Speed Arithmetic – Multiply by Twelve

Introduction · Development · Rule for multiplying by twelve (harder version) · Rule for multiplying by twelve (easy version) · Carrying a number · Exercizes · Recommended Books

Introduction

As in a previous article, we explore the connection between algebra and arithmetic, this time by applying algebra to get a new, faster way to multiply a given number by twelve.

This method was originally due to Jacow Trachtenberg. We recommend reading the previous article and mastering the material in it before moving on to the present one, since much in the previous article will not be repeated here and will be assumed.

Development

As before, since

12 = 10 + 2

we can multiply by ten, multiply by two and add the results instead of multiplying by twelve. Thus, our preliminary rule for multiplying by twelve would read:

Rule for multiplying by twelve (harder version):

Multiply by two, multiply by ten, add the results

Example

Multiply 142 by 12.

Solution

Multiply by two:

Multiply by ten:

Notice that the addition step required a carry of 1.

Let’s take a very simple example and analyze exactly how the digits are found. Suppose we want to multiply 31 by twelve. Applying the rule gives

Multiply by two:

Multiply by ten:

Here’s how we got each digit, in order from the least significant on the right to the most significant on the left:

Writing these next to each other in decreasing order of significance looks like this:

In words, we multiplied the 1 by 2 and brought down the result, 2. Then we multiplied the 3 by 2 and added the 1 that was next to the 3 to get 7. Finally, we brought down the 3. If we set this problem up the way we did multiplication by 11 and do it step-by-step, the results will look like this:

Set up the problem

Now multiply the 1 by 2, and write down the result, 2:

Next, multiply the 3 by 2 and add the 1 to it to get 7:

Finally, bring down the 3 (actually, multiply 0 by 2 and add 3, same thing):

So, we have found that 31 · 12 = 372 and we are done.

We’re now ready to state our simplified rule for multiplying by twelve.

1. Double the first digit on the far right and write it down
2. For each next digit, double it, add its neighbor to the right, and write the result down to the left of the last digit found
3. Write down the digit on the far left to the left of the last digit found

Example

Multiply 123214 by 12.

Solution

Set up the problem as above:

Double the first digit on the far right (2 · 4 = 8) and write it down:

Double the next digit (1), add its neighbor to the right (4), and write the result (1 · 2 + 4 = 6) down to the left of the last digit found:

Double the next digit (2), add its neighbor to the right (1), and write the result (2 · 2 + 1 = 5) down to the left of the last digit found:

Double the next digit (3), add its neighbor to the right (2), and write the result (3 · 2 + 2 = 8) down to the left of the last digit found:

Double the next digit (2), add its neighbor to the right (3), and write the result (2 · 2 + 3 = 7) down to the left of the last digit found:

Double the next digit (1), add its neighbor to the right (2), and write the result (1 · 2 + 2 = 4) down to the left of the last digit found:

Last, write down the digit to the far left (1) to the left of the last digit found:

We are now done and have found that 123214 · 12 = 1478568. In practice, we do the entire calculation in "one" step so that what gets written down is just

Carrying a number

As with multiplying by 11, we’ll often run across situations where we get a number greater than 9. This number can be greater than 19, but will never be greater than 29, so we’ll have to carry a 2 at most.

Example

Multiply 9876 by 12.

Solution

Set up the problem:

Double the digit on the far right (6 · 2 = 12) and write down the 2, carrying the 1:

Double the next digit (7), add the neighbor on the right (6) and the carried 1 (7 · 2 + 6 + 1 = 21); since the result is 21, write down the 1 and carry the 2:

Double the next digit (8), add the neighbor on the right (7) and the carried 2 (8 · 2 + 7 + 2 = 25); since the result is 25, write down the 5 and carry the 2:

Double the next digit (9), add the neighbor on the right (8) and the carried 2 (9 · 2 + 8 + 2 = 28); since the result is 28, write down the 8 and carry the 2:

Write down the last digit (9) plus the carried 2 (9 + 2 = 11):

So we found that 9876 · 12 = 118512 and we’re done.

Exercizes

Try these multiplications yourself, then make up more and check the correct answers with a calculator.

1. 8,475,630 · 12 = 101,707,560
2. 123,456,789 · 12 = 1,481,481,468
3. 99,998,796 · 12 = 1,199,985,552
4. 134,245,683 · 12 = 1,610,948,196
5. 1,111,111,111 · 12 = 13,333,333,332
6. 25,834,009 · 12 = 310,008,108
7. 777,888,999 · 12 = 9,334,667,988
8. 305,080,709 · 12 = 3,660,968,508
9. 89,089,089 · 12 = 1,069,069,068
10. 55,956,656 · 12 = 671,479,872

Recommended Books

The original, 1960 book that first brought Dr. Trachtenburg's work to the world's attention.  Well written, brief, with plenty of examples.

An inexpensive Dover book that covers the same kind of topics. I haven't read this one (yet), but I have many other Dover titles, and am always happy with them.