Integer Beta Function CalculatorFree online math calculator — instant results, no sign-up
Evaluate the beta function for two positive integer arguments.
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About the Integer Beta Function Calculator
The Integer Beta Function Calculator evaluates B(m, n) for two positive integer arguments using a factorial identity. With the default inputs 3 and 4, it gives B(3, 4) = 1/60, displayed as 0.0166666667.
- First positive integer
- The first argument m of the beta function is a unitless positive integer, with a default value of 3.
- Second positive integer
- The second argument n of the beta function is a unitless positive integer, with a default value of 4.
Formula and method
B(m, n) = (m − 1)! × (n − 1)! / (m + n − 1)!; B(3, 4) = 2! × 3! / 6! = 12/720 = 1/60 ≈ 0.0166666667.
- mThe first positive integer argument; the default is 3.
- nThe second positive integer argument; the default is 4.
- B(m, n)The unitless beta function value for the two arguments.
- !Factorial notation: multiply all positive integers up to the preceding nonnegative integer, with 0! defined as 1.
Check that both arguments are positive integers, then evaluate the factorial ratio (m − 1)! × (n − 1)! / (m + n − 1)!. For m = 3 and n = 4, the numerator is 2! × 3! = 12 and the denominator is 6! = 720. Dividing gives the exact value 1/60, whose decimal approximation is displayed as 0.0166666667.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- First positive integer3
- Second positive integer4
Results
- Result0.0166666667
How to use the Integer Beta Function Calculator
- 1Enter 3 in First positive integer to use the default first argument.
- 2Enter 4 in Second positive integer to use the default second argument.
- 3Ensure that both entries are whole numbers greater than zero.
- 4Run the calculation to evaluate B(3, 4) using the factorial ratio.
- 5Read the Result row as 0.0166666667, the rounded decimal value of 1/60.
Reading your result
The Result row reports B(m, n), a unitless number. For the default inputs 3 and 4, Result = 0.0166666667 is a rounded representation of 1/60. It also represents the integral of t²(1 − t)³ from t = 0 to t = 1; it is not automatically a probability or percentage.
Limitations
This tool is restricted to positive integer arguments, so zero, negative values, and fractional inputs are outside its supported domain even though the beta function has broader mathematical definitions. Decimal output may be rounded. Factorials grow rapidly, so large arguments can exceed numerical limits in direct factorial evaluation; very small results may also lose precision or underflow, depending on the implementation.
Common questions
Why does B(3, 4) use 6! in the denominator rather than 7!?+
For positive integers, the denominator is (m + n − 1)!, so the default inputs give (3 + 4 − 1)! = 6!.
Does swapping the two integer arguments change the result?+
No. The beta function is symmetric: B(m, n) = B(n, m). Therefore, B(4, 3) has the same value as B(3, 4), namely 1/60.
Can either beta function argument be 1?+
Yes. Since 0! = 1, the formula remains valid when an argument is 1: B(1, n) = 1/n and B(m, 1) = 1/m.
Is 0.0166666667 the exact value for the default inputs?+
No. The exact value of B(3, 4) is 1/60, which has a repeating decimal expansion. The displayed result is rounded to ten decimal places.
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
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Results are estimates for informational purposes and are not financial, medical or legal advice.