Speed Math

The Trachtenberg Speed System

Who was Trachtenberg?

  • Professor Jakow Trachtenberg was the founder of the Mathematical Institute in Zurich, Switzerland.
  • He was a Russian, born June 17th, 1888, and studied engineering.
  • While still in his early twenties, he became Chief Engineer with 11,000 men under his supervision.
  • After the Czar of Russia was overthrown, he escaped to Germany where he became very critical of Hitler. He was later imprisoned.
  • Most fellow prisoners around him gave up hope and died even before being sent to their death. He realized that if he wanted to stay alive, he had to occupy his mind with something else rather than focus on the hopeless conditions surrounding them. He set his mind on developing methods to perform speed mathematics.
  • With the help of his wife, he escaped from prison and fled to Switzerland.
  • There, he taught his speed math system to young children. It was very successful.

Trachtenberg developed a set of rules (algorithms) to multiply long numbers by numbers from 0 to 12. These rules allow one to dispense with memorizing multiplication tables if that is desired.
Even better, it gives a way to help memorize them, by allowing one to
work out the answer by rule if one cannot remember it by rote. We
perform each rule starting at the far right. The `number’ is the digit
of the multiplicand just above the place that we are currently
computing. The `neighbor’ is the digit immediately to the right of the
`number’. When there is no neighbor, we assume it is zero. We also
write a zero in front of the multiplicand.

Note that the following rules only use the operations of addition, subtraction, doubling, and `halving’.


Download the file trach.pdf for some examples. Also see the book: Trachtenberg, Jakow (1960). The Trachtenberg Speed System of Basic Mathematics. Doubleday and Company, Inc.

Speed Math via Vedic Mathematics

Vedic mathematics is based on sixteen sutras which serve as somewhat cryptic instructions for dealing with different mathematical problems. Below is a list of the sutras, translated from Sanskrit into English. They were presented by a Hindu scholar and mathematician, Jagadguru Swami Sri Bharati Krishna Tirthaji Maharaja, during the early part of the 20th century.

  1. By one more than the previous one
  2. All from 9 and the last from 10
  3. Vertically and crosswise (multiplications)
  4. Transpose and apply
  5. Transpose and adjust (the coefficient)
  6. If the Samuccaya is the same (on both sides of the equation, then) that Samuccaya is (equal to) zero
  7. By the Para-party rule 
  8. If one is in ratio, the other one is zero.
  9. By addition and by subtraction.
  10. By the completion or non-completion (of the square, the cube, the fourth power, etc.)
  11. Differential calculus
  12. By the deficiency
  13. Specific and general
  14. The remaining by the last digit
  15. The ultimate (binomial) and twice the penultimate (binomial) (equals zero)
  16. Only the last terms
  17. By one less than the one before
  18. The product of the sum
  19. All the multipliers

The first one is basically the multiplication algorithm by 11 discovered independently by Trachtenberg.

Let us look at the second one, which is used quite a bit in Vedic Mathematics: All from nine and the last from ten.

subtracting from a large power of ten with many columns of zeros, it is
not necessary to write the notation for borrowing from the column on
the left. One can instead subtract the last (rightmost) digit from 10
and each other digit from 9. For example:


the method is also used when finding the deficit from the next larger power
of ten when setting up a multiplication problem using the
cross-subtraction method.

The third one is Vertically and crosswise (multiplications). One use for this is for multiplying numbers close to 100.

you want to multiply 88 by 98. Both 88 and 98 are close to 100. Note
that 88 is 12 below 100 and 98 is 2 below 100. This can be pictured as


subtract crosswise to get the first two digits of the answer. It
doesn’t matter if we do 88-2=86 or 98-12=86, both give the same number.
To get the last two digits we multiply vertically: 12 x 2=24.
Therefore, the answer is 8624.

The same strategy works for
multiplying two numbers above 100. For example, 107 times 111. Quickly
we add the surplus from 107 (which is 7) to 111 to get 118, the first 3
digits of the answer. To get the last two digits, we multiply the
surplus of 107 from 100 by the surplus of 111 from 100: namely, 7 x
11=77. Thus, the answer is 11,877.

Vedic Mathematics is all about using different formulas in a variety of ways.
In the above rule we are using:
The above notation is short for:
and is often used since it’s easier to see what the number actually is.

above generalizes for numbers close to a base of 1000. Note that the
second sutra becomes quite useful for when you are computing the
deficit from the base.

Trick for computing mod 13

mod 13

  • 0. Take any non-negative integer in base 10; for example, 4523.
  • 1. Start at the most significant digit; in our example 4.
  • 2. Now follow the hollow arrow once; i.e., we go to 1.
  • 3. Now follow the solid arrows as many times as the next most significant digit; i.e., we follow the solid arrows 5 times to get 6.
  • 4. Repeat steps 2-4 until you are at the least significant digit; i.e., we follow the hollow arrow to 8, then the solid arrows 2 times to 10, the hollow arrow to 9, and finally the solid arrows 3 times to 12.
  • 5. Whatever number you end up at is your original number mod 13.
  • 6. Memorize the graph to impress your friends.
  • 7. Now figure out how to make your own for different bases and modulo different numbers.

Memorizing Pi – World Records and Techniques

How many digits of pi do you have memorized?

But seriously… is it 3? 5? 10? more than 30? If it’s more than 30 pat yourself on the back because that’s a great accomplishment! If it’s only a few, then no worries. Below we will teach you some techniques that can be used to conquer the digits of pi.

Computations of Pi

Some basic information and a brief time line on computations of the digits of pi:

  • 1540 – 1610: 35 digits determined
    • done by German mathematician Ludolph van Ceulen
    • used a geometric method (just like Archimedes did)
    • proud of his calculation that took a great part of his life
    • he had the digits engraved on his tombstone
  • 1949: 2, 037 digits computed (John von Neumann et al.)
  • 1973: Over one million digits computed
  • 1989: One billion digits computed (Chudnovsky brothers)
  • 2010: 2.7 trillion digits computed (F. Bellard)
  • In the near future: Almost all of them computed?

Who memorizes pi?

This is just a joke. It does bear a tiny bit of truth but the two sets of people aren’t mutually exclusive. I am both a scientist and a science fan!

Digits Memorized vs. Year (Graph)


Record Holders*: David Fiore
April 1st, 1979:

  • David Fiore wrote down 10,625 decimal places of pi
  • He was 18 years old at the time
  • He is known as the first person to ever break 10,000 decimal places
  • It took him three hours and five minutes

Record Holders*: Creighton Carvello (1944-2008)
June 27th, 1980:

  • Creighton Carvello recited 20,013 decimal places of pi
  • 2003: he recalled 3,500 facts about every FA Cup Final since 1872 (names of referees, goal scorers, teams, crowd attendances, scores, venues…)
  • Memorized the exact sequence of 10,000 words from Ernest Hemingway’s The Old Man and the Sea
  • Recited 17 random digits after seeing them for 2 seconds

Record Holders*: Rajan Mahadevan
July 5, 1981:

  • Rajan Mahadevan recited 31,811 digits of pi
  • He discovered his exceptional ability to memorize numbers at the age of 4 during a party hosted by his family
  • During the party, Rajan wandered to a parking lot and committed the license plate numbers of every guest’s car for recitation later
  • A quote: “I am not good at remembering words – words confuse my system of memorizing. Numbers, I have no problems at all. I put away huge numbers in something similar to a computer file and I can recall them even after decades.”

Record Holders*: Hideaki Tomoyori
March 10th, 1987:

  • Hideaki Tomoyori recited 40,000 decimal places of pi
  • Took him 17 hours 21 minutes (including breaks totaling 4 hours 15 minutes) to recite
  • Took him 10 years to memorize 40,000 decimal places

Record Holders*: Chao Lu
November 20th, 2005:

  • Chao Lu recited 67,890 decimal places of pi
  • Took him 24 hours 4 minutes to recite (with no breaks)
  • Took him 1 year to memorize 100,000 digits (he made a mistake at the 67,891th digit when going for the record)
  • He is the current (official) record holder
  • In 2006, Akira Haraguchi, a retired Japanese engineer, claimed to have recited 100,000 decimal places. This, however, has yet to be verified by Guinness World Records.

Unofficial: Andriy Slyusarchuk
June 17th, 2009:

  • A. Slyusarchuk claims to have 30 million digits memorized
  • The digits are printed in 20 volumes of text
  • He is a neurosurgeon, medical doctor and professor
  • He was able to recite randomly selected sequences from within the first 30 million places of pi
  • Reciting 30 million digits of pi at one digit a second would take 347 days (nonstop)
  • No officially documented attempt to debunk his claims has been successful as of yet

Why memorize pi? To beat Grace!
May 12th, 2008:

  • Grace Hare recited 31 digits of pi
  • It took her 18 seconds
  • She is 3 years old and the youngest record holder

How to memorize pi? Piems!
A piem is a (pi) poem where the length of each word represents a digit of pi
For example, the following piem encodes the string: 3. 141592 65358 9793 23846

I wish I could determine pi
Eureka! cried the great inventor.
Christmas pudding, Christmas pie
Is the problem’s very center.

Notice that:
Pie = 3;
I = 1; wish = 4; I = 1; could = 5; determine = 9; pi = 2;
Eureka=6; cried=5; the=3; great=5; inventor=8;
and so on. Thus, each word represents a digit of pi.

My favourite piems!
There’s over a bazillion piems and variations (lots and lots). The best ones are:

May I have a large container of coffee right now?

Hey, I need a large motorboat to rescue women and girls.

God! I need a drink,
Alcoholic of course,
After all those lectures
Involving radical equations.

3.1415 926 5358 979

Long Piems

  • The short story Cadaeic Cadenza encodes 3835 digits
  • It was written in 1996 by Mike Keith
  • Words of length 10 encode the digit 0
  • Words of length 11 (or 12) encode the two consecutive digits 1,1 (or 1,2)
  • 2010: In his book Not A Wake, Keith extends to 10,000 digits of pi

Technique: Grouping Digits

  • Split pi into small groups of digits (like 4 digits or 5, 6, 7, whatever you are comfortable with)
  • Focus on memorizing the first small group
  • Some people find singing it helps
  • When comfortable with the first group, move on to the next
  • Cons: If you lose your spot, you may have to start over.

Grouping Example: (3.14159) (26535) (8979323) (84626) (4338327) (95028) (8419716) (93993) (7510582) (09749) (4459230)

Start by memorizing (3.14159) for a minute… then add the next group (26535) and practice for two minutes. Then add the third group and practice until you are comfortable (REPEAT!!)

Classic Memory Techniques – The Major System

  • Major System: Convert numbers into sounds.
  • Sounds without numbers are used as ‘fillers’
  • Form words from the sounds
  • In practice, use 100 ‘peg words’: rat is 41; bar is 94

Classic Memory Techniques – Link System

  1. Start by converting each digit of pi to its corresponding phonetic sound
  2. Group sounds together to create a list of words
  3. Words created should be actions or objects
  4. Alternatively, use your ‘fixed’ peg words for the number
  5. Use the Link System: Link words together into a long chain by using a sequence of events, a story, or a journey. The CRAZIER the story the BETTER!!

Example: 3.14 15 92 —> 14 = door; 15 = doll; 92 = pan;

You are standing at the biggest door you have every seen.
You knock at the door and this Raggedy Ann doll answers.
Out of nowhere, she smacks you with a pan she is holding!

Coordinate Method

  • Pros: Can recite starting at any decimal spot (if you lose your spot, you don’t have to start over)
  • First 10 decimal places (1415926535) associated with 0
  • Use the Major System to encode as: turtle-pinochle-mall and link it to 0 (saw)
  • Example: Picture yourself using a saw to cut open a turtle who is playing pinochle at the mall
  • Next 10 digits (8979323846) would be linked to 1 in the same manner
  • Next 10 digits linked to 2
  • Repeat.
* Reference for World Record Holders: Pi World Ranking List and Wikipedia

Divisibility Tricks

Is the number N divisible by…. 2? 3? 5?

Everyone knows the first trick:
N is divisible by 2 if its last digit is 0, 2, 4, 6, or 8 (that is, last digit is even).

Most people know the next trick:
N is divisible by 3 if the sum of the digits is also divisible by 3.
You can repeat this rule too.

For example: Is the number 93,225 is divisible by 3? Well…

9+3+2+2+5 = 21

And, 21 is divisible by 3, hence 93,225 is divisible by 3.
N is divisible by 4 if the last two digist form a number divisible by 4.
do an example: Is the number 23894723985729316 divisible by 4? Well the
last two digits is 16 and 16 is divisible by 4, so YES!

N is divisible by 5 if it ends in 0 or 5.

For 6, we just combine the rules for 2 and 3:
N is divisible by 6 if it is divisible by both 2 and 3.

For the rest, we will stick with prime divisors p.
Consider multiples M of p until:

M*p+1=0 (mod 10)

We want the smallest such M.

n = (Mp+1)/10

Consider n and p-n, and usually we just pick the lowest.

to find out if a number is divisible by p, take the last digit of the
number, multiply it by n, and add it to the rest of the number (OR:
multiply it by (p – n) and subtract it from the rest of the number).

If you get an answer divisible by p (note that this includes 0), then
the original number is divisible by p. Repeat the rule if you don’t
know the new number’s divisibility.

Now try to see if you can come up with the rule for 7! One thing you might find interesting is the following post that discusses using a ‘divisibility graph‘ for 7.

How to Compute Cubed Roots Fast

Take a look at this video of Scott Flansburg on the Discovery Channel’s “More Than Human”:

Scott Flansburg takes cubed roots fast

In the video you see Scott Flansburg take the cubed root of 658,503 to get an answer of 87 in a matter of a second. How does he do it you ask?

This trick does require some memorization though, and also requires the
number given to be a perfect cube. You need to memorize the cubes of
the numbers 0 through 9 (or be able to figure them out on the spot).
This information is contained below:


that the last digits of the cubes on the right have all the numbers 1
to 9, but no number is repeated. Here is how to find the two-digit cube
root of a perfect cube.

Take a number, such as 658,503 which is grouped into two parts.

Looking at the number we see it ends in a 3, and according to the table
only 7^3 ends in a 3, thus the last digit of our number is 7.

Next, ignore the last 3 digits of the cube, so consider 658. Compare
these digits with the table above. Note that 658 fits between 512 and
729. You always choose the smaller one, in this case 512 which happens
to correspond to 8^3.

Thus, the last digit is 7 and the first digit is 8, giving an answer of 87.

this trick is used for six digit perfect cubes. To help understand how
this works, ask yourself – What is the last digit of (10x+y)^3? Clearly
it is y^3 mod 10 (how does this relate to #1?).

Another Example:
In 474,552 we have that 343 is the immediate smallest number from 474 so the first digit is 7.
The last digit in 474,552 is 2 and only 8^3 ends in a 2, so the last digit is 8. Hence, 78^3=474,552.

The Human Calculator – Scott Flansburg

  • Scott Flansburg was born in New York on December 28, 1963.
  • He served in the U.S. Air Force (1982-1988) and had a tour of duty with the Office of Special Investigations in Tokyo, Japan. He then returned to the United States and became an advocate of teaching math in an innovative fashion. 
  • In 1989 he drew the attention of Regis Philbin. Mr. Philbin is the individual that is credited for naming Scott, The Human Calculator during his appearance on The Regis and Kathy Lee Show. 
  • In 1991 he began working on a project called Turn On the Human Calculator In You, a series of tapes that was subsequently on Mike Levy’s Amazing Discoveries.
  • Turn On the Human Calculator In You was one of the most successful early infomercials touted as having in excess of 125 million viewings.
  • The follow-up product was called Mega Math with Kevin Tredeau (1996) which added a video and workbook and was later published by Tru Vantage International.
  • He has subsequently appeared on countless television shows including Oprah, Ellen, The Tonight Show with Jay Leno, Good Morning America, Discovery Channel’s `More Than Human’, and thousands of local radio and TV shows.

You can take a look at the infomercial here:
amazing discoveries calculator picture