Hi. Yesterday, we explored the idea of equality.
We saw that math gives us a way to recognize
when two things that look different are actually
the same. A certain number of laps equals a 5K,
or four quarters equals a dollar. A certain number
of meters actually equals a certain number of
miles. And even when the conversion is not obvious
at first, we can usually find a way to make the
relationship work. Now today, I would like to
show you a situation where the intuition completely
breaks down. Imagine a circular track with a
diameter of 100 meters. A perfect circle. Now
the diameter is simply the distance across the
circle through its center. And now let's ask
a question that is similar to yesterday's questions.
How many diameters equal... one trip around the
circle. Now, in math, we refer to the distance
around the circle as its circumference. And the
relationship is actually simple. Circumference
equals pi times the diameter. And since pi is
about 3 .14, one trip around the circle is a
little more than three diameters. So far, nothing
unusual. But... Now let's ask the same kind of
question that we asked yesterday. If I keep going
around the track, will I eventually find a whole
number of circumferences that equals a whole
number of diameters? Now for miles and meters,
the answer is yes. For quarters and dollars,
the answer is yes. For tablespoons and cups,
the answer is yes. But for a circle, The answer
is no. Never. No matter how many times you go
around the track, and no matter how large the
circle becomes, you will never find a whole number
of circumferences that exactly equals a whole
number of diameters. Hmm. The ancient Greeks
called this idea incommensurable. And what it
means is that the two distances have no common
measuring stick. And to me, that is remarkable.
A circle is one of the simplest shapes we can
imagine. And yet, hidden inside is a relationship
that can never be perfectly reconciled. The circumference
and diameter are forever linked, yet never fit
together exactly. And that mystery is wrapped
up in a number we call Pi. Hmm. Well, this week,
we've explored perspectives. We've climbed mountains,
and we've moved between viewpoints. We've gone
deeper into ideas, and we've discovered hidden
roots. We've even built a mathematical tool belt.
And we've learned to see relationships that were
hidden in plain sight. But perhaps one of the
greatest gifts math offers is the ability to
see the world differently. Sometimes it reveals
that two things are equal, and sometimes it reveals
that they never can be. And sometimes, as with
a circle, that realization opens the door to
wonder. And for me, it raises a deeper question.
Where did this relationship come from? We humans
create systems of measurement, miles and meters,
dollars and quarters, tablespoons and cups. And
with enough conversions, we can always make them
equal. But the relationship between a circle's
diameter and circumference is different. We did
not invent it, but we discovered it. So perhaps
our final perspective for the week is simply
this. Why? Why should such a simple shape contain
such a beautiful and surprising relationship?
And for me, it's questions like that are part
of what makes math so fascinating. It's not because
math answers every question, but it's because
math sometimes points us towards deeper ones.
This is Dave from Intersecting Us.