ArtsAutosBooksBusinessEducationEntertainmentFamilyFashionFoodGamesGenderHealthHolidaysHomeHubPagesPersonal FinancePetsPoliticsReligionSportsTechnologyTravel

Pascal's Triangle For Dummies

Updated on July 20, 2012
Blaise Pascal (1622-1662)
Blaise Pascal (1622-1662) | Source

Pascal's triangle is a triangle array of the binomial coefficients in a triangle. It is named after the French mathematician Blaise Pascal, but has been known from centuries. It was already known in India, China and Greece.

For the western world is known as Pascal's triangle since 1730. Just take a look how the name array changes according to the country or continent:

Khayyam triangle in Iran, after Omar Khayyam a poet and mathematician from Persia (1048-1131 A.D.)

Yang Hui's triangle in China, after Yang Hui (1238–1298)

In Italy, it is referred to as Tartaglia's triangle, named for the Italian algebraist Niccolo Fontana Tartaglia (1500–1577)


Binomial Expansion

Given (a+b)n as an algebraic binomial to the nth power, we will have an expansion with some coefficients that can be found on the Pascal's triangle. For years we have been struggling for ways of memorizing this set of numbers in a triangle. After checking the original 'building block', we came out with an idea: Mnemonics and graphics.

Consider the expansion:

(x + y)2 = x2 + 2xy + y2 = 1x2y0 + 2x1y1 + 1x0y2

Source

Pascal's triangle determines the coefficients which arise in BINOMIAL EXPANSIONS. For example we can build a triangle from scratch positioning our 'duck' on arrow #3. 'Duckier' will be surrounded by numbers 1.

Many High school students have had problems building this triangle. Our teacher would start with number 1 on top. Arrow 2 would be kind of useless but contained two number 1's as coefficients. Which now make sense to us:

(a+b)1 = 1a+1b

Using that duck in place, we can build Pascal's triangle all the way down with a solid foundation.

Bellow we have binomials to the power of 3, 4, 5,6 and 7. Notice the coefficients that follow the Pascal' triangle array with no problem.

Now that you are able to take this graphic-algebraic expansion, we see that the sequence of coefficients on the "eighth arrow" will be like this:
1, 7, 21, 35, 21, 7, 1

How would you calculate (0.999)5 ?

In the next video, a teacher will talk about Pascal and a practical example... in real time.

Tip spoiler: (0.999)5 = (1- 0.001)5 

Solving 0.0999 to the Fifth Power -- watch her trick

A Pascaline, built in 1652
A Pascaline, built in 1652 | Source

About Blaise Pascal

Blaise Pascal (19 June 1623 – 19 August 1662), was a French mathematician, physicist, inventor, writer and Catholic Philosopher. Blaise's father was a tax collector in Rouen, which made Blaise get used to calculations. Actually, at 19 he invented the mechanical calculator which was know as Pascaline. This way, young Pascal would help his father on the tedious accounting calculations. Unfortunately he died at 39 of age, due to tuberculosis and stomach cancer. His legacy was remarkable to engineering, science and philosophy.

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://corp.maven.io/privacy-policy

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)