An integer is called even if it is divisible by 2. For integers written in a numeral system based on an even number, such as decimal, hexadecimal, or in any other base that is even, divisibility by 2 is easily tested by merely looking at the last digit. If it is even, then the whole number is even. In particular, when written in the decimal system, all multiples of 2 will end in 0, 2, 4, 6, or 8.

Two is the smallest prime number, and the only even prime number (for this reason it is sometimes called "the oddest prime"). The next prime is three. Two and three are the only two consecutive prime numbers. 2 is the first Sophie Germain prime, the first factorial prime, the first Lucas prime, and the first Ramanujan prime.

Two is the third (or fourth) Fibonacci number.

Two is the base of the binary system, the numeral system with the least number of tokens allowing to denote a natural number n substantially more concise (log₂ n tokens), compared to a direct representation by the corresponding count of a single token (n tokens). This binary number system is used extensively in computing.

For any number x:

x + x = 2 · x addition to multiplication

x · x = x² multiplication to exponentiation

x^x = x??2 exponentiation to tetration

Extending this sequence of operations by introducing the notion of hyperoperations, here denoted by "hyper(a,b,c)" with a and c being the first and second operand, and b being the level in the above sketched sequence of operations, the following holds in general:

hyper(x,n,x) = hyper(x,(n + 1),2).

Two has therefore the unique property that 2 + 2 = 2 · 2 = 2² = 2??2 = 2???2 = ..., disregarding the level of the hyperoperation, here denoted by Knuth's up-arrow notation. The number of up-arrows refers to the level of the hyperoperation.

Two is the only number x such that the sum of the reciprocals of the powers of x equals itself. In symbols

${\displaystyle \underset{k=0}{\overset{\mathrm{?}}{?}}\frac{1}{2^k}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+?=2.}$ 

This comes from the fact that:

${\displaystyle \underset{k=0}{\overset{\mathrm{?}}{?}}\frac{1}{n^k}=1+\frac{1}{n?1}\text{for all}n?\mathbb{R}>1.}$ 

Powers of two are central to the concept of Mersenne primes, and important to computer science. Two is the first Mersenne prime exponent.

Taking the square root of a number is such a common mathematical operation, that the spot on the root sign where the exponent would normally be written for cubic and other roots, may simply be left blank for square roots, as it is tacitly understood.

The square root of 2 was the first known irrational number.

The smallest field has two elements.

In a set-theoretical construction of the natural numbers, 2 is identified with the set {{?},?}. This latter set is important in category theory: it is a subobject classifier in the category of sets.

Two also has the unique property such that

${\displaystyle \underset{k=0}{\overset{n?1}{?}}{2}^k=2^n?1}$ 

and also

${\displaystyle \underset{k=a}{\overset{n?1}{?}}{2}^k=2^n?\underset{k=0}{\overset{a?1}{?}}{2}^k?1}$ 

for a not equal to zero

In any n-dimensional, euclidean space two distinct points determine a line.

For any polyhedron homeomorphic to a sphere, the Euler characteristic is ? = V ? E + F = 2, where V is the number of vertices,