Watercolour on Collograph, unique state
Paper: BFK Reeves 300gsm white
Mixed media: printmaking and watercolour
One is its own factorial, and its own square and cube (and so on, as 1 × 1 × … × 1 = 1). One is the first figurate number of every kind, such as triangular number, pentagonal number and centered hexagonal number to name just a few.
Because of the multiplicative identity, if f(x) is a multiplicative function, then f(1) must equal 1.
It is also the first and second numbers in the Fibonacci sequence, and is the first number in many mathematical sequences. As a matter of convention, Sloane’s early Handbook of Integer Sequences added an initial 1 to any sequence that didn’t already have it, and considered these initial 1’s in its lexicographic ordering. Sloane’s later Encyclopedia of Integer Sequences and its Web counterpart, the On-Line Encyclopedia of Integer Sequences, ignore initial ones in their lexicographic ordering of sequences, because such initial ones often correspond to trivial cases.
One is the empty product.
One is the smallest positive odd integer.
One is a harmonic divisor number.
One is often the internal representation of the Boolean constant true in computer systems.
One is neither a prime number nor a composite number, but a unit, like -1 and, in the Gaussian integers, i and -i. The fundamental theorem of arithmetic guarantees unique factorization over the integers only up to units (e.g. 4 = 22 = (-1)4×123×22).
One was formerly considered prime by some mathematicians, using the definition that a prime is divisible only by one and itself. However, this complicates the fundamental theorem of arithmetic, so modern definitions exclude units. The last professional mathematician to publicly label 1 a prime number was Henri Lebesgue in 1899.