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Trachtenberg Speed System of Mathematics 

Introduction

Mathematics can occasionally be challenging, particularly when multiplying enormous digits. But what if we tell you there's a method that can make multiplication more effortless and quicker? That’s where the Trachtenberg Method arrives.

Trachtenberg Speed System

The Trachtenberg System is a system of rapid mental calculation. The mental math system consists of a number of readily memorized operations that allow one to perform arithmetic computations very quickly. It was developed by Jakow Trachtenberg to keep his mind occupied while being held in a Nazi concentration camp.

The Trachtenberg Speed System is a mental math method designed to help perform fast arithmetic calculations, especially multiplication, by breaking complex problems into simple steps. This system makes math easier and more fun for kids, boosting their confidence and calculation speed.

Teaching Trachtenberg techniques at home can improve a child's mental agility and make math less intimidating. Islamic teachings also encourage gaining knowledge and using intellect wisely to benefit oneself and society.

The Trachtenberg System is a system of calculation that gives a fast calculation method since it is easier and more efficient than normal arithmetic. As one would expect, it is very beneficial to those individuals, especially students and working professionals, who wish to increase their speed and efficiency in multiplication, as well as addition and subtraction. This system has features that divide complex operations into simpler ones, and the rules and patterns can be readily learned and employed.

History of the Trachtenberg System

The Trachtenberg System is an approach named after Jakow Trachtenberg, who invented it while he was imprisoned in a concentration camp during World War II. Trachtenberg, who was an engineer by training, developed this system for his use as well as that of other prisoners to enhance their ability to do mental arithmetic. After he got out of the war, he developed and wrote about how he was using those techniques, and this led to the dissemination of the methods in educational systems across the world.

Basic Benefits of the Trachtenberg Method (In Simple Terms)

The Trachtenberg Method is established on a few major concepts that make mathematics more effortless:

Cracking Down on Big Calculations: 

Rather than doing a big calculation, the method breaks it into shorter, easier steps. This makes it more effortless to crack.

Clear Rules for Each Operation: 

Every sort of mathematical function (like multiplying or adding) has its own set of easy-to-follow rules that work for any number.

Focus on Speed and Accuracy: 

The method helps you solve math troubles quickly without making blunders.

No Need for Lots of Memorization: You don’t need to memorize multiplication tables or other math facts, as the approach facilitates the operation.

Why Use the Trachtenberg Method?

Speed: This method can be faster than traditional multiplication, especially with practice.

Mental math: It facilitates you in doing mathematics in your mind, which can hone your senses and enhance your memory.

No need for calculators: In cases where you don’t have a calculator, the method can be a valuable tool.

Rules of the Trachtenberg Method

Here are the rules, with examples listed below

Multiply by 1:

Multiplying by 1 does not change any number, so just copy down each digit of the given number

Examples

1.      7 × 1 = 7

2.      45 × 1 = 45

3.      128 × 1 = 128

4.      6001 × 1 = 6001

5.      1 × 999 = 999

(Nothing changes because multiplying by 1 keeps the number the same.)

Multiply by 2:

Simply multiply each digit by two.

Examples 

        1.      6 × 2

Simply multiply each digit (6) by two.

            =12

2.       23 × 2

Simply multiply each digit (2 & 3) by two.

2 × 2 = 4  & 3 × 2 = 6

Combine both and

    =46

     3.      47 × 2

Simply multiply each digit (4 & 7) by two.

7 × 2 = 14 → write 4, carry 1

4 × 2 = 8
Add carry: carry: 8 + 1 = 9
Combine 

    =94 Ans

     4.      125 × 2

Simply multiply each digit (1, 2 & 5) by two.

5 × 2 = 10 → write 0, carry 1
2 × 2 = 4; Add the carry: 4 + 1 = 5
1 × 2 = 2
Combine 

=250 Ans

Multiply by 3:

Rule: 

          1. Subtract the rightmost digit from 10.

    2. Subtract the remaining digits from 9.

    3. Double the result.

    4. Add half of the neighbor to the right, plus 5 if the digit is odd.

    5. For the leading zero, subtract 2 from half of the neighbor.

Example: 492 × 3

Working from right to left:

(10 − 2) × 2 + Half of 0 (0) = 16. Write 6, carry 1.

(9 − 9) × 2 + Half of 2 (1) + 5 (since 9 is odd) + 1 (carried) = 7. Write 7.

(9 − 4) × 2 + Half of 9 (4) = 14. Write 4, carry 1.

Half of 4 (2) − 2 + 1 (carried) = 1. Write 1.

Therefore 492 × 3 = 1476 Ans

Example: 674 × 3

Working from right to left:

(10 − 4) × 2 + Half of 0 (0) 

= 12. 

Write 2, carry 1.

(9 − 7) × 2 + Half of 4 (2) + 5 (since 7 is odd) + 1 (carried)

 = 12  

Write 2, carry 1

(9 − 6) × 2 + Half of 7 (3) + 1 (carried) = 10. Write 0, carry 1.

Half of 6 (3) − 2 + 1 (carried) = 2. Write 2.

Therefore 492 × 3 = 2022

Multiply by 4:

Rule:

1 Subtract the rightmost digit from 10.

2 Subtract the remaining digits from 9.

3. Add half of the neighbor, plus 5 if the digit is odd.

4. For the leading 0, subtract 1 from half of the neighbor.

Example: 346 × 4

Working from right to left:

(10 − 6) + Half of 0 (0) = 4. Write 4.

(9 − 4) + Half of 6 (3) = 8. Write 8.

(9 − 3) + Half of 4 (2) + 5 (since 3 is odd) = 13. Write 3, carry 1.

Half of 3 (1) − 1 + 1 (carried) = 1. Write 1.

= 1384

Example: 568 × 4

Working from right to left:

(10 − 8) + Half of 0 (0) = 2. Write 2.

(9 − 6) + Half of 8 (4) = 7. Write 7.

(9 − 5) + Half of 6 (3) + 5 (since 5 is odd) = 12. Write 2, carry 1.

Half of 5 (2) − 1 + 1 (carried) = 2. Write 2.

=2272

Multiply by 5:

Take “half” of the neighbor. If the number is odd, add five.

Example: 73 × 5

Half of 3’s neighbor, the trailing zero, is 0

Since  3 is odd, add 5 and write  5
Half of 7’s neighbor is 1, take 1

Since 7 is odd, add 5, 5+1=6. Write 6
Half of the leading zero’s neighbor is 3

So 73 × 5 = 365

Example: 86 × 5

Half of 6’s neighbor, the trailing zero, is 0
Half of 8’s neighbor is 3
Half of the leading zero’s neighbor is 4

So 86 × 5 = 430

Example: 59 × 5

Half of 9’s neighbor, the trailing zero, is 0 → 9 is odd, so add 5 → 5
Half of 5’s neighbor is 4 → take 4 → 5 is odd, so add 5 → 9
Half of the leading zero’s neighbor is 2

59 × 5 = 295

Multiply by 6:

Add “half” of the neighbor. If the number is odd, add five.

Example: 357 × 6

Working right to left:

7 has no neighbor, add 5 (since 7 is odd) = 12.

Write 2, carry the 1.

5 + half of 7 (3) + 5 (since the starting digit 5 is odd) + 1 (carried) = 14.

Write 4, carry the 1.

3 + half of 5 (2) + 5 (since 3 is odd) + 1 (carried) = 11.

Write 1, carry 1.

0 + half of 3 (1) + 1 (carried) = 2.

Write 2

So, 357 × 6 = 2142

Example: 579 × 6

Working right to left:

9 has no neighbor, add 5 (since 9 is odd)

9+0+5= 14. Write 4, carry the 1.

7 + half of 9 (4) + 5 (since  7 is odd) + 1 (carried)

7+4+5+1= 17. Write 7, carry the 1.

5 + half of 7 (3) + 5 (since 5 is odd) + 1 (carried)

5+3+5+1 = 11. Write 1, carry 1.

0 + half of 5 (2) + 1 (carried)

0+2+1 = 3. Write 3

      So, 579 × 6 = 3474

Multiply by 7:

Double each digit, and then add “half” of the neighbor. If the number is odd, add five.

Example: 693 × 7 = 4,851

Working from right to left:

(3×2) + 0 + 5 + 0 = 11 = carry 1, write 1.

(9×2) + 1 + 5 + 1 = 25 = carry 2, write  5.

(6×2) + 4 + 0 + 2 = 18 = carry 1, write 8.

(0×2) + 3 + 0 + 1 = 4 = write  4.

So, 693 × 7 = 4,851

Multiply by 8:

Righthand digit: Subtract the righthand digit from 10, and then double it.

Subtract the middle digits from nine, and then double it. Add “half” of the neighbor.

Subtract two from the left-hand digit.

Example: 456 × 8 = 3648

Working from right to left:

(10 − 6) × 2 + 0 = 8. Write 8.

(9 − 5) × 2 + 6 = 14, Write 4, carry 1.

(9 − 4) × 2 + 5 + 1 (carried) = 16. Write 6, carry 1.

4 − 2 + 1 (carried) = 3. Write 3.

Example: 678 × 8 = 5424

Working from right to left:

(10 − 8) × 2 + 0 = 4. Write 4.

(9 − 7) × 2 + 8 = 12, Write 2, carry 1.

(9 − 6) × 2 + 7 + 1 (carried) = 14. Write 4, carry 1.

6 − 2 + 1 (carried) = 5. Write 5.

678 × 8 = 5424

Multiply by 9:

Subtract the right-hand digit from 10.

Subtract the middle digits from nine, and then add the neighbor.

Subtract one from the left-hand digit.

Example: 2,130 × 9 

Working from right to left:

(10 − 0) + 0 = 10. Write 0, carry 1.

(9 − 3) + 0 + 1 (carried) = 7. Write 7.

(9 − 1) + 3 = 11. Write 1, carry 1.

(9 − 2) + 1 + 1 (carried) = 9. Write 9.

2 − 1 = 1. Write 1.

So, 2,130 × 9 = 19,170

Example: 4,350 × 9

Working from right to left:

(10 − 0) + 0 = 10. Write 0, carry 1.

(9 − 5) + 0 + 1 (carried) =5. Write 5.

(9 − 3) + 5 = 11. Write 1, carry 1.

(9 − 4) + 3 + 1 (carried) = 9. Write 9.

4 − 1 = 3, Write 3.

So, 4,350 × 9 = 39,150

Multiplying by 10

Rule: Add 0 (zero) as the rightmost digit.

Example: 7 × 10

Add 0 to the right of 7.

7 × 10 = 70

Example: 35 × 10

Add 0 to the right of 35.

35 × 10 = 350

Example: 402 × 10

Add 0 to the right of 402.

402 × 10 = 4020

Multiply by 11:

Add the digit to its neighbor. (By "neighbor" we mean the digit on the right.)

Example: 3,425×11=37,6753,425×11

= (0 + 3) (3 + 4) (4 + 2) (2 + 5) (5 + 0)

=3 7 6 7 5

Example: 3,425×11=37,675666666  66,789×11

= (0 + 6) (6 + 7) (7 + 8) (8 + 9) (9 + 0)

=7 4 6 7 9

Multiply by 12:

Starting from the rightmost digit, double each digit and add the neighbor. (The "neighbor" is the digit on the right.)

If the answer is greater than a single digit, simply carry over the extra digit (which will be a 1 or 2) to the next operation. The remaining digit is one digit of the final result.

Example: 316×12

Determine neighbors in the multiplicand 0316:

·         digit 6 has no right neighbor

·         digit 1 has neighbor 6

·         digit 3 has neighbor 1

·         digit 0 (the prefixed zero) has neighbor 3

6 × 2 = 12 (2 carry 1)

1 × 2 + 6 + 1 = 9

3 × 2 + 1 = 7

0 × 2 + 3 = 3

0 × 2 + 0 = 0

316 × 12 = 3,792 Ans

For rules 9, 8, 4, and 3, only the first digit is subtracted from 10. After that, each digit is subtracted from nine instead.

6×2=12 (2 carry 1) 1×2+6+1=93×2+1=70×2+3=30×2+0=0316×12=3,792

The Trachtenberg Method is just one part of the larger Trachtenberg System. The system extends beyond multiplication to division, addition, and subtraction.

Conclusion

The Trachtenberg Speed System provides kids with quick, easy mental math methods that make multiplication and calculation fun and less intimidating. It improves focus, speed, and confidence in math. Combined with the Islamic emphasis on seeking knowledge and mental growth, parents can help children learn these techniques at home, fostering a love for numbers and problem-solving.