Duplation
The Duplation (from Latin duplare , doubling ' ) was once one of the basic arithmetic , but is no longer used today.
Basics
Duplation is another form of multiplication , but only multiplied by a factor of 2. In this way it is possible to solve multiplication problems by addition . However, the duplation is very time-consuming and labor-intensive, especially with larger numbers, which is why it is no longer taught today.
Historical
Duplation was already used in ancient Egypt. Since the Egyptian number system had no sign for the 0, today's methods of multiplication could not be used. Instead, the Egyptians developed their own methods. On the Rhind papyrus , named after the Scot Alexander Henry Rhind , which dates from around 1700 BC. Various examples of it have survived. In addition to duplation, the process later called Russian peasant multiplication should be mentioned.
Since the Roman number system was structured similarly to the Egyptian and therefore had the same problems, the duplation was continued for a long time. It was still taught by default in the Middle Ages. It was not until the 13th century that Arabic numerals became established in Western Europe that duplation gradually disappeared.
functionality
If you want to solve a multiplication problem with the help of duplation, you create a table with two columns. In the left column one of the two factors of the task is entered, in the right column the number "1". Now you multiply both numbers by the factor 2 or add them to yourself. The result is entered in the table in the next line. Now you continue by multiplying these numbers by 2 or adding them to yourself. Again you enter the results in the next line. The row "1, 2, 4, 8, 16, 32, ..." results every time in the right column. You continue with this process until the second factor of the original task, which was not entered in the table, can be formed by adding different numbers from the right column. It does not matter which numbers in the right column are used, it is only important that they result in exactly the other factor. As soon as you have found these numbers, you also add the numbers that are in the same row in the left column. The result of this addition is also the solution of the original problem.
The best way to explain how it works is with a calculation example.
Calculation example
In this example, task 14 * 21 is to be solved.
version 1
In variant 1, factor 1 (14) is entered in the left column of the table and factor 2 (21) is to be formed by the numbers in the right column.
| Left column | right column |
|---|---|
| 14th | 1 |
| 28 | 2 |
| 56 | 4th |
| 112 | 8th |
| 224 | 16 |
The table can end at this point, as factor 2 can now be formed using the numbers in the right column, since:
1 + 4 + 16 = 21 .
According to the rules of duplation, the numbers in the left column that are in the same row as the previous numbers are now added up.
In this example that means:
14 + 56 + 224 = 294 .
This means that 14 * 21 = 294.
Variant 2
In variant 2, factor 2 (21) is entered in the left column of the table and factor 1 (14) is to be formed by the numbers in the right column.
| Left column | right column |
|---|---|
| 21st | 1 |
| 42 | 2 |
| 84 | 4th |
| 168 | 8th |
The table can end at this point, as factor 1 can now be formed using the numbers in the right column, since:
2 + 4 + 8 = 14 .
According to the rules of duplation, the numbers in the left column that are in the same row as the previous numbers are now added up.
In this example that means:
42 + 84 + 168 = 294 .
This also shows that 14 * 21 = 294.
Variation 3
Since this method is very complex, especially for multi-digit tasks, you can multiply by a factor of 10 in intermediate steps.
As in variant 2, factor 2 (21) is entered in the left column and factor 1 (14) is to be formed by the numbers in the right column.
| Left column | right column |
|---|---|
| 21st | 1 |
| 210 | 10 |
| 42 | 2 |
| 84 | 4th |
The table can end at this point, as factor 1 can now be formed using the numbers in the right column, since:
10 + 4 = 14
According to the rules of duplation, one now also adds up the numbers in the left column that are in the same row as the previous numbers.
In this example that means:
210 + 84 = 294
This also shows that 14 * 21 = 294.