Factors and Orders of Magnitude

Numerical ranges are a very important part of many patent claim recitations, and they can be pivotal in the determination of non-obviousness, one of the three requirements of patentability (see Graham v. Deere and 35 U.S.C. 103, 383 U.S. 1, 148 USPQ 459 (1966)).

However, factors and orders of magnitude express extreme ranges of numerical values.

A "factor" is a numerical multiplier of a subject numerical value. For example, a factor of 2 implies a doubling, a factor of 3 implies a tripling, and a factor of 10 implies an increase of ten times the subject value.

Also, an increase by a "factor of 10" implies an increase of an "order of magnitude". A factor of 100 increase implies two orders of magnitude. Thus, orders of magnitude are expressed by that are exponential powers of 10.

To bring orders of magnitude into some perspective, let us compare the size of nuclei to the size of atoms. This calculation has implications in various nuclear, chemical, biological, electronic, and physical fields. The size of atoms vary according to their atomic and molecular surroundings, i.e., their interactions with neighboring and nearby atomic and nuclear entities. Atoms can be in a state of bonding, e.g., ionic or covalent bonding, which can affect the atom size when compared to a ground or nonbonded state. Also, varying pressure or temperature can affect the size of atoms, e.g., states at the critical point of a material differ from those at supercritical or subcritical conditions, perhaps even if only a small amount.

By contrast, the size of the nucleus remains substantially constant regardless of chemical bonding or physical conditions of the surrounding. This is because the nucleus is surrounded by an atmospherical "blanketing" layer of electrons rotating and spinning around the nucleus. Chemical reactions have essentially no effect on the nuclei.

The sizes of atoms range 32pm for the smallest, He or Helium (a noble gas), to 225pm for Ce or Cesium (an alkali-metal). A picometer, pm, is 10E-12 m or one-trillionth of a meter. The corresponding nuclear sizes are 4fm and 12.8fm, based on the formula d=2.5(nuclear mass)(E(1/3)) [wikipedia.org, "nuclear size"]. A femtometer, fm, is 10E-15 or one-thousand-trillionth of a meter.

Thus, the relative sizes of atom to nucleus for He and Ce, are 8000:1 and 2000:1. Similarly, for the heaviest atom, U or Uranium, the corresponding atomic and nuclear sizes are 175pm and 30fm for a relative atom-to-nuclear size of 6000:1.

In order to better visualize these relative sizes, one might use a 100m soccer field as a comparative diameter (a "metaphor", say) for the atom, a relative order of magnitude of 3, equal to 1000:1 relative size, corresponds to a nuclear size of 10cm. For He, Ce and U, the relative nuclear sizes in this metaphor are 1.2cm, 5cm and 1.7cm!


Comparing results in terms of factors, or especially or orders of magnitude differences, when arguing unexpected results to support unobviousness of an invention, is quite difficult for a patent examiner or administrative judge to ignore and dismiss.

Other comparisons might be made with the sizes of leptons, such as electrons, or of neutrinos, or of quarks, or even of macro-objects such as galaxies or other celestial or astronomical groups. However, the electrons and neutrinos represent a size that might be beyond comparison with current technology.

If not already, then perhaps the concepts of factors and orders of magnitude should be added to students Standards of Learning (SOLs) for eighth graders in the U.S.? Also, adequate metaphors should be applied as well to enable the students to adequately visualize these kinds of relationships. Consider the trillion dollar U.S. Budget, terabyte and petabyte storage drives, the 10-100 trillion cells in a human adult, etc as further examples for new comparisons and metaphors. Or is it already so? At least one book has already been published that gives plenty of such comparisons for teaching school students.

Francis "Fran" Lorin
siberkhem.com