Everyone
knows it to be true that excellence with language and excellence with math are
mutually exclusive gifts. It is an idea
as old as Plato, who sided with the mathematicians by—so the legend goes—writing
above the entrance to his Academy the ward and warning: “Let none ignorant of
mathematics enter here” (It’s much catchier in Greek). When he got around to imagining his perfect society
in his Republic, he went further,
having poetry expelled from that imagined realm. Yes, everyone knows one cannot be good with
both letters and numbers.
But what everyone knows—on this
point, as on so many others—is wrong.
I begin this meditation with the
confession that I am an English professor, with all my degrees in the
humanities; and yet, this does not mean that I am a stranger to the ways of
numbers. On the contrary, I balance my
checkbook without a calculator, I figure my change in the few seconds between
handing over my cash and having the change amount show up on the register
readout, and I occasionally enjoy doing other sorts of math in my head. It is simpler than many think. The secret—which I will seek here to explore
by examining one very real example from my own hardly excessively mathematical
existence—is to forget the algorithms and formulas and all the machinery of
symbol manipulation that schools pass off as education in mathematical
thinking, but which is really just training in how to avoid it.
To do math in your head, you must
truly think about numbers and their relationships; numbers, and not necessarily
their symbols, which we call “numerals” (Our failure to teach people to
distinguish between the two is one
reason people cannot do mental math). If
you learn to do this, much can be done with no pen or paper, and using nothing but
the basic arithmetic one learns in grade school—which brings me to Sunday
morning.
I was driving to church one recent
Sunday morning, and I was running low on gas.
It was near the start of the term, so I was trying to make each penny
count—by, among other things, counting my pennies—as we trundled towards the
end of our first month and payday. I
had, I reckoned, enough to make it through town to the North end, where I could
worship, then head back south, stopping to fill up with the cheapest gas in
town just before exiting the city and heading for home. Arriving on the North end, however, I thought
it wise to get a few dollars’ worth of fuel there, before risking the drive back
through town, where all those traffic signals and slow motorists would no doubt
conspire against me to make my return trip as wasteful as possible. Better safe than sorry.
And so, I sought to buy just five
dollars’ worth of gas before church.
Afterwards, I would largely fill up ($30 ought to do it, I
thought). It turned out that both times
I bought gas for $2.54/gallon, and each time I nearly hit my mark: my expenses
were $4.98 and $30.03, respectively. After
the second stop, my gas gauge read practically full.
As I rolled home, I wondered exactly
how much gas I had bought, so during the 15-minute highway drive, I meditated
on the numbers, and by the time I got home, I not only had an estimate, but
consultation of my receipts revealed a precise match between my accounting and
theirs.
Here is how I did it:
First, I added the 3 cents from my
second purchase to the total from my first, getting $5.01. That, plus the $30 remaining from the second
purchase makes a total expense of $35.01.
Now I know how much I spent, and how much it cost ($2.54/gallon), now
for some questions. They will allow me,
not to calculate a precise answer in the mechanical way they taught us in
school, but to home in on an approximation that will be as precise as I need.
Question
1: (Approximately) how much gas did I buy?
Step 1:
The price, I noticed, was just over $2.50/ gallon (which is half of $5), at
which price $5 would thus buy me two gallons.
Seven times that would get me 14 gallons (7 x 2=14) at $35 (7 x
5=35). My total expense (2 pennies shy
of five, plus 3 pennies over 30) is $35.01.
So 14 gallons is a good estimate for how much gas I bought (and jibes
with my nearly full reading on my 15-gallon tank), but the real answer (since
each gallon would actually cost an extra 4 cents beyond $2.50) would be
slightly less.
Question
2: How much less?
Step 2:
Since each gallon costs 4 cents more the my original estimate of $2.50, 14
gallons would cost me slightly more than my original estimate of $35. Now, I could try to multiply 4 cents by 14,
but that is beyond the 12x12 multiplication table they taught us in school, so
it is easier if I break it down into some smaller steps, so…
Step 2a:
2 gallons would cost me not $5, but $5.08 (5 + .04 [for gallon 1] + .04 [for gallon
2]).
Step 2b:
Seven times two is 14, so to estimate the extra cost in fourteen gallons, I
would multiply those extra eight cents by seven (7 x 8=56; notice how I use no
math more complex than the basic multiplication table.). This means that fourteen gallons would cost,
not $35 flat, but 56 cents beyond that ($35.56). Which is clearly 55 cents beyond what I spent
(56-1=55).
Step 3:
This makes more calculation easy. Clearly,
since the price difference is only 55 cents, the difference in gas will be much
less than a full gallon (since that costs 254 cents, about 4 or 5 times as
much). Also, I notice that 55 is a
multiple of 5, and $2.54 is also a multiple of five—if I round up one
penny. So, I round up, and divide 255
cents by 5 cents to see that 5 cents will buy me 1/51 of a gallon (5 goes into
25 5 times flat; 5 into 5 once; so, 255/5=51—no pencil needed).
Step 4:
Now, I round again: 1/51 is ugly, but it is close to a clean 1/50, which is
2/100, or 2%. (for % reads as “percent,”
and “per cent” just means “per 100”—see, language and math can be friends). So we
say that 5 cents will buy 2% (2 percent) of a gallon (a bit less, but,
again, we are estimating to make the math easier; this is mental arithmetic,
not engineers launching a space probe).
Step 5
(This could be part of step 4, but let us keep things simple): Thus, if 5 cents
buys 2 percent, then 55 (5 x 11) cents will buy 22 (2 x 11) percent of a
gallon. So I know that my purchase was
55 cents short of $35.56 (the cost of 14 full gallons at $2.54/gallon), and so
0.22 (22%) of a gallon shy of 14 gallons.
Subtract 20 from 100, then two from that, and you see that 100% minus
22% equals 78%, so that my purchase was 13 gallons and 78% of a gallon, or
13.78 gallons.
Step 6:
Because we rounded off several times, this is an approximation, so we round off
a bit and call it 13.8 gallons. Now we
go to the receipts, and I find that my first purchase that day ($4.98) got me
1.963 gallons, while my second ($30.03) got me 11.827 gallons. I use my calculator for a precise answer, and
I find the answer is smack between my two estimates: 13.790 (This, despite the
fact that their measurements, to the thousandths place, was 10 times more
precise that my estimating of percentages).
And so ended my adventure that
Sunday in mental mathematics. It is
bracing every now and then to stretch the mind thus, like composing a poem, or
expressing your thoughts in sentences of purely Anglo-Saxon monosyllables. Let us not, in our rush to take advantage of
the technological trappings of the twenty-first century, forget to remain familiar
with numbers themselves; their meanings, and not just their symbols. Reading, writing, and reckoning: the three R’s
of traditional pedagogy; we would all do well to do them better.
So, was Plato wrong? Not really.
Mathematics is an essential part of a good education, but so is language
and poetry. Let us not forget that,
while Plato criticized Homer, he also knew him well enough to quote him. Any critic who has read your work closely
enough to quote it by heart—that critic is not your real enemy. Plato himself was, after all, not a geometer,
but a writer, and one of great skill.
Part of his claim to fame is that he invented a new literary form: the philosophical
dialogue, and a great deal of the dialoguing therein involved wrangling over
the meaning of words, like “justice.” As
a final irony, Plato is now read much more often by poets and writers than by
scientists and mathematicians. So,
whether poets, philosophers, or essayists; mathematicians, scientists, or
what-have-you; there is no need for us to hate, fear, or expel each other. Instead, we should all try to be more
familiar with all the tradition that is our heritage on this great and glorious
globe. So far as we know, human culture
is the only one that exists. Let’s all
try to appreciate it a little better.
