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Speed Arithmetic

© Copyright 1998, Jim Loy

Addition of two numbers:

    2166
    9327
    ----

Let's add these two numbers in our heads (i.e., without paper). Can you do that? Our first attempt is to do it like most of us do on paper: 6+7=3 carry the 1 (13), 1+6+2=9, 1+3=4, 2+9=11. The answer is . . . now what were those numbers? The problem here is memory, not mental arithmetic. That's why people use paper (or an abacus, or their fingers), to help out their memories.

Memory is why speed arithmetic experts (I call them "arithmetickers") usually add numbers like these from left to right. It is a little more complicated that way (you have to back track). But, you end up saying the answer from left to right, just as it is normally said. It is easier to remember a number from left to right.

Let's try again: 2+9=11, 1+3=4, 6+2=8, 6+7=3 and that previous 8 should have been a 9 (because of the carry). I actually remembered the answer, 11493, that time. It's still a test of my memory, but not bad. It may take you a little practice to be able to do that.

Addition of columns:

    29
    37
    15
    21
    32
    85
    44
   ---

How about this addition problem? Add these up in your head. It's not too tough to add up the right column, remember the carry, and add up the left column, just as you would do with a pencil.

In the right column, a speed arithmetic person might group the 9 and the 1 (10), the two 5's (10), and then the 7+2+4 (13) to get 33. Something similar works for the left column. A few people group elevens instead of tens (see The Trachtenberg System of Speed Arithmetic).

Instead, what I do is add 29+37=66, then 66+15=81, then 81+21=102, 102+32=134, 134+85=219, and 219+44=263. Isn't that slower? Maybe. But I have little to remember, just the sum so far. It becomes very fast, if you practice doing it that way. That is how a person with an abacus (or a person counting with fingers, as in Chisanbop) would do it. And an abacus is just a way of remembering the latest sum. That's something you can easily do without an abacus.

Multiplication:

    24
   x48
  ----

Speed arithmetic people multiply from left to right, too. No matter how you do it, there's plenty to remember here. Let me show you how this is done. It takes practice to get good at it. But, just watch me do this:

    2x4 =   8
    2x8 =   16
           ----
            96
    4x4 =   16
           ----
           112
    4x8 =    32
           ----
           1152

See how that is done? The order in which this is done is the trick: left digits, opposite corners, right digits. Bigger numbers can be done in a similar, but more complicated way.

Why?

Why should a person learn to do this? Is it just a way to show off? No, it helps a person in math and sciences. It makes a person more accurate (even when using paper or a calculator). You won't get bogged down doing the arithmetic. You can get on to other, even more interesting ideas.

Addendum (tricks):

Older speed arithmetic books dwelt almost exclusively on tricks. Here are some of those tricks (which you can deduce on your own, instead of memorizing this table):

• Multiply by 5: Multiply by 10 and divide by 2.
• Multiply by 6: Sometimes multiplying by 3 and then 2 is easy.
• Multiply by 9: Multiply by 10 and subtract the original number.
• Multiply by 11: See The Trachtenberg System of Speed Arithmetic for a very easy method.
• Multiply by 12: Multiply by 10 and add twice the original number.
• Multiply by 13: There is no easy trick method. Multiply by 3 and add 10 times original number.
• Multiply by 14: Multiply by 7 and then multiply by 2 (or vice versa, whichever seems easier).
• Multiply by 15: Multiply by 10 and add 5 times the original number, as above.
• Multiply by 16: You can double four times, if you want to. Or you can multiply by 8 and then by 2.
• Multiply by 17: There is no easy trick method. Multiply by 7 and add 10 times original number.
• Multiply by 18: Multiply by 20 and subtract twice the original number (which is obvious from the first step).
• Multiply by 19: Multiply by 20 and subtract the original number.
• Multiply by 24: Multiply by 8 and then multiply by 3. A similar method works for other numbers that can be factored, like 32 or 45 or many others.
• Multiply by 27: Multiply by 30 and subtract 3 times the original number (which is obvious from the first step).
• Multiply by 45: Multiply by 50 and subtract 5 times the original number (which is obvious from the first step).
• Multiply by 90: Multiply by 9 (as above) and put a zero on the right.
• Multiply by 98: Multiply by 100 and subtract twice the original number.
• Multiply by 99: Multiply by 100 and subtract the original number.

There are a lot more tricks for multiplication, division (divide by 5 by multiplying by 2 and dividing by 10), addition, subtraction, and squaring.

Some books on speed arithmetic are (click on a book to go to the amazon.com page for that book):

• Math Magic - by Scott Flansburg.
• Rapid Math Tricks and Tips - by Edward H. Julius.
• The Trachtenberg Speed System of Basic Mathematics translated by Ann Cutler and Rudolph McShane.
• Miracle Math - by Harry Lorayne (author of The Memory Book), not listed at amazon.com.
• How to Calculate Quickly - by Henry Sticker
• Doing Simple Math in Your Head - by W. J. Howard

Square a two-digit number that ends in 5 (like 85) by multiplying the left digit by the next highest number (8x9=72), and tack on "25" on the right (7225). Square any two digit number (like 87) by squaring the left digit and tacking on two zeros (6400), double the product of the two digits and tack on one zero (1120), add those (6400+1120=7520), square the right digit and add (7520+49=7569). You don't need to tack on zeros, if you can keep the columns straight. This is just an application of the multiplication method described above.

When you multiply a number by two, you can do it just as you do it on paper, but do it in your head. You just double each digit, remembering to add a carry when that comes up. You should be able to double almost anything in your head. Dividing by two should be almost as easy.

Also see Day Of The Week Of Any Date.

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