'Pascal's' Triangle ELEVENTH CENTURY AD
Blaise Pascal (162342) gave his name to a triangular array of numbers such as the one below:
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
If you study this array of numbers, you will see that every number in it is equal to the sum of the two numbers above it on either side, except for the I's of course. Thus, 15 is the sum of the 10 above it to the left and the 5 above it to the right. And 35 is the sum of the 15 above it to the left and the 20 above it to the right.
But Pascal's Triangle is not just an intriguing oddity for people who like to play around with numbers. It actually gives the numerical coefficients (the numbers which go beside algebraic letters) of the series of solutions to the raising to successive higher powers of a binomial. A binomial consists of two numbers added together, represented as (a + b). In the triangle, each successive line across gives the numbers which go with the solutions. Thus, if (a + b) is raised to the power of one, it stays exactly as it is, and the first line gives the coefficients of the result: I and 1, for @ plus b. But if (a + b) is raised to the second power, meaning that it is squared, (a + b) times (a + b) gives us the answer a 2 + 2ab + b 2, and it will be immediately obvious that the numbers beside the letters of this answer are given by the second line across in the triangle, namely, 1, 2, 1. And so on as (a + b) is cubed, then raised to the fourth power, and fifth power, on indefinitely.
Now, it may seem that this too is merely an oddity. But not so. As one raises a binomial to higher and higher powers, one soon can lose one's way and wonder what the numerical coefficients are actually going to be in the solution. But a glance at an extended Pascal Triangle can give the answer instantly, thus providing one with solutions to the problems without having to multiply them out. The Triangle is a wonderful time-saver, and one of the fundamental steps in getting mathematics really on its feet. But although it bears the name Pascal's Triangle, it was by no means invented by Pascal. He merely put it in a newer form in the year 1654. In fact, this Triangle was invented in China. It may be seen depicted in a Chinese book of 1303 AD by Chu Shih-Chieh, entitled Precious Mirror of the Four Elements. Even here, it is called 'The Old Method'.