by Norman Costa

As a child, arithmetic came easily to me. My father, though not well educated, had a facility with numbers. He had a fascination for tricks and short-cuts to computations. One day he brought home a book, "The Trachtenburg Speed System of Basic Mathematics," by Jakov Trachtenberg. He is a survivor of The Holocaust, who worked out his system in the camps, and saved his sanity in the process.

I tried some of the techniques. They worked very nicely, except that I lost interest, quickly. His system was great for very fast arithmetic calculations, but it did nothing to convey an understanding of numbers. That was the 8th grade. From another source I learned how to speed multiply, mentally, 2 numbers ending in 5. I still use it today because I developed an understanding of how the numbers worked.

"Anybody got a calculator? I need to know how much is 165 times 35." People are looking for a calculator or pencil and paper. About 15 to 20 seconds pass. I am quiet and unconcerned with finding anything. "Fifty-seven seventy five." I say. Silence. A few moments later the one in need of emergency calculation asks, "You sure." As I smile at him I tap my right temple with my right index finger. Someone finds a calculator, enters the information, and says, "Wow! You're good."

Another speed trick is multiplying any arbitrarily long number by 25. As long as I can look at the number I can start reciting the answer, beginning with the left most digit. "OK, smarty pants. How about 9 billion, 880 million, 981 thousand, 445." I right down the number so I can see it. I reply, "2 4 7 0 2 4 5 3 6 1 2 5."

As a junior in high school I fell in love with the book, "One Two Three ... Infinity," by George Gamow, and first published in the 1940s. It has been updated and improved and still selling. It did two things for me. It gave me a feel for numbers including an introduction to infinity. Also, it was the beginning of a life-long interest in Einstein's theories of Special and General Relativity. That life-long interest expanded into Cosmology and Quantum Mechanics.

In high school I read a number of the popular books on Einstein. Understanding the Lorentz transformation formulas was a near spiritual experience. College was statistics and calculus. Graduate School brought me into advanced statistics and a start on Bayesian Statistics. For lack of a better way to say this, statistics gave me a love for number play.

This came to the forefront one day when Bill O'Reilly ("The Factor" on Fox) responded to an AAAS scientist who said that scientists do not regard present knowledge as absolute. O'Reilly smirked at the scientist and thought he 'owned' him by saying that, of course, there are absolutes. There are only 24 hours in a day (actually more and getting longer, I was thinking,) there are only 4 seasons in a year (arbitrary demarcations, I thought,) and 1 + 1 is always equal to 2 (not in Boolean math, and not in summation of near light speed velocities, I said to myself.) It was a wonderful moment. The man is an idiot.

Today I read the non-mathematical explanations of both Cosmology and Quantum Mechanics. My calculus is too old and too rusty to take me any further. Yet, I am intrigued by how much more I would understand if I took the time relearn and surpass my earlier mastery of calculus. Hmmm.

Due to non-use most of my science and math knowledge is now rusty or forgotten. Of everything that I have forgotten, I most regret the loss of calculus and statistics. Perhaps one of these days I should sit down and crack open an old text book and begin learning again.

Norm, why did you just show off your computational skills and not tell us the actual "tricks?"

Posted by: Ruchira | August 22, 2011 at 07:51 PM

@ Ruchira:

I was waiting for someone to ask. I'll tell you the second procedure. It is so easy it is embarrassing.

To multiply an arbitrary number of any length by 25:

1. You have to be able to look at the number. (If you can do this procedure by simply holding the number in your head, you will really amaze your friends and confuse your enemies.)

2. Mentally, place 2 zeros at the end of the number.

3. HERE'S THE MAGIC!!! Now divide the number by 4. Proceed as in long division, but you should be able to do the arithmetic and carry numbers in your head. A divisor of 4 is relatively easy to handle in your head.

Try this with the example above.

9,880,981,445

988098144500

4 Divided into 9 8 8 0 9 8 1 4 4 5 0 0

Give it a try.

9 8 8 0 9 8 1 4 4 5 0 0

_ _ _ _ _ _ _ _ _ _ _ _

Posted by: Norman Costa | August 22, 2011 at 08:17 PM

@ Ruchira:

Have you tried the above example. It hardly takes any practice at all, and friends and family will gaze in wondrous admiration - as long as you don't tell them your secret.

Do you want to try the multiplication of numbers ending in 5? It's a little more taxing on your memory and concentration and doing some calculations in your head. With a little bit of practice you will amaze your friends and confuse your enemies.

Posted by: Norman Costa | August 23, 2011 at 10:31 PM

I did try it, Norman. Thanks! It's a neat little trick. And yes, do tell me about multiplying numbers ending in 5.

Posted by: Ruchira | August 23, 2011 at 11:47 PM

Let me guess, to multiply arbitrarily large numbers by 5, you just add one 0 at the end, and divide the whole thing by 2?

Posted by: Sujatha | August 24, 2011 at 10:36 AM

@ Ruchira and Sujatha:

Sujatha, you are generalizing from the earlier example, correctly, if the multiplier is only a single digit, 5.

I have a more general solution when the numbers being multiplied have any length, but the units digit is 5, in both cases. However, the mental arithmetic beyond two or three digits is difficult.

Let's take the example of 165 x 35.

1. Both numbers end in 5.

2. Ignore, completely, the units digits 5 and 5.

Here's where do everything in your head.

3. Multiply 3 x 16 = 48.

4. Now add 3 + 16 = 19.

5. Take half 19, dropping any numbers after the decimal point. 19 / 2 = 9.5 (drop the .5 and you are left with 9)

6. Add 9 + 48 = 57.

7. All that is left is to determine the final two digits of the answer. If the sum of 16 + 3 = 19 is an odd number, the last two digits are 75. If the sum is even, the last two digits are 25.

8. So we concatenate 57 with 75. 5,775 is the answer.

This speed calculation becomes quite easy after only a little practice. I use it rarely, that's why I need about 15-20 seconds. I have to remind myself of what to do. Otherwise, I would knock them off in about 5-10 seconds.

The length of the numbers you can handle is dependent on your facility with mental multiplication and adding.

How about 155 X 155....24,025

How about 185 x 115....21,275

Posted by: Norman Costa | August 24, 2011 at 07:58 PM

Quick. What's 95 x 35?

Posted by: Norman Costa | August 27, 2011 at 08:22 AM