Number 0 in Math

A page of the Numeropedia - the Special Encyclopedia of Numbers

All Numbers  & [0-9] 0 - 1  - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9  &  [...<0] - [0<...<1]

1  - 10 - 100  - 1000 - 10,000 - 100,000 - 1M - 10M - 100M1B10B - 100B - 1T - 10T - 100T - 1Q - ...

... In Arithmetic, 1+1 = 2, in Computer Science, 1+1 = 0, but in Love, 1+1 = 1!

00          None/Never         Number 0         Number 0 in Math.  (More Math. Pages)

The value of some basic mathematical functions: sin(0o) = sin(180o) = cos(90o) = cos(270o) = tan(0o) = tan(180o) = cot(90o) = cot(270o) = sinh(0) = tanh(0) = log(1) = 0.

The Dirac delta function d(x) vanishes everywhere except in an infinitesimal interval around the origin x = 0, its values are so large, such that its normalized integral over that interval is 1.

The 0th power of any non-zero number n: n0 = 1, but the “0th power of 0”, i.e. 00, is undefined.

“0 factorial”: 0! = 1 (by definition) and subfactorial of 1: !1 = 0.

The derivative of a constant function is always equal to 0.

"Zero vector" is a vector whose magnitude is zero; its direction is undefined.

A root (or a zero) of a function f(x) is a value of x at which the function f equals 0.

A number ending with 0 is divisible by 2, 5 and 10.

The 0o angle is identical with the 360o angle and all integral multiples of 360o.

The probability of an event is a function whose values are only between 0 and 1.

"Zero-sum game" is a game in which players make payments only to each other: one player’s loss is the other player’s gain, so the total amount of “money” available remains constant.

Popular symbol for the unit in a additive group is 0.  Popular symbol for the unit in a multiplicative group is 1

Euler numbers (or secant numbers Ei) are defined by the infinite series: sec(x) = 1/cos(x) = 1+E2(x2/2!)+E4(x4/4!)+E6(x6/6!)+…, where E0 = 1, E2 = –1, E4 = 5, E6 =  –61, E8 = 1,385, E10 = –50,521, E12 = 2,702,765, E14 = –199,360,981, E16 = 19,391,512,145, E18 = –2,404,879,675,441… The odd-indexed Euler numbers are all 0. The even-indexed Euler numbers have alternating signs.