Indian multiplication
Multiplication 435 times 12.
We draw a table with three columns and two rows, which we mark with multiplicand and multiplier digits. Divide each window of the table with a diagonal into two triangles. We fill in the table by multiplying the digits for each cell, which indicate the rows and the column in which the cell is located.
If the number is less than ten, we write it in the lower triangle. If the result is greater than or equal to ten, we write tens in the upper triangle and units in the lower triangle.
Finally, we read the numbers from right to left along the diagonals. We compile units and remember dozens and transfer them. The result is 5220.
Chinese graphic multiplication
It developed thanks to the Chinese love for calligraphy and painting, and its subsequent application in mathematics in multiplying smaller numbers.
Multiplication 123 times 21.
We draw one parallel for hundreds, two parallels for tens and three parallels for units for multiplicands in the direction from SW to NE. Then we draw two parallels for tens and one for units for the multipliers from NW towards SE. Then we enclose the intersections corresponding to thousands, hundreds, tens and units in disjoint rectangles and find out their numbers.
In this case we have 2 intersections in the first rectangle, 5 in the second, 8 in the third and 3 in the fourth. Result 2583. In this way we would continue to multiply larger numbers. If the sum of the intersections in a rectangle came out to be greater than 9, we would add the number of tens to the next rectangle.
Gypsy multiplier
– used by medieval merchants
– allows to multiply by fingers up to 9 times 9 (condition is knowledge of multiplier up to 5 times 5)
Multiplication 8 times 7.
Procedure: „Eight and what is ten?“ „And two.“ We bend two fingers on the first hand (c = 2). We’ll do the same with number seven. So we hide three fingers on the other hand (d = 3). At the position of tens we write the sum of erect fingers (a + b = 3 + 2 = 5) and at the position of units we write the product of crouched fingers (c x d = 2 x 3 = 6). This is the correct result 56.
This procedure is only applicable to multiplying numbers that are both greater than or equal to 5, and that works with the following equations:
(10-c)*(10-d)=100-(c+d)*10+cd,
=10*(10-c-d)+cd,
=10*(a+b)+cd.
There are pitfalls in some calculations. For example, at a multiplication of 7×6, c x d = 3×4 = 12.
That’s more than 10. In this case, we leave the number 2 in place of the units and transfer 1 to the tens.
Figure 2 shows the procedure for multiplying 14 x 9. If we want to multiply these numbers, we crouch the fourth finger on the left hand – the ring finger. From the left, the first finger – thumb – represents hundreds (a = 1), the remaining fingers before the crouch represent tens (b = 2) and the fingers following the crouch represent units (c = 6). Result 126. The algorithm works:
Multiplying (10+ d) times d, where d = 2,… .., 9, holds:
(10+d)*9=90+9d,
=100+10*(d-2)+(10-d),
=100a+10b+c.
Chinese counting on fingers
In China, only one hand is used to show numbers 1-10. Some symbols are the same as us, others are related to the Chinese character.
Interesting
The trouble can happen when you raise your thumb in China and want to order one apple. This is a fundamental mistake, because this gesture – meaning number one to us – is something completely different in China: with that thumbs up, you have shown the Chinese that he is stupid or worse.
For the clever
Multiply two identical numbers by selecting a factor with the left foot, and setting the right foot to a square symbol.
The number 12 indicates a dozen. 12 dozen (12 times 12) is called a gross. Can you use a multiplication monkey to calculate how much is a gross?
Why is the square symbol used for the squared numbers?