**Summary**

In PicoPhysics the record of observation is important. The quantitative expressions in the record of observation consist of a number along with unit. It is important to study the linkage between units and number used in record of observation. This discussion is about concept of numbers and there manipulation using arithmetic operations. A mapping operator is defined that generates numbers and units for record of observing objects composed for more than one identity.

Besides numbers representing quantity, we find other numbers in pure mathematics. These are Infinite, Zero and Negative numbers. Negative numbers represent transactions abstracted thru use of arithmetic operators. It is seen that the result of Mathematical abstraction can only be selectively applied to physical reality.

A reality can not be measured in-absentia. The binary zero has no place in record of observation. A record of observation, without mathematical processing (except scaling) will have numbers represented by square of a Real number. To address in-absentia condition, the observed reality can be conceptualized to be composed (minimum) two mutually independent parameters, each represented by mutually exclusive real number. The exclusiveness together with two parameters is enough to exclude the in-absentia results.

The usable number concept in Pico-Physics is derived from Set theory. Set theory extends the range of numbers to set of transfinite numbers. Real numbers express the observed reality. The real number it-self is obtained by transforming a transfinite number ‘representing the object’ using a transformation number representing unit reality. The set of usable numbers are subset of transfinite number set.

The available series of transfinite numbers is an intrinsic characteristic of object and units of observation. The reality can be represented by a minimized set of constituents, parameters or properties. The cardinality of representative set of numbers represents dimensions of reality. It also determines the infinite order of transfinite number representing the observed object. The infinite order is dimensions of the reality.

The infinite order of the transfinite number in observation record can be computed by subtracting the infinite order of unit of measurement from reality. It is generally zero. Reality, set theory and Pico-math all provide finite real numbers as the set to record reality.

To measure one identity we need units from known identity with infinite order greater than that of unknown identity minus two. A mapping operator is defined that generates numbers and units for record of observing objects composed for more than one identity.

The measure of two independant identities is same when they are can be mapped 1:1 Conformal on each other.To execute measurement a Conformal Mapping is required between . object and unit of measure. The measure in this case is determined by the mapping density and unit of measure.

The number space (in the represented case, number plane,) connects arithmetic, algebra, geometry and co-ordinate geometry together into a unified strata of knowledge. It provides for geometrical visualization of nature.

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