarters, fifths, sixths and tenths. By contrast the "logical" decimal choice of 100 minutes per hour can be divided into halves, quarters, fifths and tenths. Not quite as useful if you have to do division in your head.

Another interesting example is Daniel Fahrenheit's thermometer scale, which modern folks often consider less "logical" than Anders Celsius' scale in which the difference in temperature between freezing and boiling water is set to 100 C. Fahrenheit, in his older scale, originally set the difference between freezing water and human body temperature equal to 64 F.

Why 64? Imagine making a thermometer: you fill the tube with your working liquid (mercury, or in Fahrenheit's day alcohol). You put it one of your references (say freezing water) and mark the level. Then you put it in another (in your mouth, for Fahrenheit's scale) and mark the level again. Now you need to divide the space between into equal divisions. If you have to make 100 divisions, you need a precision steel ruler. But if you have to make 64 divisions, you just need a compass and straightedge, because you just need to divide the distance exactly in half 5 times (2^6 = 64).

Furthermore, Fahrenheit took advantage of an interesting coincidence to set the freezing point of water at 32 F, and assign a third scale point, the temperature of a stable mixture of ice, salt and water, to 0 F. Since 2^5 = 32, this distance, too, can be marked off very easily with compass and straightedge.

Hence Fahrenheit's scale lent itself far easier to the manufacture of accurate thermometers at a time when precision instruments were very expensive.

There are often similar reasons partly underlying other "nondecimal" subdivisions of our traditional measurement units. Basically, it turns out powers of 10 are not a very convenient way to subdivide things, inasmuch as they have at best a merely typical number of factors smaller than themselves. For example, among two digit numbers 10 has a slightly lower than average number of factors smaller than itself (2 versus 2.8). More convenient numbers are 12 (4 factors), 16 (3 factors), 24 (6 factors), 36 (7 factors), and of course the champ 60 (10 factors).

Hardly a surprise, therefore, that we have 12 inches per foot, 16 ounces per pound, 24 carats in pure gold, 36 inches in a yard, and vide supra 60 minutes in an hour.
2.6.2006 6:53pm
John H. Costello (mail) (www):
Although I generally agree with you, I would argue that an abilitiy to decide what should be multiplied when generally comes after or during the development of raw number crunching, which tends to be downplayed. Before you can work at word problems you generally must have the memorized knowledge that 7*7 = 49 or 8*9 is 72. I work as a cashier, and constantly encounter people who cannot do that level math. Some years ago I worked at one end of a department store that continually ran sales. Teenagers would come up to me and ask: How much is this item? The sign above the item said that it was 10% off the tagged price.
2.6.2006 7:01pm
Splunge (mail):
Oops, I should add that Fahrenheit later redefined the upper end of his scale to set 212 F equal to the boiling point of water at 1 atm, and that shifted human body temperature (he used axillary) from 96 F, the original, to about 97 F. Also, I should mentioned that you put the new thermometer under your arm, not in your mouth, for the second reference, since oral temperature is higher, about 98-99 F.

As for why he didn't use the boiling point of water right away, he did not orginally know why the boiling point varied as much as it does. Only later did folks realize the importance of atmospheric pressure, i.e. altitude and weather. Once this was understood, and it was possible to measure barometric pressure, one could defined the boiling point at 1.000 atm (760 mm Hg) and it would be reproducible.
2.6.2006 7:03pm
cathyf:
I am amazed at the current fad of using calculators in calculus classes. When I took calculus, the only thing that you would need a calculator for was converting fractional scores to decimal/percent. They are fancy graphing calculators, and that only makes things worse. My husband spends lots of time in physics lab teaching students how to make graphs -- like the origin isn't usually {0,0}, and the shape of the graph depends on how you make your axes.

Back when I took calculus in high school, Sr. Rita Cecile made us calculate integrals by counting rectangles. We had to take a piece of graph paper, plot the function by calculating a reasonable sampling of points, then count the square, and estimate the triangles at the function line. She only made us do this twice, but it was enough to give us two extraordinarily valuable pieces of knowledge. The first is that an integral really is the area underneath a curve. The second is that we really really really wanted the Fundamental Theorem to be true...

cathy :-)
2.6.2006 7:31pm
David Cohen (mail):
If you Google "number of seconds in one year" it will tell you.
2.6.2006 7:34pm
Eugene Volokh (www):
Nyejm: You wrote "123 x 456 + 789 = 56877!" I don't need a calculator to say that 56877! is much greater than 123 x 456 + 789, or even 123 x 456 + 789!
2.6.2006 7:39pm
nyejm (mail) (www):
Yes, very clever. A factorial joke.
2.6.2006 8:21pm
reneviht (mail) (www):
Up until I was about 20, I thought multiplication was repeated addition. I only recently learned that multiplication is not repeated addition; that's just how you calculate it. Multiplication is function composition. 3 times 2 means you have 2 three times. This seems completely trivial, until you learn to extend multiplication to non-numerical objects. Every physicist is capable of multiplying physical units by numbers and other units, and this definition remains unchanged when you consider multiplication in groups.
2.6.2006 8:53pm
CEB:
Telford Work-
Thanks for the correction; that's what I get for believing something I read on the internet! So how does one (no pun intended) multiply binary on paper? It does work sometimes to just do it like decimal (101 (5) x 10 (2)= 1010 (10) binary &101 x 10 = 1010 decimal); is there an intuitive way to do it right?
2.6.2006 8:56pm
reneviht (mail) (www):
Actually, it is just like decimal, except with the following tables used for 1-digit operations:

0 + 0 = 0
1 + 0 = 0 + 1 = 1
1 + 1 = 10

0 * 0 = 0 * 1 = 1 * 0 = 0
1 * 1 = 1

You do multiple digit addition/multiplication the same regardless of the base. Just remember to carry when the sum reaches the base (in this case, 2). Binary multiplication takes less memorization, but a heckuva lot more writing.
2.6.2006 9:02pm
Mark Liberman (mail) (www):
Knowing what to multiply when -- and what quantities NOT to multiply and when NOT to multiply them -- is also important in framing a statistical model. Failing to think about this in common-sense terms can result in a very implausible model, and possibly serious artifacts when the model is fitted to real-world data. I argue here that this flaw applies to Groseclose and Milyo's much-discussed investigation of media bias.
2.6.2006 9:21pm
Zach (mail):
Now, for the next scientific breakthough -- can anyone figure out how to inject the missing genetic mathematical material in to the brains of non-number oriented autistics?"

For most everyday purposes when you don't want to physically punch the problem into a calculator, you can get a good estimate of how big an answer is by throwing out all the least significant digits. If you have to multiply 28,472 by 13,287, figure it's pretty close to 3*1 x 10^8 (actual answer:378307464) The biggest advantage to mental math is in getting a "number sense" of about how big the answer _should_ be so that you can pick out the obvious mistakes. You only need a calculator if you need to know the answer with a great amount of accuracy.
2.6.2006 11:10pm
tfellow:
As an example this link from Yahoo Finance says the World's Biggest Photovoltaic Plant in Nevada "will develop a total of 36 MWs of PV projects in Nevada, enough energy to power 36,000 homes". With division (a little off topic) we see that they're really counting on energy conservation since that gives each home 1000 Watts to use anyway they want! That's about enough to run a hair dryer (on low). Most new houses have 100 amp service which provides ~11000 Watts so these guys are off by at least an order of magnitude as we used to say in m(ath) school

In the new blog age (blogage?) maybe they'll need pay more attention to multiplication in j-school and f(inance)-school.
2.6.2006 11:54pm
Tracy Coyle (mail) (www):
Just to add to the multitude of fun: "millions of Americans have been spied on without warrants" I believe was the phrase. Followed by, all they had to do was ask for the FISA warrant. Ok. 1,000,000 Americans, 4 years, 365 days, 24 hours. The NSA would have to draft, present and obtain 1 warrant every 2 minutes...24 hours a day, 365 days a year, for 4 years.

Too much math, not enough cents...
2.7.2006 12:10am
Bruce Lagasse (mail):
Tony and Silicon Valley Jim (re RPN):

I agree entirely about the superiority of RPN, and my conviction came about in an odd way. Many years ago, my company bought a number of Wang desk calculators, and by pure dumb luck, the very first one purchased wound up on my desk. Therefore, I was forced to learn its "backwards" computation logic, that I found out much later was called RPN. I seemed to have a lot less trouble picking it up than most of my other colleagues; perhaps RPN happens to be hard-wired into my brain. Also, the Wang calculator was the first and only electronic device I've ever seen that used Nixie tubes for its display. It was no doubt pretty inefficient, but my gosh, they looked cool!
2.7.2006 3:02am
alkali (mail) (www):
Traveler: belated thanks for catching that typo.
2.7.2006 9:22am
David Chesler (mail) (www):
Was there anybody else who saw the movie Proof and had to confirm (in his/her head, while watching the movie) that 1729 really is the sum of two different numbers cubed? (10^3 + 9^3 and 12^3 +1^3)

I didn't see Proof, but I know that not only can it be expressed as the sum of distinct cubes two different ways, but it is the lowest such number that can be so expressed.


My father came close to physically striking me when he learned that in high school I was still doing subtraction the New Math way ("borrow 1 from the tens place and change it to 10 in the ones place...") -- fine for theory, but in practice so much easier to increment the subtrahend.

He also was well familiar with the 9 rule for accounting errors that Silicon Valley Jim mentions, which he promptly explained works because "9 is a magic number". (He claims he got through GI basic training with his sanity intact by calculating and memorizing the powers of various numbers, up to 9 raised to the 9th power.)

Word problems in my older son's 3rd-grade class are of the form "Carmen says that 32 is divisible by 5 because 3 plus 2 equals 5. Why is she wrong?" and he wants to answer "Because she's retarded!" The kid is going to go places.

Unlike my daughter, I'm afraid. In the 5th grade she's still trying to "remember" and "figure out" her times table. I repeat to her what a co-worker's aero prof said (about the equations of flight): "Do your 'remember' your name? No, you know it. Similarly you must know" (the fundamental equations of your field of study, or your times table.) I fear she gets this from her mother, who once described an exercise ball as being "about 2 feet high and 3 feet across". (And yet I understood what she meant: it's about as high as object she'd describe as 2 feet high, and about as wide an an object she'd describe as 3 feet wide, and to hell with the relationship between all diameters of a sphere. Just as at exactly the same height, I'm as tall as men who claim to be 5'8", while she's as tall as women who claim to be 5'6".)

tfellow writes:
With division (a little off topic) we see that they're really counting on energy conservation since that gives each home 1000 Watts to use anyway they want! That's about enough to run a hair dryer (on low). Most new houses have 100 amp service which provides ~11000 Watts so these guys are off by at least an order of magnitude as we used to say in m(ath) school

That's peak usage (100 amps) versus average usage. Both are important. I was concerned about an increase in my electric bills (turns out it was the new dehumidifier, not the new meter, oh well) so I got all my old electric bills off the supplier's web interface, and divided kilowatt-hours by 24 times the number of days in the billing cycle, to express my usage as average kilowatts drawn, and sure enough my typical around-the-clock average draw is about 1 kilowatt. (Or for the less numerate, 1 kilowatt-hour per hour.) Up to 1.3 kw when the dehumidifier is running. (Efficiency is kw-h per liter of water condensed, and it's rated IIRC 0.8, and it's rated 50 pints per day, and I'm emptying the 2.5 gallon bucket about twice per day -- I think that all worked out.)

Of course you can't use that daylight-only plant alone to power all those homes, but you can factor out that many homes from the expected fuel consumption of your existing generation plant.
2.7.2006 9:35am
SeaDrive (mail):
Further on the comment above about the divisors of 60, it really would have been better if we had 6 fingers/hand, and used base 12 rather than 5 fingers/hand and base 10. We actually use the easy divisors and multiples of 12 for a lot of things such as 6-packs, cartons of 12, etc. Often this is because these quantities allow for convenient packaging.
2.7.2006 11:21am
CompaniaHill:
One of the most important everyday math tips I learned is a trick to help me keep track of the units. For example, somebody says he can throw a ball 100 feet per second, and you want to convert that to miles per hour. You may already know that there are 5280 feet in a mile and 3600 seconds in a minute. But do you multiply by 5280/3600? 3600/5280? 5280x3600? 1/5280x3600?


Here's how I do it (forgive my lousy HTML):



100_feet___60_seconds___60_minutes______1_mile___68.2_miles

--------
_x_----------_x_----------_x_---------_=_----------

__second____1_minute_____1_hour______5280_feet________hour


My high school math teacher Mr. Metzger (see, after all these decades I still remember the names of the teachers who were important in my life) taught us this trick. Treat the units as if they were part of the number, multiplying and dividing and canceling them as appropriate.

2.7.2006 12:40pm
Mary Katherine Day-Petrano (mail):
"Why should the stigma be any less - or absent completely - for innumeracy than for illiteracy?"

Because "stigma" constitutes a vio of the Americans with Disabilities Act. "innumeracy=mathematics disorder, DSM IV. "iiteracy" (peope who can't read)=dysexia.
2.7.2006 3:27pm
Splunge (mail):
SeaDrive, you want to talk to these people. I think there are some ancient societies (in Africa, maybe) that are thought to have used base-12 counting systems. Certainly it survives in weights and measures, and in special words like dozen. The Sumerians and Babylonians used base-60, of course, which is why we have 60 minutes in the hour, vide supra, and 360 degrees in a circle. I've read an unconvincing argument this had something to do with a compromise between ancient base-12 and base-5 counting systems.

I believe most experts think a people get their base from how they count on their fingers (and toes, maybe). You can get base-5, base-10, base-12 and base-60 counting systems all from how you count on your fingers (base-20, like the Mayans, you presumably get from counting on your fingers and toes). Base-5 and base-10 are obvious. Base-12 you can get by using each of the three segments (outer, middle, inner) in each of the four fingers on a hand. Base-60 you can get by using each of the five digits of the right hand to point to one of the 12 finger segments on the left hand.
2.7.2006 3:32pm
Barry Kearns (mail):
Silicon Valley Jim wrote (re: cubing 12 in your head)

It's also one of those numbers that you remember if you attended Catholic schools earlier than, say, 1965. It's the volume in square inches of a cubic foot.

I hope you don't ever try to measure a volume in square inches... there are an infinite number of square inches in a cubic foot, right?

I understood that you meant cubic inches in a cubic foot, but in a thread about math errors, why propagate more?

I mention this in part because I had a junior-high chemistry teacher who would routinely answer mismatched units questions with either "infinity" or "zero"... we learned to take that as a cue that we had a fundamental error in unit expression that we needed to fix before proceeding further. (I actually had someone at my table ask him how many cubic feet were in an acre, and he answered them with "zero"... perfectly straight-faced.)
2.7.2006 4:35pm
Silicon Valley Jim:
Barry:

Touche. You're absolutely right. Homer nods, and all that.
2.7.2006 7:27pm
Splunge (mail):
A late comment on Mr. Volokh's original post:

All of life is word problems!

This is liberal artsy nonsense. I knew plenty of people in college (MIT) and graduate school (Berkeley) who were poor at what most people mean by "word problems," but they were nevertheless gifted engineers and scientists. They just weren't any good at stories and narratives.

These people certainly did not lack for any ability to solve "real world problems" -- but they didn't use an intermediate representation in words during the process of turning the real-world problem into mathematics. Some of them used wordless mental images, no doubt, but quite a number "saw" physical processes directly in terms of the math required to quantify them. Indeed, I had one professor who had profound insight into many-body quantum mechanics -- one of the best men in the world in this field -- but who was so famously inarticulate his students used to joke that he had never learned about nouns (he used pronouns and verbs, largely). Unless he could say it in equations, this man found it difficult to explain even the simplest of things. Needless to say, his social skills were not the best, but in solving certain "real world problems" he had no equal.

I'm sure Mr. Volokh can rescue his argument by re-defining "word problem" to mean "any problem not originally expressed in math." But this is not what most people mean by "word problem." And I suspect some (although by no means all) of the people Mr. Volokh finds frustrating, who are good at math but not at "word problems," are the brilliant engineers and scientists of tomorrow.
2.7.2006 8:30pm
Eugene Volokh (www):
Jeez, Splunge -- "liberal artsy nonsense"! You sure know how to wound a math-computer science major, one who was pretty good for his age at math long before he got into law and other mushy stuff. Life is word problems in the sense that the sentences immediately following that one elaborate:
All of life is word problems! A problem in the real world never comes to you as "figure out 123 x 456 + 789." Understanding how to translate the real-world condition into numbers is the important skill, not the calculation. And while most school-level word problems are usually only a very first step in that direction, if people can't master that first step, their knowledge of raw arithmetic or algebra won't help them actually deal with the world.
Some people may be so good at translating real-world conditions into numbers that it happens seamlessly. But few are, and those who are, I suspect, mostly took some time to learn it.

If you don't want to view "real-world problems" as essentially "word problems," in the sense of problems that beginning students originally think about using words, and then translate into numbers, that's fine. But it seems to me that equating the two is pretty sound, and hardly nonsense, liberal artsy or not.
2.7.2006 10:56pm
JGR (mail):
About bases..

With very few exceptions, all the base systems from natural cultures are from five different bases: 2,5,10,20,60. There is almost universal agreement that these derive from human digits. The primary exceptions are systems of four or eight. One theory is that they used their four fingers, but not their thumbs; Another is that they used the spaces between their fingers. The latter has anthropological support in at least some cultures.
The book that I recommended above - John D Barrow's Pi In the Sky - has a fascinating chapter on this, and gives some interesting examples of other bases:
" ... there exist whole counting cultures which are based upon the human anatomy. One of the most elaborate examples was that practiced by the Torres Strait Islanders as late as the nineteenth century. Starting on the right side of their body, they touched first five fingers for the numbers 1 to 5, then the wrist(6),elbow(7), shoulder(8), chest(9), left shoulder(10), left elbow(11), and so on, down to the left fingers which took them to 17. Then they continued with the left little toe(18), through to the left big toe which takes them to 22, left ankle(23), left knee(24), left hip(25), right hip(26),and then back down the right side in the same way, finishing with the right little toe(33). One imagines that maths lessons might have resembled a form of St Vitus's dance."
There are also several variations of finger-counting that involve bending the fingers, some in elaborate ways (think of sign language).
Why 60 is often a natural base is something that there is no consensus on. Hypotheses involve astronomy (An early belief was that a year had 360 days, which divided by 12 equals 60.) Another has to do with geometry, although this is more improbable. The commingling of bases which is also not very probable. (The theory I read was on the mixing of base 6 and base 10, but it's entirely possible that Splunge might have read a different theory than me). Then the fact that 60 has a lot of divisors, although this is rather putting the cart before the horse. The truth is that it's all guesses as to 60 - which I may market as a slogan for a t-shirt. "IT'S ALL GUESSES AS TO 60"
Mary, I'll throw in the copyright to my t-shirt for another five grand, and sure you can keep the rights and profits - My generosity is legend.
A final thought: Imagine what feminists would say if male cultures had developed bases of 11?
2.8.2006 3:39am
RGT (mail):
Splunge,

"This is liberal artsy nonsense. I knew plenty of people in college (MIT) and graduate school (Berkeley) who were poor at what most people mean by "word problems," but they were nevertheless gifted engineers and scientists."

Can't agree with you on the engineering (no idea about pure math or pure science). If I'm talking to the RF engineer (I'm a Mech. E) about a new invention, we need to start from concepts, words and verbal descriptions to work toward the correct formulation of the relevant equations.

My latest patent came about after a long brainstorming session which produced nothing but a clear, one sentence description of what was required to solve a particular problem. Only then could we start to generate ideas (verbal and 'sketches on napkins'). The math came later when it was time to decide 'how big does A need to be to get us B'.

RGT
2.8.2006 9:15am
ctw (mail):



"Imagine what feminists would say if male cultures had developed bases of 11"

don't have to - I know per a friend of my wife's: "oh, that's just another male behavior that's [alliterative profanity] driven"

it's surprising that the origins of 60 as a base are so obscure given its prevalence in cultures (sec/hr, min/hr and min/deg, mph (:>)). but in any event, as a codger I'd recommend that your t-shirt should read:

IT'S ALL GUESSES TO 60
then, wisdom
2.8.2006 1:23pm