Pascal's triangle
The Pascal triangle shows the coefficients of the binomial theorem with rational exponents in the form
Explanation
You can expand the table with binomial coefficients with broken powers. A number is the sum of the number there just below and to the left of that. You can see it in the penultimate row by
All rows have infinitely many terms.
m/n 0 1 2 3 4 5 k → ∞ ··· ··· ··· ··· ··· ··· ··· ··· 5/2 ··· 3/2 ··· 1/2 ··· 0/2 ◦ ◦ ◦ ◦ ◦ ··· −1/2 ··· −3/2 ··· ··· ··· ··· ··· ··· ··· ··· ···
For broken exponents we have the formula
The top fraction m / n is the exponent of the binomial theorem, the bottom number k is the current number of the term in the outcome.
HistoryIn 1676 Isaac Newton gave in a letter the following information about his formula in which A, B, C, … always indicates the immediately preceding term. With this we are going to calculate the square root of and get Nowadays we work with Taylor series. The series for the square root gives of course exactly the same solution. |