1 (one) is a number, numeral, and the name of the glyph representing
that number. It represents a single entity, the unit of counting or
measurement. For example, a line segment of "unit length" is a line segment
of length 1.
Mathematically, 1 is
For any number x:
x·1 = 1·x = x (1 is the multiplicative identity) x/1 = x (see division) x1 = x, 1x = 1, and for nonzero x, x0 = 1 (see exponentiation)
Using ordinary addition, we have 1 + 1 = 2.
One cannot be used as the base of a positional numeral system; sometimes tallying is referred to as "base 1", since only one mark (the tally) is needed, but this is not a positional notation.
The logarithms base 1 are undefined, since the function 1x always equals 1 and so has no unique inverse.
In the real number system, 1 can be represented in two ways as a recurring decimal: as 1.000... and as 0.999... (q.v.).
Formalizations of the natural numbers have their own representations of 1:
In a multiplicative group or monoid, the identity element is sometimes denoted "1", but "e" (from the German Einheit, unity) is more traditional. However, "1" is especially common for the multiplicative identity of a ring, i.e. when an addition and "0" are also present.
In Boolean algebra, 1 corresponds to true.
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.
In computing, many programming languages and computer systems use 1 to represent theBoolean value true, but this is not as common as using 0 for false. The reverse also occurs.
One is neither a prime number nor a composite number, but a unit, like -1 and, in theGaussian 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 professionalmathematician to publicly label 1 a prime number was Henri Lebesgue in 1899.
One is one of three possible values of the Möbius function: it takes the value one forsquare-free integers with an even number of distinct prime factors.
One is the only odd number in the range of Euler's totient function φ(x), in the cases x = 1 and x = 2.
One is the only 1-perfect number (see multiply perfect number).
By definition, 1 is the magnitude or absolute value of a unit vector and a unit matrix (more usually called an identity matrix). Note that the term unit matrix is usually used to mean something quite different.
One is the most common leading digit in many sets of data, a consequence of Benford's law.
The ancient Egyptians represented all fractions (with the exception of 2/3 and 3/4) in terms of sum of fractions with numerator 1 and distinct denominators. For example, . Such representations are popularly known as Egyptian Fractions or Unit Fractions.
The Generating Function which has all coefficients 1 is given by
This power series converges and has finite value if, and only if, | x | < 1.
Evolution of the glyph
The glyph used today in the Western world to represent the number 1, a vertical line, often with a serif at the top and sometimes a short horizontal line at the bottom, traces its roots back to the Indians, who wrote 1 as a horizontal line, like the way in which it is written inChinese script. The Gupta wrote it as a curved line, and the Nagari sometimes added a small circle on the left (rotated a quarter turn to the right, this 9-look-alike became the present day numeral 1 in the Gujarati and Punjabi scripts). The Nepali also rotated it to the right, but kept the circle small. This eventually became the top serif in the modern numeral, but the occasional short horizontal line at the bottom probably originates from similarity with the Roman numeral I. In some European countries (e.g., Germany) the little serif at the top is sometimes extended into a long upstroke, sometimes as long as the vertical line, which can lead to confusion with the glyph for seven in other countries. Where the 1 is written with a long upstroke, the number 7 has a horizontal stroke through the vertical line.
While the shape of the 1 character has an ascender in most modern typefaces, in typefaces with text figures the character usually is of x-height, as, for example, in .