Section 30.5 Going Deeper: Alternative Multiplication Techniques
On Worksheet 5, we looked at how the standard multiplication algorithm (multiplication in columns) has all of the same components as looking at multiplication using the grid method we used for algebra. This grid method also has a connection to the base-10 diagrams that were used on Worksheet 4. We're going to look at a couple other multiplication methods. The first is another representations of the same idea, and the second one is a surprising method discovered by the ancient Egyptians that uses only multiplication by with addition.
Subsection 30.5.1 The Lattice Method
The lattice method is an organizational scheme for calculating products. This method is convenient because it tracks the place values for you, so that you don't need to keep track of the trailing zeros in your numbers. Interestingly, this method of calculation was independently discovered by Arab, European, and Chinese mathematicians.
To use the lattice method, you start by creating a grid where the number of columns is the number of digits in the first number of the product, and the number of rows is the number of digits in the second column. Then write the digits of the numbers in the correspond positions. Lastly, draw the up-right diagonals through all of the boxes. Here is an example:
From here, you write out the products of each pair of numbers in the corresponding square, where the tens digit goes in the upper-left region and the ones digit goes on the lower-right. If your product gives a one-digit number, use a in the tens digit.
Next, you add along the diagonals down and to the left. Start from the lower-right and work your way to the upper-right. If you end up with a number larger than carry the tens digit to the next diagonal. The final answer is the collection of digits read from the top-left down, then to the right.
Although the presentation of this calculation feels significantly different, it's actually the exact same concept that we used with the grid. To see the connection, we simply need to rearrange how we calculate the products. The primary distinction between these two methods is that the place values are being tracked by the diagonals in the lattice method, whereas the grid method requires you to keep track of that yourself. This makes the final addition step significantly more compact.
Ultimately, neither method is necessarily better than the other. They both accomplish the same thing, and there are costs and benefits to each. The lattice method is similar to the traditional multiplication algorithm in that there are some ideas that get lost in the process. If you were just following the "rules" of the calculation, you may not necessarily see the connection between the diagonals and the place values On the other hand, it's hard for the grid method to compete with the organization provided by the lattice method.
Subsection 30.5.2 The Egyptian Multiplication Method
The Egyptian multiplication method is named this way because they were the first known culture to have adopted this method. There is a similar method called the Russian peasant multiplication method, which was a rediscovery of the method by Russian peasants in the 19th century. This method is significantly different from our other methods because the only multiplication is multiplication by 2. The simplicity of the calculations is part of the appeal of the method.
We will perform the same calculation as before: \(126 \times 23\text{.}\) We start by making two columns. The first column starts with the number 1 and the second column starts with the larger number in our product.
From here, we're going to double each entry moving down the column, stopping when the first column will exceed the second number in the product.
Next, we look for a combination of numbers in the first column that add up to the second number in the product. The method is quite simple, but explaining it can be a little wordy. The basic idea is that you work your way back up the table, keeping a running total as you go. If keeping the value puts you above your target, skip it. Otherwise, you keep it. At some point, you will hit your target number, and you can skip any remaining values. The process for this example is performed in steps below.
The last step is to add up the values on the right side of the chart corresponding to the selected rows. For clarity, we're going to rewrite the desired values before adding.
For most people, it's quite surprising that this works. Let's take a look at what's actually happening. In our example, notice that the right side is always 126 times the left side. For example, the row with on the left side has \(8 \times 126 = 1008\) on the right. We then picked the rows on the left that add up to 23 giving us \(23 = 16 + 4 + 2 + 1\text{.}\) We can take this and multiply both sides by 126:
We can see that the third line consists of the same values that appeared in the right-most column.
From a mathematical perspective, there's not much difference between the groupings that are being used in this method and the base-10 groupings. All we've done is broken up the product into smaller pieces. In fact, we could even go as far as saying that the only difference between this method and the multiplications we've done previously is that this method is using a binary (or base-2) approach. So even though it looks very different, the underlying concepts are actually still the same.
The point of showing you these examples is to further expand your perspective of multiplication. What is normally taught as a very rigid process turns out to have many different approaches.