Pascal's Triangle
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About Pascal's Triangle
Pascal's Triangle was known long before Blaise Pascal wrote about it. It was known to both Chinese and the Arabs, but it seems that Pascal was the first to discover the importance of all of the patterns it contained.
At the top of Pascal's Triangle is the number 1 - this corresponds to the 0 th row. The first row (1 & 1) contains two 1's, both formed by adding the two numbers above them to the left and the right, in this case 1 and 0. Do the same to create the 2nd row: 0+1=1; 1+1=2; 1+0=1. And the third: 0+1=1; 1+2=3; 2+1=3; 1+0=1. Pascal's triangle doesn't end - you can continue doing this to infinity.
The sums of numbers in the nth row
The sum of the numbers in the nth row is equal to 2 to the nth power or 
. For example:
Can you think of the ways in which you can use this?
Prime numbers
If the 1st element in a row is a prime number (remember, the 0th element of every row is 1), all the numbers in that row (excluding the 1's) are divisible by it. For example, in row 7 (1 7 21 35 35 21 7 1) 7, 21, and 35 are all divisible by 7.
Powers of 11
If a row is made into a single number by using each element as a digit of the number (carrying over when an element itself has more than one digit), the number is equal to 11 to the nth power where n is the number of the row.
Row no. |
Power of 11 |
Result for the corresponding power of 11 |
Numbers in the actual row |
Row 0 |
0 |
1 |
1 |
Row 1 |
1 |
11 |
11 |
Row 2 |
2 |
121 |
1 2 1 |
Row 3 |
3 |
1331 |
1 3 3 1 |
Row 4 |
4 |
14641 |
1 4 6 4 1 |
Row 5 |
5 |
161051 |
1 5 10 10 5 1 |
Row 6 |
6 |
1771561 |
1 6 15 20 15 6 1 |
Row 7 |
7 |
19487171 |
1 7 21 35 35 21 7 1 |
Row 8 |
8 |
214358881 |
1 8 28 56 70 56 28 8 1 |
Can you find the Fibonacci Sequence in Pascal's Triangle? Have a look again
See the page on figurate numbers, then come back to this page to see whether you can find any figurate numbers in Pascal's Triangle. Can you describe any such patterns?
Blaise Pascal was born in Clermont on June 19, 1623, and died in Paris on August 19, 1662. He was considered to be a mathematical child prodigy and at the age of fourteen he was admitted to the weekly meetings with Roberval, Mersenne, and other French geometricians, from which, ultimately, the French Academy sprung. He eventually abandoned mathematics in favour of theology.
You can get quite a few Pascal's Triangle worksheets from the Math Forum web site.
Download a simple version of Pascal's Triangle here.
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