Pascal's Triangle
Pascal's Triangle (below) has 1 at the top (apex), followed by two 1s ('1 1') in the second row, then '1 2 1' in the third row, then '1 3 3 1' and so on. Each row starts and ends with 1, and every other number (in every row) is the sum of the two numbers diagonally to the left and right of it in the previous row.
As stated on the main
Maths page, the triangle represents the binomial coefficients. To
mathematicians, this means that it shows 'the family of positive integers
that occur as coefficients in the binomial theorem'.
The binomial theorem, in short, gives the coefficients for raising the sum 'a + b' to the power n, where n is any positive integer.
For example, (a + b) raised to the power 1 is simply (a + b), which is represented in the second row of the triangle ('1 1', as stated above – one a and one b). (a + b) raised to the power of 2 – which we write as (a + b)2 – is equal to a + 2ab + b. This is represented in the triangle as '1 2 1': one a2, two times a*b, and one b2. (a + b) raised to the power of 3 – (a + b)3 – is a3 + 3a2b + 3ab2 + b3 – represented as '1 3 3 1'; and so on.
Note that the top row, which is referred to as Row 0 (zero), shows the binomial coefficients for (a + b)0 – (a + b) to the power zero. As every schoolboy should know, anything raised to the power zero is 1 (one); hence the solitary 1 in the top row of Pascal's triangle.
I don't expect non–mathematicians to understand all this – I don't fully understand it myself – but at least you now know (if you didn't already) what Pascal's triangle looks like, and you should have some idea of why.
© Haydn Thompson 2017