Interesting Facts About Pascal's Triangle

Updated on June 12, 2020
David3142 profile image

I am a former maths teacher and owner of Doingmaths. I love writing about maths, its applications and fun mathematical facts.

Blaise Pascal (1623 - 1662)
Blaise Pascal (1623 - 1662)

What Is Pascal's Triangle?

Pascal's Triangle is a number triangle which, although very easy to construct, has many interesting patterns and useful properties.

Although we name it after the French mathematician Blaise Pascal (1623–1662) who studied and published work on it, Pascal's Triangle is known to have been studied by the Persians during the 12th century, the Chinese during the 13th century and several 16th-century European mathematicians.

The Triangle's construction is very simple. Start with a 1 at the top. Each number below this is formed by adding together the two numbers diagonally above it (treating empty space on the edges as zero). Therefore the second row is 0 + 1 = 1 and 1 + 0 =1; the third row is 0 + 1 =1, 1 + 1 = 2, 1 + 0 =1 and so on.

Pascal's Triangle
Pascal's Triangle | Source

Hidden Number Patterns in Pascal's Triangle

If we look at the diagonals of Pascal's Triangle, we can see some interesting patterns. The outside diagonals consist entirely of 1s. If we consider that each end number will always have a 1 and a blank space above it, it is easy to see why this happens.

The second diagonal is the natural numbers in order (1, 2, 3, 4, 5, ...). Again, by following the construction pattern of the triangle, it is easy to see why this happens.

The third diagonal is where it gets really interesting. We have the numbers 1, 3, 6, 10, 15, 21, .... These are known as the triangle numbers, so called as these numbers of counters can be arranged into equilateral triangles.

The First Four Triangle Numbers
The First Four Triangle Numbers | Source

The triangle numbers are formed by each time adding one more than was added the previous time. So for example, we start with one, then we add two, then add three, then add four and so on giving us the sequence.

The fourth diagonal (1, 4, 10, 20, 35, 56, ...) is the tetrahedral numbers. These are similar to the triangle numbers, but this time forming 3-D triangles (tetrahedrons). These numbers are formed by adding consecutive triangle numbers each time, i.e. 1, 1 + 3 = 4, 4 + 6 = 10, 10 + 10 = 20, 20 + 15 = 35, etc.

The fifth diagonal (1, 5, 15, 35, 70, 126, ...) contains the pentatope numbers.

Binomial Expansions

Pascal's Triangle is also very useful when dealing with binomial expansions.

Consider (x + y) raised to consecutive whole number powers.

(x + y)1 = x + y

(x + y)2 = x2 + 2xy + y2

(x + y)3 = x3 + 3x2y + 3xy2 + y3

(x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4 etc.

The coefficients of each term match the rows of Pascal's Triangle. We can use this fact to quickly expand (x + y)n by comparing to the nth row of the triangle e.g. for (x + y)7 the coefficients must match the 7th row of the triangle (1, 7, 21, 35, 35, 21, 7, 1).

The Fibonacci Sequence

Take a look at the diagram of Pascal's Triangle below. It is the usual triangle, but with parallel, oblique lines added to it which each cut through several numbers. Let's add together the numbers on each line:

  • 1st line: 1
  • 2nd line: 1
  • 3rd line: 1 + 1 = 2
  • 4th line: 1 + 2 = 3
  • 5th line: 1 + 3 + 1 = 5
  • 6th line: 1 + 4 + 3 = 8 etc.

By adding together the numbers on each line, we get the sequence: 1, 1, 2, 3, 5, 8, 13, 21, etc. otherwise known as the Fibonacci sequence (a sequence defined by adding the previous two numbers together to get the next number in the sequence).

Fibonacci in Pascal's Triangle

Patterns in Rows

There are also some interesting facts to be seen in the rows of Pascal's Triangle.

  • If you sum all the numbers in a row, you will get twice the sum of the previous row e.g. 1, 1 + 1 = 2, 1 + 2 + 1 = 4, 1 + 3 + 3 + 1 = 8 etc. This is down to each number in a row being involved in the creation of two of the numbers below it.
  • If the number of the row is prime (when counting rows, we say the top 1 is row zero, the pair of 1s is row one, and so on), then all of the numbers in that row (except for the 1s on the ends) are multiples of p. This can be seen in the 2nd, 3rd, 5th and 7th rows of our diagram above.

Fractals in Pascal's Triangle

One amazing property of Pascal's Triangle becomes apparent if you colour in all of the odd numbers. Doing so reveals an approximation of the famous fractal known as Sierpinski's Triangle. The more rows of Pascal's Triangle that are used, the more iterations of the fractal are shown.

The Sierpinski Triangle From Pascal's Triangle

Source

You can see in the image above that colouring in the odd numbers on the first 16 lines of Pascal's Triangle reveals the third step in constructing Sierpinski's Triangle.

© 2020 David

Comments

    0 of 8192 characters used
    Post Comment

    No comments yet.

    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, owlcation.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://maven.io/company/pages/privacy

    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)
    ClickscoThis is a data management platform studying reader behavior (Privacy Policy)