3.3
Pascal’s Triangle
Blaise Pascal was a French philosopher and mathematician who lived during the 1600’s. Pascal is known for among other things, making the first calculating machine and for writing a number of books in mathematics.
Pascal’s triangle was named after Blaise Pascal but the construction was known long before. Pascal investigated the creation of numbers in his writing Traité du triangle arithmétique (1653), and because of this it was named Pascal’s triangle.
What is then Pascal’s triangle?


Simply put, you can say that it is the creation of numbers that follow a certain pattern. This pattern looks like the form of a triangle.
The upper triangle is the number 1 and every new row under it contains a number more than the row above. The newer numbers are calculated by the sum of the numbers to the left and right in the row above. If there is not a number both to the right and left in the row above, the number becomes the same as the only number to the left or right above. This means that every row begins with the number 1.
How are Pascal’s triangles used?
Pascal’s triangles aren’t just interesting triangles of number, but even have other important uses within mathematics. One area is algebra.
Imagine that you want to increase the binomial (a + b) by a certain power (for example 1, 2, 3, 4, 5...). The various powers of (a + b) will look like this:
(a + b)^{0} = 


1 




(a + b)^{1 }= 



1a 
+ 
1b 



(a + b)^{2} = 


1a^{2} 
+ 
2ab 
+ 
1b^{2} 


(a + b)^{3} = 

1a^{3} 
+ 
3a^{2}b 
+ 
3ab^{2} 
+ 
1b^{3} 

(a + b)^{4} = 
1a^{4} 
+ 
4a^{3}b 
+ 
6a^{2}b^{2} 
+ 
4ab^{3} 
+ 
1b^{4} 
As you can see, the numbers above marked in red coincide with Pascal’s triangle.
