Technical Supplement 2C
The Ratio Notation and the The Power of 1
This section starts with more philosophical thoughts about the isotope Ratio or R
notation. This isotope notation leads to calculations that have many rather odd properties
centered around the number 1. As we shall see, these properties account for the fact that
when switching from the  notation to the R notation, isotope algebra changes so that
subtraction becomes division, multiplication = addition, and 1 = 0. This section illustrates
some of the math involved in the R algebra, and concludes with practical examples that
show how to set up parallel fractionation equations in the two different  and R notations.
Let’s start with the philosophy.
Sometimes the isotope world seems to revolve around the number 1. After all,
reaction rates of heavy and light isotopes are almost the same, 1:1 so that the ratio of the
rates is usually very nearly 1. And fractionation is expressed as a small deviation from 1.
Near the value 1, a special sort of mathematics seems to apply as the following example
shows. If you take 1.01 and divide 1 by this number, you obtain 0.99, the same fractional
difference of 0.01 from 1 that 1.01 is different than 1. Are you beginning to see the
pattern here? Mathematics near 1 has a special flavor, where inverses are nearly the same
as their converses. This is not usually true. The world near 1 has a special quality, where
values exist as near-mirror images.
Take another example. A current trend in defining  isotope values is to avoid
multiplying by 1000, so that the  values remain as fractions close to….you guessed it, 1.
A value of -25 becomes -0.025 different than 1, and this math for deviants or deviations
from 1 is catching hold in some texts dealing with isotopes. Perhaps this trend reflects the
experience that it is fun to work near 1 where simplifications seem to arise very easily.
But these simplifications disappear farther away from the magical number of 1. After all,
if you write the complex-looking formula  = (-+ 1000)/(1000 - ) from Box 2.1 that
expresses the relationships between two kinds of fractionation factors  and , who
would ever guess that this simplifies to  = -? But when the ratio (-+ 1000)/(1000 - )
is near 1, this simplification holds. Or, who would ever suspect that ln[(1+1)/(2+1)] = 1
- 2 when the  values are expressed as fractions? This is not usually true, but does apply
when 1 and 2 are small so that the ratio in the brackets is near…1. Amazing, but true.
Simplicity seems to have an odd claim and charmed life for mathematics conducted near
1, and here you might get used to expecting the unexpected.
This all becomes a little more interesting when you think about using  or R
notations to calculate fractionations. The  notation involves additions and subtractions
using permil  fractionation factors, e.g., for carbon isotopes -8o/oo CO2 with 20o/oo
fractionation = -8 -20 = -28o/oo plants or, using the fractional definitions of  and
fractionation factors, -0.008 - 0.02 = -0.028. The alternative math involves ratios “R”
instead of , with R values multiplied and divided by  when fractionation occurs,
recalling that  = (1000 + /1000 (see Box 2.1). For example, if R = 0.992 and  = 1.02,
fractionation is given as 0.992/1.02 = 0.97255, which is about -0.028o/oo vs. 1 and also
28o/oo lower than 1. Although there are some slight differences in these results, the point
is that the same basic result emerges with the R and  notation as with the  and 
notation. If you work further with these types of examples and examine the two sets of
equations closely, i.e., the additions and subtractions involving  and  vs. those
multiplications and divisions involving R and , you will find some differences,
especially when values depart greatly from…..you guessed it, 1. The exact equations for
fractionation are actually those for R and , but are closely approximated by the  and 
algebra....as long as you are close to 1.
(Technical Note: Mixing is not truly exact for either set of notation (see Table 2.2
above), and partly for this reason and partly because scientists normally report values in
terms of , this book uses the -based notation for most calculations. Truly exact
equations that incorporate both mixing and fractionation are presented in Section 4.6 and
listed in the Appendix at the end of this printed book).
But let’s return to our theme, the power of 1. You may have noticed that the 
scale does not look like it is close to 1. But if you look at the definition =
(RSAMPLE/RSTANDARD – 1)*1000, and strip away the “*1000” and the “-1” in the
definition, what do you find at the core of a  value? It is a complex-looking ratio of
ratios, but it simplifies to, you guessed it, a value near….1. So, because  values contain
RSAMPLE values that are near the RSTANDARD values, the ratio RSAMPLE/RSTANDARD has the
special advantages of being near 1. You just have to embrace the dialectic that a 0o/oo 
value means that a value of 1 is present at the ratio-of-ratios core of the  definition, i.e.,
0 = 1 at the core of isotope math. If you embrace this mystical bit of reasoning, you are
well on your way to understanding isotope maths.
But what is the point of all of this? Perhaps this is just a bit of isotope
appreciation, a special mathematical glow that accrues from working with 1. The power
of 1 is not obvious in the beginning, but much of the math of isotopes swivels and swirls
around 1, reflecting the basic similarity in the heavy and light isotopes. And poised in this
special place, it turns out that there are many somewhat magical mathematical properties
when numbers are near 1. Especially important for isotopes is that inverses and reverses
come out about the same no matter which side of 1 you are on, the top side >1 or the
bottom side <1. That is the bottom-dweller’s opinion from Mr. Polychaete.
For your reference, here are some equations written both ways, in terms of  using
subtraction and addition, and in terms of the ratio (R) notation using division and
multiplication. Being able to use both approaches allows you easier access to the wider
isotope literature where both sets of approaches are commonly used.
1) Fractionation during formation of a product from a source material
PRODUCT = SOURCE -  and

RPRODUCT = RSOURCE/
where  is the permil fractionation factor for the faster reaction of the light-than-heavy
isotopes expressed in positive units,  is the fractionation factor for the faster reaction of
the light-than-heavy isotopes and is typically slightly greater than 1,  = (1000 +
/1000, and R is the ratio of heavy-to-light isotopes in the source material.
2) Fractionation during a two-way exchange reaction between two substances, A
and B, i.e., A reacts to form B in a forward reaction while B is also reacting to form A in
a reverse reaction:
A – 1 = B – 2
RA/1 = RB/2
where the subscripts 1 and 2 respectively represent the forward and reverse reactions.
3) Fractionation in a steady-state reaction chain consisting of a source and a
downstream pool, where inputs from the source equal outputs from the pool:
Input to Pool = Output from Pool
SOURCE – 1 = POOL – 2 so that POOL = SOURCE – 1 + 2
RSOURCE/1 = RPOOL/2 so that RPOOL = RSOURCE2/1)
Overall, there is an important parallelism at work in each of these examples. The
parallelism is that each time when moving from the  notation to the R-based notation,
division substitutes for subtraction and multiplication substitutes for addition. In a larger
sense these following parallelisms hold across the notations:  equates to R,  equates to
, subtraction equates to division, and addition equates to multiplication.
More examples of these parallelisms are available in Fry (2003) where  is used
in a slightly different sense, i.e.,  denotes the slower reaction of heavy-than-light
isotopes and has values typically slightly less than 1. If you think this practice of
switching fractionation definitions is confusing, you are right. Unfortunately, there is
really no solid consensus in the isotope literature about which fractionation terminology
is correct, and authorities caution that readers must always check the individual study to
see how terms are defined (Hayes 2004). Perhaps this is the disadvantage of 1, that it is
easy to switch from one side of 1 to the other in these R and  definitions, leading to the
extra burden of keeping track of definitions as you read the isotope literature.