Different Methods of Multiplication

Everyone knows how to multiply column-wise, fewer people know about multiplication with lines, but there are other interesting methods too.

Multiplication is a very simple operation, in fact, it’s essentially the same as addition. Of course, this holds true until the numbers get larger.

2×3 = 2 + 2 + 2 (three times two) or 24×6 = 24 + 24 + 24 + 24 + 24 + 24 (six times 24)

So, is knowing the multiplication table not necessary? Yes, but it’s more convenient with it. For instance, with the multiplication of 235×4596, you would have to add 4596 a total of 235 times! Or, alternatively, add 235 a total of 4596 times…

The word “add” is used for a reason. Here’s a simple way to confirm this. Take a piece of paper, fold it 5 times in one direction, and then fold it 3 times in the other. This results in a 5×3 operation. Count the resulting rectangles from the folds — there are 15. This is the same as if you took 3 strips of fabric (or anything else) of length 5 and combined them.

No matter how you look at it, the result is 15!

Unusual Ways to Multiply

At school, we are taught to use two tools: the Pythagorean table (it’s believed that this multiplication table was invented by the Greek mathematician Pythagoras) and column multiplication. Are these really the most efficient tools? There are other interesting ways to multiply numbers. Maybe one of them will be easier, and you won’t have to memorize the multiplication table?

Peasant Method

This was used to determine the area of a piece of land. For example, let’s say we have a field that is 6 units long and 5 units wide.

To find out what 6×5 is, we do the following: divide the left number by 2, and multiply the right number by 2, until the left number becomes 1.

4 | 5

4/2=2 | 5×2=10

2/2=1 | 10×2=20

4×5=20, which is correct, just like 1×20=20.

What happens with this method? We split the rectangle in half until its width becomes 1. Dividing by 2 is not hard.

But what if one of the sides cannot be divided by 2? The process will be long and not as simple.

6 | 2 → 12

6/2=3 | 2×2=4 → 12

3/2=1.5 | 4×2=8 → 12

1.5/2=0.75 | 8×2=16 → 12

If there is an even number in the left part — skip that line, and if the value is less than one — disregard it as well, leaving the second and third lines, which result in 8+4=12. But what if you need to multiply 173 by 735? No, this method of multiplication is not the easiest and simplest.

Eastern Method

Either the Chinese or Japanese multiplication method, also called “graphical”. The idea is that the digits of the first number are represented by parallel lines, while the digits of the second number are perpendicular to them. The number of intersections equals the result of the multiplication. That means you don’t need to know the multiplication table, just the ability to sum. For example:

2 × 3 and even 15 × 12

Japanese or Chinese method, the essence is the same

How Does Line Multiplication Work?

The first number (purple in the image) is drawn from bottom to top, left to right, starting with thousands, then hundreds, tens, and units. The second number (blue in the image) is drawn in the opposite direction: from top to bottom.

In the first example, it’s simple: 2 and 3. Two lines intersect with three others, resulting in 6 points. In the second example, we first draw 15 — one line for ten (one ten), then five lines representing 5 (five units). Then (12) is drawn perpendicularly to it with the second one and 2 lines.

Then, we count the intersections in the opposite direction, starting from the right. In this case, it is 10, 7, and 1. The result is summed up vertically:

10

7

1

180

Compared to the traditional “column” method, it may initially seem that the Japanese-Chinese method is simpler…

Line multiplication method

But what do you do if you need to multiply 10 by 12? How do you represent “zero” with a line? You can’t, it doesn’t participate, but you can draw it as a dashed line and not count the intersection. Simple…

But if we need to multiply 853×951, drawing and counting the points will take quite a while. The good old column method will again prove to be more convenient. Anyone can try multiplying 9878 and 8794 using the “Japanese method” and time how long it takes.

Chinese multiplication method with zero

This technique is not universal, it’s not suitable when the numbers are large, but it’s very easy to explain to young children who don’t yet know the multiplication table.

Shutters

This is also known as “grids” or the Indian method of multiplication. It’s easiest to believe in the Indian origin if you recall who invented your mathematics in ancient times. So, to multiply two numbers, you need to create a matrix (or table, we’re trying to keep it simple).

Multiply 45 by 82

Since each number has 2 digits, the table will be 2×2. Each cell must be crossed out diagonally. Then, we write the digits 4, 5, 8, 2 from left to right, and top to bottom, opposite each cell. We begin by multiplying the digits across from each other: 4 by 8, 5 by 8, 4 by 2, and 5 by 2.

Well, here’s the multiplication table again, otherwise you’ll be adding numbers for a long time.

The results are recorded in the cells cleverly, with tens above the diagonal and units below it. But if the value is less than 10 (i.e., it’s one digit rather than two), instead of writing the tens above, we write “zero” or leave the space empty.

Now, to find the result, you need to sum the numbers in each diagonal, as shown in the picture. From top to bottom:

3

0+2+4=6

8+1=9

0

The result is 3690.

This is quite simple, but for three-digit numbers, you’ll have to draw a 3×3=9-cell table.

How to multiply with column method?

And now, we come to the traditional method long multiplication or column multiplication. This is how we’re taught to multiply large numbers in school. How does it work, and is it simpler than all the exotic methods? Write the two numbers one below the other: 23×12:

23

12

—

46

23

276

We start multiplying “from the end”. Take the last digit of the second number, 2. Multiply it by 3, getting 6, and by 2, getting 4. Write them down from the end, 46. Do the same with the second digit, 1.

Multiply 1 by 3 and by 2. Write 23 under 46, shifting one position to the left so that the 6 is empty and the 2 lines up with the 4.

All that’s left to do is add up all the digits, starting from the top down: 6+0, 4+3, 0+2. If any pair of digits results in more than 9, carry over the remainder. So, 4+3=7, but if we added 5+6=11, we’d write down 1 and carry over 1 to the next column.

You probably know how to do all this already, but now, knowing all the other multiplication methods, you can judge whether column multiplication is easier than all the other methods.

Which Multiplication Method is Best?

After trying all the multiplication methods, it becomes clear that all the alternative methods are just variations of the familiar “column method”. The operations break down into smaller parts: first multiplication, then addition.

Only in the so-called Chinese/Japanese method does multiplication as such not occur (instead, it’s the intersection of lines), and this method really allows you to avoid using the multiplication table, but you’ll have to do a lot of drawing, which is not easy.

The Indian method is a bit simpler in that you don’t need to draw as many lines as in the Chinese one, but it still requires quite a bit of visual thinking.

If you’re going to multiply numbers by hand, it makes the most sense to use the column method. It is simple and convenient, and requires less effort to learn. All the other methods are good, but in everyday life, multiplication is usually done with a calculator, and this traditional approach works best. However, using any of the methods above can be useful for your children as it develops logical thinking.