ArtsAutosBooksBusinessEducationEntertainmentFamilyFashionFoodGamesGenderHealthHolidaysHomeHubPagesPersonal FinancePetsPoliticsReligionSportsTechnologyTravel

Multiplication: Do It Like the Egyptians

Updated on October 27, 2014
Scribe Statue of Min-nakht - Walters 22230 - Three Quarter.jpg Wikipedia Public Domain
Scribe Statue of Min-nakht - Walters 22230 - Three Quarter.jpg Wikipedia Public Domain | Source

I just came across a new hub by Cooldudemaths and it reminded me I have a few (interpret several) really fun Math projects I have come across as an adjunct instructor for a local community college. I found over the years that my students relate to topics that can engage them from unexpected sources. This particular lesson can be used in a high school classroom and adjusted to be included in a beginning analysis class. This example can be used to present topics from the Associative Property to Factoring and the Distributive Property. For higher levels one can use this to talk about binary operations and other philosophical topics.

It is always best to start at the beginning. For simplicity let's only talk about multiplying positive whole numbers. The Egyptians did not know about negative numbers or what 0 meant. In your best Egyptian style headdress explain to your class that they multiplied by a series of doubling steps. Then they would sum up the relevant parts to give the correct answer.

The following table will show how to set up a table of values for multiplying 23 by 42.

Find 23 * 42 = ???

 
23
42
start with a single 42
1
42
double the above row
2
84
double the above row
4
168
double the above row
8
336
double the above row
16
672
notice how the fist column is larger than 23
32
stop
Rene Descartes painted by Frans Hals Wikipedia public domain image
Rene Descartes painted by Frans Hals Wikipedia public domain image | Source

Finding the Solution

Now the Egyptians didn't add they way we do. They didn't know what base 10 was. Their number symbols were different than ours too. They did know how to double amounts though. They simply matched up the amount and then portion the total in known quantity amounts. So their method is to double upwards from a unit amount until the multiple factor is exceeded.

To solve start in the column headed up by 23. Starting with the largest number less than 23 or 16. Now add numbers backwards until you total exactly 23. I can't add 16 + 8 because this is > 23. The first two lines to add together then are 16 + 4 = 20. Keep going. 16 + 4 + 2 = 22 OK so the last addition will be the right combination. 16 + 4 + 2 + 1 = 23

To find the answer add together the numbers in the right hand column under the 42 that are associated with the numbers found in the left hand column. This turns out to be: 672 + 168 + 84 + 42 = 966

As long as we may be talking about properties of numbers let's demonstrate that the Commutative Property holds even for the Egyptian scribes. Here is the table for 42 * 23 = ???


Find 42 * 23 = ???

start with a single 23
1
23
double the above row
2
46
double the above row
4
92
double the above row
8
184
double the above row
16
368
double the above row
32
736
this double makes 64 > 42
64
stop
Sophie Germain Wikipedia Public Domain
Sophie Germain Wikipedia Public Domain | Source

Solving 42 * 23 = ???

In this example we start with 32. But 32 + 16 = 48 > 42. This means we need to use this combination 32 + 8 + 2 = 42 That means if we add the numbers in the right hand column in the same rows you get 736 + 184 + 26 = 966 SURPRISE! You got the same result.

Multiplying this ancient way works every time. All the student need to do is set up a table. It is easy to see that the left hand column is the base 2 or exponential series with 2 as the base and the exponent in whole numbers. This means the left hand column will always follow the series 1, 2, 4, 8, 16, 32, 64, 128, 256, 512 and so on. They will still want to make tables because it is the right hand column that will not be obvious.

Concepts

Now you can work in the ties between how the ancient Egyptians method and our method are really compatible. This will include at least the Distributive Property and Factoring. You can link in other properties if you really have their attention. These two are the obvious ones to show.

The trick is to use the left hand binary series. This is the Factoring skill. For example let's look at 55 * 105. Notice that 55 becomes 32 + 16 +4 + 2 + 1 = 55.

55 * 105 becomes (32 + 16 + 4 + 2 + 1 ) * 105

The Distributive Property turns this in to:

32*105 + 16*105 + 4*105 + 2*105 + 1*105 =

3360 + 1680 + 420 + 210 + 105=

5775


Gottried Wilhelm von Leibniz Wikipedia Public Domain
Gottried Wilhelm von Leibniz Wikipedia Public Domain | Source

Math History

There are countless other historical approaches to Math that we have replaced with modern techniques. We have had thousands of years to design our number system not to mention the style formatting the numerals. There are ancient examples of how preceding cultures developed various types of problem solving solutions that are just as interesting.

Sometimes it is not even the remotely ancient you should consider. It is interesting how people solved problems before Algebraic symbols and concepts existed. That was just a little over 500 years ago. It was cumbersome before more modern systems were introduced. Presenting Math in a non traditional way can present a different face on understanding. These historical discussions let those who have a stumbling block with normal teaching processes to see how the many facets fit together. Often it is just the alternative method that is enough for them to realize what is happening and why.

Comments

    0 of 8192 characters used
    Post Comment

    • hostaguy profile imageAUTHOR

      frank nyikos 

      3 years ago from 8374 E State Rd 45 Unionville IN 47468

      Thank you Melissa. I'm gearing up for another Math related article. I want to make it useful to beginning Algebra students so I am struggling a bit. It should be ready in a couple of weeks I hope.

    • melissae1963 profile image

      Melissa Reese Etheridge 

      3 years ago from Tennessee, United States

      This is such an interesting article. I know several students who would be interested.

    working

    This website uses cookies

    As a user in the EEA, your approval is needed on a few things. To provide a better website experience, hubpages.com uses cookies (and other similar technologies) and may collect, process, and share personal data. Please choose which areas of our service you consent to our doing so.

    For more information on managing or withdrawing consents and how we handle data, visit our Privacy Policy at: https://hubpages.com/privacy-policy#gdpr

    Show Details
    Necessary
    HubPages Device IDThis is used to identify particular browsers or devices when the access the service, and is used for security reasons.
    LoginThis is necessary to sign in to the HubPages Service.
    Google RecaptchaThis is used to prevent bots and spam. (Privacy Policy)
    AkismetThis is used to detect comment spam. (Privacy Policy)
    HubPages Google AnalyticsThis is used to provide data on traffic to our website, all personally identifyable data is anonymized. (Privacy Policy)
    HubPages Traffic PixelThis is used to collect data on traffic to articles and other pages on our site. Unless you are signed in to a HubPages account, all personally identifiable information is anonymized.
    Amazon Web ServicesThis is a cloud services platform that we used to host our service. (Privacy Policy)
    CloudflareThis is a cloud CDN service that we use to efficiently deliver files required for our service to operate such as javascript, cascading style sheets, images, and videos. (Privacy Policy)
    Google Hosted LibrariesJavascript software libraries such as jQuery are loaded at endpoints on the googleapis.com or gstatic.com domains, for performance and efficiency reasons. (Privacy Policy)
    Features
    Google Custom SearchThis is feature allows you to search the site. (Privacy Policy)
    Google MapsSome articles have Google Maps embedded in them. (Privacy Policy)
    Google ChartsThis is used to display charts and graphs on articles and the author center. (Privacy Policy)
    Google AdSense Host APIThis service allows you to sign up for or associate a Google AdSense account with HubPages, so that you can earn money from ads on your articles. No data is shared unless you engage with this feature. (Privacy Policy)
    Google YouTubeSome articles have YouTube videos embedded in them. (Privacy Policy)
    VimeoSome articles have Vimeo videos embedded in them. (Privacy Policy)
    PaypalThis is used for a registered author who enrolls in the HubPages Earnings program and requests to be paid via PayPal. No data is shared with Paypal unless you engage with this feature. (Privacy Policy)
    Facebook LoginYou can use this to streamline signing up for, or signing in to your Hubpages account. No data is shared with Facebook unless you engage with this feature. (Privacy Policy)
    MavenThis supports the Maven widget and search functionality. (Privacy Policy)
    Marketing
    Google AdSenseThis is an ad network. (Privacy Policy)
    Google DoubleClickGoogle provides ad serving technology and runs an ad network. (Privacy Policy)
    Index ExchangeThis is an ad network. (Privacy Policy)
    SovrnThis is an ad network. (Privacy Policy)
    Facebook AdsThis is an ad network. (Privacy Policy)
    Amazon Unified Ad MarketplaceThis is an ad network. (Privacy Policy)
    AppNexusThis is an ad network. (Privacy Policy)
    OpenxThis is an ad network. (Privacy Policy)
    Rubicon ProjectThis is an ad network. (Privacy Policy)
    TripleLiftThis is an ad network. (Privacy Policy)
    Say MediaWe partner with Say Media to deliver ad campaigns on our sites. (Privacy Policy)
    Remarketing PixelsWe may use remarketing pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to advertise the HubPages Service to people that have visited our sites.
    Conversion Tracking PixelsWe may use conversion tracking pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to identify when an advertisement has successfully resulted in the desired action, such as signing up for the HubPages Service or publishing an article on the HubPages Service.
    Statistics
    Author Google AnalyticsThis is used to provide traffic data and reports to the authors of articles on the HubPages Service. (Privacy Policy)
    ComscoreComScore is a media measurement and analytics company providing marketing data and analytics to enterprises, media and advertising agencies, and publishers. Non-consent will result in ComScore only processing obfuscated personal data. (Privacy Policy)
    Amazon Tracking PixelSome articles display amazon products as part of the Amazon Affiliate program, this pixel provides traffic statistics for those products (Privacy Policy)