Zero to the power of zero
The value (1 − 1)0 = 00 = 1 arises in the development of Pascal's triangle.
Explanation
Pascal's triangle shows the development of the coefficients in the binomium in the form
(a − b)n = an − n an − 1b + ½ n (n − 1) an −2b2 + ···
When you write the binomium (a − b)n as (1 − 1)n all a's and b's in the development disappear and only the binomial coefficients remain. The sum of the rows are powers of zero, as (1 − 1)n = 0n.
0 1 → 00 ≝ 1 1 1 −1 → 01 = 0 2 1 −2 1 → 02 = 0 3 1 −3 3 −1 → 03 = 0 4 1 −4 6 −4 1 → 04 = 0 5 1 −5 10 −10 −5 −1 → 05 = 0 6 1 −6 15 −20 15 −6 1 → 06 = 0 7 1 −7 21 −35 35 −21 7 −1 → 07 = 0 8 1 −8 28 −56 70 −56 28 −8 1 → 08 = 0
In row number three applies 03 = 1 − 3 + 3 − 1 = 0. For row number zero you get 00 ≝ 1.
HistoryThe name of this triangle is a tribute to the French mathematician Blaise Pascal (1623 - 1662). |