Hi, this week we've been exploring perspectives.
And today I'd like to share a perspective that
we use all the time without even thinking about
it. The perspective of equality. So this morning
I am running a 5K race on a standard 400 meter
track. Nothing serious, just a workout. But it
made me wonder, how many laps is a 5K? Now, since
5 ,000 meters divided by 400 meters is 12 .5,
the answer is 12 .5 laps. Now, in math, we would
write 12 .5 laps equals one 5 -kilometer race.
Now, at first, that seems pretty ordinary. But
notice something. A distance measured in laps
became equal to a distance measured in kilometers.
Two different descriptions. but one reality.
Now, suppose I do not like the decimal. I can
multiply both sides by two, and now I have 25
laps equals two 5 -kilometer races. Again, same
relationships, different perspective. And now
everything is a whole number. And what makes
this interesting and kind of cool is on a track,
if I run enough full laps around the track, I
will eventually reach an integer multiple of
5K distances. Now, we do this kind of thing all
the time. Four quarters equals one dollar. Sixteen
tablespoons equals one cup. Meters convert to
miles. Inches convert to centimeters. And math
gives us a way to recognize when two things that
look different are actually equal. Now, what's
even more interesting... is that many of these
relationships are human inventions. We created
dollars and cups and miles and kilometers. And
because we created them, we can always find a
conversion between them. If we keep scaling the
relationship, eventually we can often express
it using whole numbers. For example, one lap
around a 400 -meter track is about one quarter
of a mile. But if I keep running enough laps,
eventually I can find a whole number of laps
that equals a whole number of miles. Now the
details may be messy, but the conversion exists.
And to me, that's one of the beautiful things
that mathematics does. It helps us to see equivalents.
It helps us to recognize when two different descriptions
are really talking about the same thing. But
tomorrow. Tomorrow, we're going to look at a
famous example involving a circle. And at first,
it will seem like just another conversion problem,
like today. But it leads to one of the most surprising
discoveries in all of mathematics. We will discover
that two distances that can never be converted
into matching whole numbers. And that simple
fact changed mathematics forever. I really hope
to see you tomorrow.