The Trachtenberg System of Speed Arithmetic
While he was in a German concentration camp, during
World War II, Jakow Trachtenberg invented his system
of speed arithmetic. His methods can be divided into
two parts, (1) multiplication by small numbers (2
through 12), and (2) addition, multiplication, and
division as other speed arithmetic experts do it. All
of this can be very helpful for any student, and has
been very successfully used by students who have
difficulty with numbers. In my opinion, the first
part, multiplication by small numbers, is successful
mainly because it is fun, not because it is a
particularly fast method. I will show how Trachtenberg
multiplied by 7, below. The one major exception to my
observation (that these are not particularly fast) is
multiplication by 11. For 11, Trachtenberg's method is
a dream. You, and everyone else, should consider using
it.
Multiplication by 11: Trachtenberg did this from right
to left (See the "note" below). I'll start with a
simple example (with no carries):
4253 x 11 = 46783
Starting at the right, we write down the first digit
(3). Then we add the first to the next digit (5) and
write that down (
. Then we add the second digit to
the third, etc. Finally we write down the left-most
(last) digit (4). In most cases, we have to deal with
carries:
4683 x 11 = 51513
Starting at the right, we write down the first digit
(3). Then we add the first pair of digits (8+3=11) and
write down the right digit of this sum (1), and carry
the one. We add the next pair of digits to the carry
(6+8+1=15) and write down the right digit of this sum
(5), and carry the one, etc. Finally, we write down
the left-most digit plus the carry.
A popular English language book on this subject is The
Trachtenberg Speed System of Basic Mathematics
translated and adapted by Ann Cutler and Rudolph
McShane. Click on the name of the book (above) to go
to that book on amazon.com.
Note: As I said above, Trachtenberg multiplied by 11
from right to left. It can be done from left to right.
This is slightly harder (sometimes), but has the
benefit of producing the answer from left to right,
just as you would read off the answer. And so, it can
be slightly faster, for that reason. Speed arithmetic
experts do most of their arithmetic (including
multiplying large numbers by large numbers) from left
to right.
Multiplication by 7: Going from right to left, we use
this rule: Double each number and add half the
neighbor (digit to the right, dropping any fraction);
add 5 if the number (not the neighbor) is odd. And of
course, we have to deal with carries:
3852 x 7 = 26964
Starting at the right (2), we double the first number
(it has no neighbor) and write down the right-most
digit of that (4) and we have no carry. Then we double
the next number (2x5=10), add five (+5=15), and add
half the neighbor (+1=16), and write down the right
digit (6) of that and carry the 1. Then we double the
next number (2x8=16), and add half the neighbor
(+2=1
, and add the carry (+1=19). Then we double the
next number (2x3=6), add five (+5=11), add half the
neighbor (+4=15), and add the carry (+1=16). Now we
double a zero off to the left of our 3852
(Trachtenberg wrote the zero out there: 03852) and add
half the neighbor (0+1=1), and add the carry (+1=2).
And we have our answer.
Notice that the carries are smaller than they were in
normal multiplication by 7. The above rule is not
simple, but once mastered, it is easy to use. It
should be about as fast as multiplying normally (which
requires memorizing the multiplication table).
Multiplication by other small numbers (3 through 12)
uses similar rules.
Addendum:
I'm going to try something here. I am going to put all
of the Trachtenberg rules for multiplying by 2, 3, 4,
5... up to 12 in a table. As in the above example
(3852 x 7=26964), we start on the right. The current
digit is called the active digit; I'll call it A. To
the right is the neighbor (if the active digit is on
the right, then the neighbor is 0), which I will call
N. Where it says + N/2, drop the fraction. Handle
carries just as in normal multiplication. Some of the
methods deal with the right digit differently than
other active digits. Most do not. After dealing with
the left digit as an active digit, deal with it as a
last digit. Here is the table:
multiply by right digit (R) other digits (A)
(including last digit) last digit on left (L)
2 x2.
3 10-R. x2. +5 if R is odd. 9-R. x2. +5 if A is odd. +
N/2. L/2. - 2.
4 10-R. +5 if R is odd. 9-R. +5 if A is odd. + N/2.
L/2. - 1.
5 N/2. +5 if A is odd. L/2.
6 A. +5 if A is odd. + N/2. L/2.
7 2A. +5 if A is odd. + N/2. L/2.
8 10-R. x2. 9-R. x2. + N. L-2.
9 10-R. 9-R. + N. L-1.
11 A+N. L.
12 Ax2. + N. L.
Long Addition
The major advantage of this method is that you procede
from left to right. Additionally the checking method
applied is quite different from the addition method
itself, so that mistakes are catched more easily.
Rule:
1. Allign the decimal points.
2. Add the left-most column first
3. Never count higher than 11 (when 11 is reached make
a dot or close a finger in the left hand and reduce
the right digit by one).
4. Add a leading zero to the digit and dot lines in
the answer (so you do not forget to compute the last
dot count.
5. For the final answer, from right to left add each
digit, the dot count below and the dot counts neighbor
(right neighbor).
Example:
9 4 2 1
+ 3 1 1 4
+ 1 3 8 2
Start down the left (9+3 is 12 reduce to 1 and one dot
+ 1 is 2) then second row (4+1+3 is
, third (2+1+8
is 11, reduce to 0 and one dot) and (1+4+2 is 7).
Your table will look like this:
9 4 2 1
+.3 1 1 4
+ 1 3.8 2
Digits 0 2 8 0 7
Dots 0 1 0 1 0
So from right to left : 7 + 0 + no dot to right = 7
0 + 1 + 0 = 1
8 + 0 + 1 = 9
2 + 1 + 0 = 3
0 + 0 + 1 = 1
Digits 0 2 8 0 7
Dots 0 1 0 1 0
1 3 9 1 7
Checking Addition
For checking we use the cloumn of numbers, Digit and
Dot totals and the answer.
If an error is found only that column needs to be
readded.
Rule:
1. Check the column of numbers (direction does not
matter - here left to right).
a) Add until you get a two-digit number, then add
the two digits together
b) Ignore any 9 digits.
c) Ignore any two touching digits that add up to 9
(Skip them both).
2. Check digits and dots. When you add the dot counts,
count them twice.
3. Check the answer. If problem check digit equals
both Digit/Dot check-digit and the answer check-digit,
the answer is correct.
Example:
9 4 2 1 - skip9,4+2=6,6+1=7 (7)
+.3 1 1 4 - 3+1+1+4=9, ignore (0)
+ 1 3.8 2 - 1+3+8=12 (1+2=3);3+2=5 (5)
Digits 0 2 8 0 7
Dots 0 1 0 1 0
1 3 9 1 7
7+5=12 (1+2=3); The check digit is 3.
9 4 2 1
+.3 1 1 4
+ 1 3.8 2
Digits 0 2 8 0 7 - 2+8=10 (1+0=1);1+0+7=8 (
Dots 0 1 0 1 0 - 1+1=2 doubled is 4 (4)
1 3 9 1 7
8+4=12 (1+2)=3; The check digit is 3.
1 3 9 1 7 - 1+3=4, ignore 9, 4+1+7=12, 1+2=3;
The check digit is 3.
All check digits are 3 - the answer is correct.
Introduction
The following methods can help to increase your
multiplication skill. Find out which method suits your
purpose most and practice. Please find the list of
abbreviations on the preceding site.
The word "neighbor" in the following paragraphs refers
to the digit at the right hand side of the digit you
are working on.
Number Multiplication
1. always place one zero in front of the first
multiplier for each digit of the second multiplier.
2. apply the rule acording to the second multiplier,
working through the digits of the first multiplier
from right to left
The rule will call for doubling or taking half. Taking
half of even numbers presents no problem. Uneven
numbers ("odd" numbers) are divided as: half of 1 is
0, half of 3 is 1, half of 5 is 2, half of 7 is 3,
half of 9 is 4.
For odd numbers the rule will sometimes call for a
special procedure.
Rules:
Multiply by Rule
11 Add the neighbor
12 Double, add the neighbor
6 Add half the neighbor plus 5 if number is odd
7 Double, add half the neighbor, add 5 if number
is odd
5 Take half of the neighbor, add 5 if the number
is odd
9 1. Last digit: subtract from 10
2. Middle digits: subtract from 9 and add the
neighbor
3. First digit: (the leading zero) subtract 1 from
the neighbor.
8 1. Last digit: subtract from 10 and double
2. Middle digits: subtract from 9 and double, add
the neighbor
3. First digit: (the leading zero) subtract 2 from
the neighbor.
4 1. Last digit: subtract from 10, add 5 if odd
2. Middle digits: subtract from 9, add half
neighbor, +5 if odd
3. First digit: (the leading zero): half the
neighbor less 1.
3 1. Last digit: subtract from 10, double, +5 if
odd
2. Middle digits: subtract from 9, double, add half
neighbor, +5 if odd.
3. First digit: (the leading zero) half the neighbor
and subtract 2.
Examples:
times 12
6452 x 12: (Double, add the neighbor)
006452
x 12
4 (2x2 + no neighbor)
2 (2x5 + 2) carry 1
4 (2x4 + 5 + 1[carry]) carry 1
7 (2x6 + 4 + 1[carry]) carry 1
7 ( 0 + 6 + 1[carry])
0 ( 0 + 0)
77424
6452 x 12 = 77424.
times 6
4958 x 6: (Add half the neighbor plus 5 if number
is odd)
04958
x 6
8 (8 + no neighbor, not odd)
4 (5 + half(
+ 5) carry 1
7 (9 + half(5) + 5 + 1[carry] carry 1
9 (4 + half(9) + 1[carry])
2 (0 + half(4))
29748
4958 x 6 = 29748.
times 9
4958 x 6: (1. Last digit: subtract from 10
2. Middle digits: subtract from 9
and add the neighbor
3. First digit: (the leading zero)
subtract 1 from the neighbor.)
052457
x 9
3 (10 - 7)
1 ( 9 - 5 + 7) carry 1
1 ( 9 - 4 + 5 + 1[carry]) carry 1
2 ( 9 - 2 + 4 + 1[carry]) carry 1
7 ( 9 - 5 + 2 + 1[carry]) carry 1
4 ( 5 - 1)
472113
52457 x 9 = 472113.
Direct Multiplication
This method is ideal for multipliation with a number
that consists of numbers between 1 and 3. It can be
used on any numbers, but with the second number higher
than 3 speed multiplication should be prefered.
2-digits multiply
1. multiply down right side
2. cross multiply
3. repeat using a pair of digits one to the left.
Examples:
24 x 23:
0024
x 23
2 multiply right
4x3) carry 1
5 cross multiply
2x3 + 4x2 + 1[carry]) carry 1
5 cross multiply
0x3 + 2x2 + 1[carry])
0 cross multiply
0x3 + 0x2)
552
24 x 23 = 552.
32344 x 32:
0032344
x 32
8 multiply right
4x2)
0 cross multiply
4x2 + 4x3) carry 2
0 cross multiply
3x2 + 4x3 + 2[carry]) carry
2
5 cross multiply
2x2 + 3x3 + 2[carry]) carry
1
3 cross multiply
3x2 + 2x3 + 1[carry]) carry
1
0 cross multiply
0x2 + 3x3 + 1[carry]) carry
1
1 cross multiply
0x2 + 0x3 + 1[carry])
1035008
32344 x 32 = 1035008.
3-digits multiply
1. multiply down right side
2. cross multiply the last two digits
3. cross multiply the last three digits
4. repeat using a triplet of digits one to the left.
Examples:
123 x 321:
000123
x 321
3 multiply right
3x1)
8 cross multiply
2x1 + 3x2)
4 cross multiply
1x1 + 2x2 + 3x3) carry 1
9 cross multiply
0x1 + 1x2 + 2x3 +1[carry])
3 cross multiply
0x1 + 0x2 + 1x3)
0 cross multiply
0x1 + 0x2 + 0x3)
39483
123 x 321 = 39483.
n-digits multiply
1. multiply down right side
2. cross multiply the last two digits
3. cross multiply the last three digits
4. cross multiply the last four digits.
(...)
n-1. cross multiply the last n-1 digits.
n. cross multiply the last n digits.
n+1. repeat using a "n-plet" of digits one to the
left.
checking
add the digits making up each multiplier and the
answer using following rules:
1. when the sum goes over 9 reduce the two digits
2. ignore any 9
3. ignore any two digits touching each other that
add up to 9
The digit sum of the multiplied numbers should equal
the digit sum of the answer
Example:
427691 x 918 = 392620338.
for 427691 figure:
4 + (2 touching 7 is 9, ignore) + 6 is 10 reduce to 1
+ (ignore 9) +1 is 2
for 918 figure:
(ignore 9) + (1 + 8 skip to 0) is 0
for 392620338 figure:
3+ (ignore 9) + 2 is 5 + 6 is 11 reduce to 2 + 2 is 4
+ 0 is 4 + 3 is 7 + 3 is 10 reduce to 1 +8 is 9 reduce
to 0
2 x 0 = 0.
Speed Multiplication
This method is ideal for multipliation of large
numbers. In this method all numbers are considered
two-digit numbers with a left-hand ´tens´ and a
right-hand ´units´ digit.
When multiplying the right hand digit we are anly
concerned with the units digit of the answer. With the
right hands digit we are anly concerned with the tens
digit of the answer.
Pair Product
To get the pair product of 47 x 4 procede as following
U T U T
4 7 4 7
x 4 x 4
16 28 6 2
^ ^
add 6 and 2 to get the pair product 8
The Pair Product needs to be well practiced to allow
you to write down the finished answer with no
intermediatre computation.
The big difference between the Speed Multiply and
Direct Multiply method is that with Speed Multiply the
highest number to carry from one pair product to the
other is a 1 or 2. This is why problems can be solved
mentally.
Single-digit multiplication
1. add the ´units´ of the left hand number and the
´tens´ of the right hand number (Pair Product)
2. each number is one digit to the answer of the
problem
Example:
4386 x 6:
04386
x 6
.
6 6x6 = 36
. .
1 8x6 = 48 and 6x6 = 36 3+8=1 carry 1
. .
3 3x6 = 18 and 8x6 = 48 8+4+1(c)=3 carry 1
. .
6 4x6 = 24 and 3x6 = 18 4+1+1(c)=6
. .
2 0x6 = 00 and 4x6 = 24 0+2=2
26316
4386 x 6 = 26316
Double-digit multiplication
Here we need to work with two numbers at a time. For
convenience mark the digits using your fingers. Here
these numbers will be market with dots ( .. ).
1. the left-hand working number is where the next
answer will be written.
2. the left-hand is also where you start multiplying
with the lower right hand number.
3. cross multiply the lower two numbers with the
current pair ( .. ).
a) multiply the lower-right number first
b) all we care about is the sum of the left-hand
´units´ and the right-hand ´tens´.
To go smoothly, it is essential to practice
Pair-Products.
Rule:
Cross multiply each bottom number twice. Once with the
crossed number, and once with the neighbor (right
neighbor).
Example:
84 x 62:
0084
x 62
..
0084
x 62
.
8 (2x4 = 0
+ (2x(no neighbor))= 8
no numbers for 6 to work with
..
0084
x 62
. .
0 (2x8 = 16) + (2x4 = 0
= 6
.
6x4 = 24 + 6x (no neighbor)= 4; 4+6=0 carry 1
..
0084
x 62
. .
2 (2x0 = 00) + (2x8 = 16)= 1
. .
(6x8 = 4
+ (6x4 = 24)= 10; 10+1+1(c)=2 carry
1
..
0084
x 62
5 only zeros for 2 = 0
. .
(6x0 = 00) + (6x8 = 4
= 4; 0+4+1(c)=5
5208
84 x 62 = 5208
This method needs a lot of pratice. Try 2-digit
numbers with numbers from 0-3. Once the method is
learned well proceed to higher and more numbers.
Triple (or more) digits multiplication
Try to get fluent in double-digit multiplication. For
triple-digit multiplication increase the number of
dots to three. Nevertheless we still do work
exculsively with the number and its neighbor. The
results-digits will derive from one more lane of cross
multiplications.
Procede accordingly for more-digits multiplication.
Division
Introduction
Trachtenberg division only requires addition and
subtraction. Unfortunately you need to write down a
lot of notes.
Rules with Example:
38,682,547 : 53
1. Write 1-10 in the left column
2. Next to the one write 53. Add 5 and 3 to get a
check digit. Write 8 in the check column.
3. Fill in the check column by adding 8 to itself in
each row. As before cancel out all 9s.
4. Fill in the divisor column by adding the divisor to
itself for each row. Veryfy each check digit with the
one written in the check column.
5. Write down the dividend in its column.
6. Make an empty column for the answer.
Digit Divisor Check Dividend Answer
38682547
1 53 8
2 106 7
3 159 6
4 212 5
5 265 4
6 318 3
7 371 2
8 424 1
9 477 0
10 530 8
7. Subtract the largest number you can use in the
divisor column to subtract from the dividend column.
Repeat.
Digit Divisor Check Dividend Answer
38682547 729859
1 53 8 371
2 106 7 158
3 159 6 106
4 212 5 522
5 265 4 477
6 318 3 455
7 371 2 424
8 424 1 314
9 477 0 265
10 530 8 497
477
20 (remainder)
Checking
All that needs to be checked is the subtrations used
to get the remainder.
Rule:
1. Subtract the remainder from the dividend and take
the digit-sum of the number you get.
2. Multiply the digit-sum of the answer by the
digit-sum of the divisor.
3. The digit-sums should be equal.
Example:
Dividend 38682547
Remainder -20
38682527 (digit sum is 5)
Answer 729859 (digit sum is 4)
Divisor 53 (digit sum is
4x8= 32 (digit sum is 5)
Square 2-digit
Rule:
1. Square the second digit
2. Multiply the two digits and double
3. Square the first digit
Example:
342
6 42 is 16; write 6, carry 1
5 3x4x2 is 24, +1(carry)is 25; write 5,
carry 2
11 32 is 9, +2(carry)is 11
1156
342 = 1156
Multiply just-under numbers
Example:
98
x 99
100 - 98 = 2 picture 98 2
100 - 99 = 1 picture 99 1
97 98 - 1 (upper left - lower right)
02 2 x 1 (multiply two right numbers)
9702
98 x 99 = 9702
Multiply just-over numbers
Example:
104
x106
104 -100 = 4 picture 104 4
106 -100 = 6 picture 106 6
110 98 - 1 (upper left + lower right)
24 4 x 6 (multiply two right numbers)
11024
104 x 106 = 11024
Square just-under numbers
Example:
962
100 - 96 is 4
92 96 - 4 is 92
16 4 squared is 16
9216
962 = 9216
Square just-over numbers
Example:
1062
106 - 100 is 6
112 106 + 6 is 112
36 6 squared is 36
11236
1062 = 11236
Squaring end-in-5 numbers
Example:
852
72 first digit x one larger (8x9)
25 always 25
7225
852 = 7225
Squaring tens-digit-5 numbers
Example:
562
31 25 + last digit (25+6)
36 sqare of last digit (62)
3136
562 = 3136
.....
good one !!! -
26-04-2005, 12:48 AM
Linear Mode
