Life is hardly more than a fraction of a second. Such a little time to prepare oneself for eternity. ~ Paul Gauguin
Fractionalization surrounds us. We
have been witnessing prominent and all too public splintering of
socio-political viewpoints on our phones, our computers, our radios, our TVs and at our brew pub eight days a week. Some virtuosos insist that until we can gain more
tolerance of each other we cannot conquer or at least live with our encompassing
socio-political fractionalization. In an age of global warming surrounded by
polarized factions of public zealots, tolerance remains a fitfully-distributed
skill to counter this fractionalization.
Fractionalization is based on the
Latin word fractus, meaning breaking into parts. About 3000 years ago, ancient
Egyptian hieroglyphs contained fraction notations. They used the Eye of Horus to represent
different unit fractions, illustrated below in two figures.
Figure 1: Fractions in the Eye of
Horus
Figure 2: Other fractions at the Eye
of Horus
Horus’ Eye itself symbolically
represented prosperity, protection and health. Horus was characterized as a
falcon, often as a peregrine falcon, or as a human with a falcon head. The
first figure shows the specific series of fractions, a geometric sequence of 2,
that the Eye of Horus itself represented.
The second figure illustrates how
Egyptian hieroglyphists represented other fractions with the Eye, here for the
fraction 1/5. The Egyptians’ means of falcon fractionalizing was far older and
more straightforward than the Romans, who used a duodecimal rather than a
decimal system for fractions.
Moving east, in 100 BCE the
Chinese not only developed a way to use fractions for comparisons, but how to make
calculations using them. The Chinese also created a notation of fractions that
is analogous to how we report fractions. These fractional procedures were presented
in the Nine Chapters on the Mathematical Art. There’s a concept,
mathematical art.
Leonardo Fibonacci was the first
European to use the fraction bar in the early 13th century, coincident with his
discovery of what later became called the Fibonacci sequence (aka, Fibonacci
ratio). The first recorded use of the word fraction in the West was in the
mid-14th century. Decimal fractions were introduced by an Islamic scholar in
952. European numeric intellectuals re-invented decimals in the late 16th
century. Afterwards, more than mathematicians realized that writing fractions
as decimals made arithmetic far easier.
Fractionalizing has thus ensued
for a very long time; way before we first learned about them in primary school.[1]
Unsurprisingly, even in 2023 not
everyone is happy with fractions. Eva Moskovitz, an American education reform
leader, stated “Schools can ebb and flow. It can be phenomenal one day, and
then you hit fractions and it falls apart.”
Erudite mathematicians have
characterized fractions in 9 ways: proper and improper, mixed, like and unlike,
terminating and non-terminating, and recurring and non-recurring.
As you may recall from
grade-school math – although I didn’t – a proper fraction is one which
has its numerator value less than the denominator. For example, ⅔ and ¼ are
proper fractions. An improper fraction has its numerator greater than
the denominator, such as 5/2
or 9/7. A fraction
represented with its quotient and remainder is a mixed fraction; 3 ⅔ is
a mixed fraction, where 3 is the quotient, ⅔ is the remainder.
If and when two fractions have
the same denominator, they are said to be like fractions; 7/2 and ½ are like fractions, so we can
easily perform addition and subtraction operations on them. When two fractions
have different denominators, they are said to be unlike fractions. For
example, 5/2 and 3/5 are unlike fractions so we
need to rationalize their dissimilar denominators before performing proper
addition and subtraction. Like many, my now very distant memories of
denominator rationalization aren’t exactly exhilarating. So it goes.
To determine whether a fraction
(and its equivalent decimal) is terminating or non-terminating you just
need to determine the prime factors of the denominator when the fraction is in
its simplest form. If these factors are made up of 2s and/or 5s, the decimal
will terminate; if not, it will not terminate. The fraction ½ is a terminating
fraction; its decimal equivalent is a ceasing 0.500.
Finally in the fraction
sweepstakes, there are recurring and non-recurring fractions. A
recurring fraction/decimal exists when decimal numbers repeat forever. An
example of a recurring and non-terminating fraction is 1/3, which when decimalized becomes 0.33333
forever and ever. A well-known nonrecurring, non-terminating
fraction/decimal is pi (π, in Greek notation). Pi is the fraction derived from
dividing a circle’s circumference by its diameter; very approximately, 3.14159.
To date, the most accurate value of π uses 62,831,853,071,796 digits, which was
achieved by University of Applied Sciences of the Grisons in Switzerland two
years ago. Their computer system completed this calculation within 108 days. OMG.
There are several political
fractions that remain noteworthy. Joe Biden’s electoral college victory margin
for the presidency was 26/538 or a significant 4.8%.[2]
The Dems’ Senate vote margin – including 3 Independents who vote in their
caucus – is a mere 2/100 or 2%. The fully-factional Repubs’ House vote margin is
a paltry 9/435 or 2.1%. These slender margins require constant management and
cajoling. Senate Majority Leader Schumer has burned much midnight oil. House
Speaker McCarthy also has worked like a dog, likely because of his oxymoronic
Freedom Caucus. Flexible tolerance for cross-party votes in the Congress remains
an endangered action.
What if we consider using these 9
mathematical types of fractions to describe our severe political fractionalization.
Could such melding between math and politics somehow allow for greater
tolerance and less upset between factions’ fractionalization?
If we are to achieve any success
in merging math with politics, we will first need to eliminate improper and proper
fractions. No faction will ever consider their position improper; so out go
fractions that have their numerators greater than denominators. Those proper
ones, the ones with smaller numerators, should not be included either because each
and every politician considers their votes always proper, even if it’s quite
unseemly. Instead, we’ll need to transform all such improper and proper
fractions into mixed ones, which shouldn’t be too hard.
I suspect mixed, like, unlike, non-recurring
and non-terminating fractions don’t have the same fraught connotative concerns
as improper ones, so they’re worthy of usage in the political realm. This is especially
true for terminating ones that need to have nothing to do with 2-, 4- or 6-year
tenures of political service.
After all, mixed drinks have
become far more popular than they were in the late 1960s through the beginning 1980s
when wine and craft beer became liquid royalty. Nevertheless by the mid-2000s
cocktail culture rose again. During the last decade there’s been nothing like a
Manhattan to ease tensions, even though you’ve never lived there. Perhaps
that’s true for mixed political factions as well. For the greater good, let’s
have mixed, recurring, non-terminating discussions among many folks that might
lead to fading fractionalization.
[1] At
California primary schools fractions are first taught in third grade.
[2] In
presidential elections the 3 normally non-voting House members from Washington,
DC vote in the electoral college.
