Number 1 in Math

 A page of the Numeropedia - the Special Encyclopedia of Numbers

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The first positive integer.  The first positive odd number.

 

Neither a prime number or a composite number.

 

A prime number is divisible only by 1 and the number itself.

 

The only one palindromic prime number of even number of digits: 11.

 

Two numbers are called relatively prime when their greatest common divisor is 1 (or no common divisor other than 1).

 

The most basic formula in trigonometry: cos2x+sin2x = 1, for any value of x.

 

The value of some basic mathematical functions:

sin(90o) = cos(0o) = tan(45o) = tan(225o) = cot(45o) = cot(225o) = cosh(0).

 

Interesting formula: arctan1+arctan2+arctan3 = 180o.

The 0th power of any non-zero number n: n0 = 1, while 00 is undefined.

 

loga(1) = 0 for any base a ¹ 0.

 

1 factorial = 1! = 1 = 0!. (0 factorial = 1, by definition).

 

Subfactorial of 2: !2 = 1 and subfactorial of 1: !1 = 0.

 

The number of surfaces (or sides) of a Möbius strip (created by taking a strip of paper, twisted once and joined together its ends).

 

There is exactly one circle passing through any 3 non-linear points in the 2-dimension Euclidean plane.

 

The first and 2nd Fibonacci numbers.

 

The first Lucas number.

 

The set of 4 numbers (1, 3, 8, 120) has a property that the product of any 2 numbers is always equal a square number minus 1.

 

A prime number cannot consist of only repeated digits other than repeated digits 1. The only known 7 prime numbers consisting of the repeated digits 1 are the numbers of 2, 19, 23, 317, 1031, 49,081 and 86,453 digits 1. 

 

One-way function is a function f that given variable x, it is easy to compute the value y = f(x), but vice versa, given value y, it is very hard computationally to find a variable x, such that f(x)= y.

 

A one-to-one mapping f maps any 2 different values of x and y in its variable domain to 2 different values f(x) and f(y) in its range.

 

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 integral over that interval equals a unit (1).

 

A Mersenne number written in binary form is a string of bits 1.

 

The string of bits 1 is the multiplicative identity, 1 = (1,1,…,1), in a normal basis representation of a Galois finite field GF(2m) over the base field GF(2).

 

The simply-stated by powerful “Principle of Pigeon Holes”: If the number of pigeons is more than the number of holes available, than at least 1 hole must have at least 2 pigeons.

 

For any two relatively prime numbers a and b, there are two numbers x and y (positive or negative) such that ax+by = 1.

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

 

The imaginary quadratic fields Q((–1)1/2) = Q(i) is one of only 9 UFDs (unique factorization domains) of the form Q((–d)1/2), d = 1, 2, 3, 7, 11, 19, 43, 67 and 163.

 

The first Markov numbers are 1, 2, 5, 13, 29, 34, 89, 169, 194, 233…

 

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

 

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.

 

1.13…

Divakar Viswanath’s constant generates the randomized Fibonacci sequence: 1.13198824

 

1.154…

A set of diameter less than or equal to 1 can be covered by a circle of diameter 2/(31/2) » 1.154700… (Jung’s theorem).

 

1.5

4356 = (1.5)×6534 = (3/2)×6534 and

2,352,941,176,470,588×(3/2) = 2,352,941,176,470,588×(1.5) = 3,529,411,764,705,882.

 

1.551…

Eugenio Calabi’s triangle: Given any equilateral triangle with its three equal largest squares that fit inside it, there is uniquely a triangle in which there equally large squares can fit. The ratio of the largest side to the other two (equal) sides is an algebraic number x = 1.55138752455… satisfying the equation 2x3–2x2–3x+2 = 0.

 

1.618…

The golden ratio ½(1+51/2) = 1.618034… in the Fibonacci and Lucas sequences.

 

1.902…

Sum of the reciprocals of the twin primes converges to Brun’s constant (proved in 1919). By calculating the twin primes up to 1014 (and discovering the Intel’s Pentium microprecessor infamous bug along the way), Thomas Nicely heuristically estimates Brun’s constant to be 1.902160578…