Multiplying without Memorizing

A few weeks ago I stopped by a Montgomery County Palestine Solidarity Network picnic to grab a meal and chat with some like-minded teachers and parents. My favorite conversation was with E, who was making her little brother a “Free Palastine” bracelet at the craft table. We worked together to teach him to say the word Pal-is-tine and E told me about her hopes and fears for the new semester. She’s excited to see her friends, go to art class, and have recess. She’s nervous about long division. All her older friends told her how hard it is and E doesn’t have her higher times tables memorized. E’s eyes lit up when I told her that there’s a way to multiply big numbers that doesn’t require memorizing her times tables- just doubling numbers and adding them together.

Ancient Egyptian Multiplication

The three number systems of Ancient Egypt- hieroglyphic, hieratic, and demotic. The first three columns show 1-9, the second three columns show tens places 10-90, and so on.

Today this technique is mostly used in rural Russia and Ethiopia but the earliest known application of this process is in Ancient Egypt around 1900 BCE, four thousand years ago. The source of this method also demonstrates systems for solving linear equations, calculating volumes and areas of difficult geometric objects, and computing right triangle sides. Pythagoras wasn’t born for thousand years- whoever said the Greeks invented math was missing some information. Most of Classical Greek math and science was likely inherited or stolen from exchange with and colonization of Egypt.

Because this process worked alongside other mathematical techniques of the time, it doesn’t work as smoothly in reverse for division, or as quickly as exponentiation requires. However, it can be easily adapted for multiplication of both decimals and negatives, and offers kids who need more time to memorize tables a way to stay on track in the meantime.

The Process

The steps to multiply two numbers using the Ancient Egyptian Method:

1. Choose one number to be the multiplicand (the other is the multiplier) and lay out a table with 1 on the left and your multiplicand on the right

2. Multiply both sides x2 until the left is as close as possible to the multiplier (don’t go over!)

3. Find the numbers from the left side that add up to the multiplier

4. Add up the corresponding numbers on the right side

5. Check your work

Example: 17 x 23

Example multiplication problem

1. 17 is the multiplier and 23 is the multiplicand. Lay out the table with 1 on the left and 23 on the right.

2. Multiply both sides by 2 until you get as close to 17 as you can without going over.

3. Find the numbers on the left that add up to 17- this will always include the last number, and will sometimes be three or more numbers from the left! Our last number is 16- when we take that away from 17, all that’s left is 1 so we will use 16 and 1.

4. Add the numbers next to them, 368 and 23, to get the answer.

5. Check the work- since 17 and 23 are both close to 20, I would expect our answer to be about 400- and so it is!

Why this Works

10 groups = 8 groups + 2 groups

Multiplication is about adding groups. Imagine you need lollipops for your whole class of 31 kids. If lollipops are sold in bags of 3, will 10 bags be enough?

Essentially I’m asking how many lollipops total are in 10 groups of 3. We can lay those groups out flat and count each one, we can memorize the multiplication fact 3 x 10, or we can remember the x 10 property. Using the Egyptian Multiplication Method though, we would essentially subdivide those 10 groups into 8 groups and 2 groups, then add them back together. This relies on the mathematical theorem that every positive integer can be represented as a sum of powers of 2.