Newton's binomial theorem
The binomial theorem (bi means the 2 variables a and b) is the formula
(a + b)n = an + n an − 1b + ½ n (n − 1) an −2b2 + ···
where n is a natural number.
Explanation
We write the binomial theorem and find
⁰ = 1.gif){kind=small}
¹ = a + b.gif){kind=small}
² = a² + 2ab + b².gif){kind=small}
³ = a³ + 3a²b + 3ab² + b³.gif){kind=small}
⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴.gif){kind=small}
⁵ = a⁵ + 5a⁴ b + 10a³ b² + 10a² b³ + 5a b⁴ + b⁵.gif){kind=small}
It contains a clear regularity. Each term has the general form
Coefficient × an−r br
where
n = Power of the binomium
r = Number of the term
The development of the binomial coefficients follows Pascal's triangle.
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1
A number is the sum of the number on the left and the right, just above it. In the rows these numbers increase at first, and decrease accordingly afterwards. There is of course a formula for that.
HistoryThe name is a tribute to the English mathematician Isaac Newton (1642 - 1727). |