1

1 (One) is a number denoting either a cardinality of a single object, or a single standard unit of measure of some quantity of a substance. When one has 1 of something, such as "1 apple" it means the same thing as saying one has "an apple". For this reason, 1 was not always recognized as a proper number as a "group of 1" may seem a redundant concept.

## In Mathematics

In Mathematics, 1 is in variably what is called the Multiplicative Identity, in several number systems. It is the Multiplicative Identity of the Integers, Rational Numbers, Real Numbers, Complex Numbers, and even the Quaternions. What it means for 1 to be a multiplicative identity, often shortened simply to **Identity**, is that when multiplied with another element the product returns that element. That is:

a\*1 = 1\*a = a

1 is also important in the definition of the **reciprocal**. For any element, a, of a system it's multiplicative inverse, a^-1, is the number such that:

a\*(a^-1) = (a^-1)\*a = 1

0, is a notable exception in that, unlike all other integers, rationals, reals, and complex numbers, it does not possess an inverse within those number systems. Because the definition of a **Group** depends on every element having an Inverse, 0 is often excluded to form a Multiplicative Group. For example we have the Multiplicative Group of non-zero rationals, reals, and complex numbers. In fact these are also Abelian Groups because multiplication is commutative in these systems as well as associative. This is something that can not be said of number systems such as the quaternions.

In the Surreal Number system, +1 is the "simplest number" to the right of 0 on the Surreal Number Line. It may be denoted by {0| } or alternatively may be thought of as R(0) (the root node of the right portion of the transfinite binary tree that forms the Surreal Number System). The Surreal Numbers form what is called an **Ordered** **Field**, which means addition and multiplication is defined over it. Here once again, 1 is the Identity Element, and 0 has no reciprocal.

## In Zeroid Theory

Although Zeroid Theory is primarily interested in the development of micro-zeroids, 1 has a surprisingly important role for several reasons. In Zeroid Theory, 1 is called "The Norm", or more technically the **Identity Normoid**. A Normoid is a number that does not change the size of a number, but may or may not change the sign or direction of a number. The Identity Normoid is special in that in changes neither. When the number system is extended to include Zeroids, and possibly even other strange things, the Identity status is preserved for all "standard numbers". These are numbers with a definite size and a sign expressible in the form of cos(x)+i\*sin(x).

In short, 1 is an identity to Zeroids as well as all the other numbers that are assumed to exist within Zeroid Theory.

The second important property of 1, which is also connected to the notion of reciprocals, is that 1 is used as the dividing line between the small and large numbers. This is important because small and large numbers, under multiplication, behave in analogous but opposite ways.

A Number whose size is strictly larger than 1 is said to be a **Large Number**, and a number whose size is strictly smaller than 1 is said to be a **Small Number**. Numbers which are said to be the same size as 1 are called **Normal Numbers**. It might be assumed that 1 is the only normal number, but in fact there is a continuum (at least) of such normal numbers. The most obvious example is -1, which does not actually change the size of a number, but rather inverts its direction, pointing it exactly in the opposing direction. This is true of the real numbers, where it has the effect of flipping the real number line around 0, and the complex numbers, where it is equivalent to a rotation of 180 degrees. But there is more. The imaginary unit *i* can also be thought of as a normal number which doesn't change the size of a number but instead merely rotates it through an angle of 90 degrees counterclockwise.

1 is in fact part of a family of numbers called **Normoids**, that have this invariant property of not changing size, but only possibly changing the rotation or direction of a number.

In Zeroid Theory it is still true that:

1\*a = a\*1 = a

For all numeric elements, a, in Zeroid Theory. This in fact is regarded as one of it's **Scalar Postulates**. However, despite still being the Identity Element, it is not true that every element in Zeroid Theory has a Multiplicative Inverse.

One way in which 1 is treated differently in Zeroid Theory from normal mathematics is that technically 1 must be treated as +1, because every number has a sign. The omission of sign is usually indicative that a number is "signless".

## See Also

0

Zeroid Theory

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