# Pascal's Triangle For Dummies

**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} = *x*^{2} + 2*xy* + *y*^{2} = 1*x*^{2}*y*^{0} + 2*x*^{1}*y*^{1} + 1*x*^{0}*y*^{2}

*x*+

*y*)

^{2}=

*x*

^{2}+ 2

*xy*+

*y*

^{2}= 1

*x*

^{2}

*y*

^{0}+ 2

*x*

^{1}

*y*

^{1}+ 1

*x*

^{0}

*y*

^{2}

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 }= **1**a+**1**b

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

## 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.

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## Comments 13 comments

Very Useful information!!! Great Hub!!

I could kind of hang in there with the algebra hubs... but.. this is a new beast. What is cool about you is you are a genius and you don't rub it in. Excellent!

Nice work! Good to meet fellow math nerds on Hubpages!

Joseph,

You and Janine are busting out those Math hubs huh? :) Very informative and thanks! I was decent at Math in high school, but that was quite some time ago. So, I could use the refreshing! :)

Tammy you are so right Lord is a genius.. I mean look at all that algebra.. I have no earthly idea.. Lol.. Joseph you are too awesome my friend

hugs from Tennessee

Debbie

Funny Josh, Isaw this topic when I woke up and immediately clicked to read. I am so a math nerd at heart and this one helped to give this math nerd a bit more of an education. This one did not disappoint and very good explanation of Pascal's Triangle. Voted and shared too!!

This is going back in time for me! Struggling to understand Pascal's brain... lol! Enjoyed reading it, thank God I don't have to study this any more.....

Voted up and interesting!

I remember learning about this in high school - I found it interesting then and now too. I did not know about the different names like you explain in the first part of the hub. Vote up, interesting, and sharing too :)

Now, just let me say this up front.....I love you, I love your writings, but after reading the very first sentence on this article, I knew I was in way over my head!

I will just have to leave this one for the more educated people to comment on, because it is like Greek to me.

I can't help but wonder what your I.Q. is Lord. It has to be HIGH.

I'm looking forward to a nice sweet juicy poem about love next.

13