I remember a friend’s post: “Another day without using Pythagoras’
Theorem.” The first thing that came into my mind was that day, she crossed the
street in a diagonal to catch the bus, so I tried to remind her that the
hypotenuse is less than the sum of its sides. Maybe she made no calculation,
but instinctively we apply mathematics almost every day.
During a session on the operative systems module, our
lecturer mentioned that every task had a consecutive number assigned. Later,
with that number, it was sequentially assigned to a particular processor. E.g.,
the first one goes to the first processor, the second one to the second, and the
third to the third one; however, as there were more tasks than processors, if
there were only 7 processors, the eighth task had to return to the first
processor, the ninth to the second and so on.
He asked what processor would be assigned for task 1234 if only
seven processors existed. I knew it was a division we had to calculate, but we had
to use the properties of the remainder of an integer division. The remainder
was between 0 to 6. For the first task, 1, the remainder was 1. For the sixth
task was six, and for the seventh, it was 0. It meant that we only had to divide
by seven, but instead of focusing on the quotient, we had to look at the remainder
as it would help to compute the processor number. For task 1234, the assigned machine should be 2 (seven processors).
Later, I also saw many other applications of the remainder,
to calculate prime numbers on a computer program or just a fixed part of a large
integer number.
When asked about how mathematics is used nowadays (out of
the classroom), we need to be more mindful (and less ungrateful) as we
use several applications based on average data. E.g., the time to go from one
place to another based on average user data.
But other more “advanced” applications might be worth a try
from time to time. One day, I had to send a parcel using stamps, and I had to
choose what stamps to use to cover the total price I had to be careful as I had
a limited set of stamps but from different denominations, including 1st,
2nd class, and others with a fixed price. Framing the situation as a
combinatorial problem made me find a way to send any parcel using the minimal quantity
of my available stamps for any delivery, covering the total price. I remember
the face of the teller every time I appeared with my parcel full of stamps
matching the price.
Recently, while watching old TV series, an episode where a
suspect challenged the detective with a “little” problem, a “minimum
information” problem: In a room, are several sacks of gold pieces, as many as
you like. Each sack contains several of these gold pieces. One sack, however,
is full of artificial gold pieces, and they weigh differently. Now, the solid
gold pieces weigh, let’s say, a pound each. And the artificial pieces weigh a
pound and an ounce. Now, you have a penny scale, so you put the penny and get a
card with the printed weight. But you only have one penny. Now, which sack has
the artificial gold pieces?
While you think about the solution, let me share other
applications of basic math to solve math problems. I remember a day someone was
asking me about the transformation from Fahrenheit to Celsius and vice versa,
and I couldn’t remember the formula; however, I remembered that -40°F was -40°C
and 212°F was 100°C. I also knew a linear formula was behind it, so I just made
a linear regression with my calculator. Then, I could get the precise formula
just by applying the basic concept that you can define a line using only two
points.
So, maths is everywhere, for instance, when we have to calculate
the amount on our balance to cover next month’s rent and utilities, the suitable
hours to call home if they are in a different time zone, or the time we should leave
home if we want to be on time for our lectures. Sums and additions, also just
the angle of the watch’s minute hand, tell us if we will be on time.
Recently, I remembered a student that wanted a simple
explanation of the expected value. After some examples, she said it is just
another way to say “mean” or “average.” Later I told her that the idea of an average
might be misleading if we restrict it to only a simple but not a weighted
average. Expectations are based on probability distributions, making one event
more “probable” than another.
It is also the same when we try to describe data using only the
mean. After some years of dealing with datasets, I ask: ok, this is mean, but
what about the standard deviation, any information about the range, the mode,
and the median? Sometimes those questions might be annoying, but that’s essential
if we want to interpret the data. However, a chart might be more informative.
Back to the minimum-information problem, I won’t give you
the solution, but please, think you can enumerate the bags, and later you can
take a different number of pieces of each bag. You already know the phony
pieces weigh an ounce more than the original. Does it help?
Now let's suppose you need to calculate the approximate height of a tree. Can you use basic maths to compute the value on a sunny day?
