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Touch and swipe

Drag the slider to generate the multiplication factors (aka the multiplicand and the multiplier).

Aka Ancient Egyptian Multiplication, displayed for the number range +100. The main advantage of this technique is that it makes use of only addition, subtraction, and multiplication by two.

- Create 2 columns for A and B
- Column A: halve A and repeat below until you get to 1 (ignore remainders)
- Column B: double and repeat until you get the same number of lines
- Strike out all lines with even numbers on the left
- Add up the remaining numbers on the right.

In mathematics, there are numerous methods and algorithms to perform arthimetic operations. Even for multiplication, a "basic arithmetic operation", there is a ton of methods and tricks. There is no best way since the method and strategy to tackle a multiplication problem depends on the tasks (large or small, integers, fractions or decimal, let alone many fixed and float formats for computer programs) and from the conditions and requirements, such such as speed, precision, skills and experiences (if a human) or computing power. A further criterion would be scalabilty. Elegant algorithms may become awkward in a higher number range. Others may become inefficient for small numbers because other ways. For instance, the Egyptian Method we are dealing with is inefficient for small number multiplications, such as 11 * 6, or for numbers near powers of two with simple multipliers, such as 254 * 11.

The Russian Multiplication works well in a number range up to 100, that is, for multiplying 2-digit numbers. In particular for numbers greater then 20, since in the range 11 to 19, there is another tricky way to multiply.

Often, the expressions "Russion Multiplication" and "Egyptian Multiplication" and used synonymously, but, to be precise, our subject is the Russian Multiplication, since in the Egyptian multiplication, columns are upside-down. Anyway, the ancient Egyptians are the inventors. Many of their mathematical methods were found on the Rhind Papyrus, which dates to 1550 BC.

There are numerous descriptions of how it is done and what is going on or, how to proof that the method really works. Turns out, not suprisingly, that the most concise and elegant proofs are those requiring a wider horizon, namely the knowledge about binary numbers.

Russian Multiplication - To multiply two whole numbers: [after MAA] - swapped multiplier and multiplicand

- Place each number at the top of side-by-side columns, with the multiplcand in the left column and the multiplier in the right column (...)
- In the multiplicand column, keep dividing by two, omitting the remainder, until you reach 1.
- In the multiplier column, keep doubling until 1 is reached in the multicand column.
- Add the entries in the multiplier column that are next to odd numbers in the multiplicand column to get the product.

On this page, you can try and test the Russian Multiplication. Drag the handles of the slider on top of this page, if you have not already done so. You will be able to watch the entire calculation in real time when to move the handles.

This app ist part of mathbydoing.app for practicing arithmetic operations [summary]

References and other sources

- "The algorithm instructs us to create a column beneath each of the multiplicands. We start by dividing the first number by 2 (and dropping the remainder if any) and recording the result in the first column. Then we divide by 2 the recorded number and write the result below. The process of division by two of the successive results continues until 1 is reached. In the second column we write the numbers obtained by successive multiplication by 2 that starts with the second multiplicand." Cut the Knot - Peasant Multiplication, by Alexander Bogomolny
- Howto (YouTube) Russian Peasant Method of Multiplication
- Proof and howto video mindyourdecisions, by Presh Talwalkar
- "The math wizards at Numberphile have brought back an age-old multiplying algorithm known as halves and doubles, peasant math, Egyptian math, or — as math presenter Johnny Ball describes it — Russian multiplying."popularmechanics.com with a video by numberphile
- "The multiplicative system ... is probably the very earliest mathematical system worthy of the name and was doubtless invented, reinvented and forgotten innumerable times ... Since it does not require any form of writing and involves only three operations, pairing off, halving and doubling, ... the system remained extremely popular with peasants the world over and became known as Russian Multiplication because, until recently, Russia was the European country with by far the largest proportion of innumerate and illiterate peasants." Mathematical Association of America: Russian Multiplication, Microprocessors, and Leibniz / Ethiopian Priests and Russian Peasants
- A-Z Teacher Stuff (Egyptian, not Russian version)
- "first, the combination of powers of two which add up to the number to be multiplied by was isolated, and then the corresponding blocks of counters on the other side yielded the answer."Story of Mathematics: Ancient Egyptian Number System
`for(p=0;p+=(-(a&1))&b,a!=1;a>>=1,b<<=1);`

Stackoverflow, Russian Peasant Multiplication- Interactive demonstration from 2011, similar to MathByDoing, but with Flash which is not supported anymore and won't run smoothly on Windows WOLFRAM Demonstrations Project
- researchgate.net Historical Methods of Mulitplication

This math trick lets you quickly multiply two numbers (10 to 19 each) in your head. The easiest way! Prerequisite: you should know your times tables reasonably well up to 10x10.

- Add A and the last digit of B
- Multiply by ten (add a "0")
- Multiply the two last digits and
- add the product. Done.