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In mathematics, the factorial of a non-negative integer , denoted by , is the product of all positive integers less than or equal to . The factorial of also equals the product of with the next smaller factorial: For example, The value of 0! is 1, according to the convention for an empty product. [1]
= ((((3×5 + 4)×4 + 1)×3 + 0)×2 + 1)×1 + 0 = 463 10. (The place value is the factorial of one less than the radix position, which is why the equation begins with 5! for a 6-digit factoradic number.) General properties of mixed radix number systems also apply to the factorial number system.
To put in perspective the size of a googol, the mass of an electron, just under 10 −30 kg, can be compared to the mass of the visible universe, estimated at between 10 50 and 10 60 kg. [5] It is a ratio in the order of about 10 80 to 10 90 , or at most one ten-billionth of a googol (0.00000001% of a googol).
[1] [2] [3] One way of stating the approximation involves the logarithm of the factorial: (!) = + (), where the big O notation means that, for all sufficiently large values of , the difference between (!
The value of each is taken to be 1 (an empty product) when =. These symbols are collectively called factorial powers. [2] The Pochhammer symbol, introduced by Leo August Pochhammer, is the notation (), where n is a non-negative integer.
If each book had a mass of 100 grams, all of them would have a total mass of 10 93 kilograms. In comparison, Earth's mass is 5.972 × 10 24 kilograms, the mass of the Milky Way galaxy is estimated at 2.5 × 10 42 kilograms, and the total mass of all the stars in the observable universe is estimated at 2 × 10 52 kg. [4]
Double factorial. The fifteen different chord diagrams on six points, or equivalently the fifteen different perfect matchings on a six-vertex complete graph. These are counted by the double factorial 15 = (6 − 1)‼. In mathematics, the double factorial of a number n, denoted by n‼, is the product of all the positive integers up to n that ...
Wilson's theorem. In algebra and number theory, Wilson's theorem states that a natural number n > 1 is a prime number if and only if the product of all the positive integers less than n is one less than a multiple of n. That is (using the notations of modular arithmetic), the factorial satisfies. exactly when n is a prime number.