WEBVTT

00:00:02.730 --> 00:00:05.389
Hi, I'd like to finish this week with a math

00:00:05.389 --> 00:00:08.910
problem. Not because there's a lesson, but because

00:00:08.910 --> 00:00:12.890
it's about something to wonder about. So, imagine

00:00:12.890 --> 00:00:17.329
the positive x -axis and positive y -axis forming

00:00:17.329 --> 00:00:21.710
a 90 degree angle. Now, imagine every possible

00:00:21.710 --> 00:00:24.850
line that begins at the origin and extends somewhere

00:00:24.850 --> 00:00:27.910
into that first quadrant. There are infinitely

00:00:27.910 --> 00:00:33.219
many of those lines. Okay. Next, draw the 45

00:00:33.219 --> 00:00:36.679
degree line exactly halfway between the two axes.

00:00:37.460 --> 00:00:41.219
And that line divides our 90 degree angle perfectly

00:00:41.219 --> 00:00:44.899
in half. So it may seem reasonable to say that

00:00:44.899 --> 00:00:47.700
there are just as many possible lines below the

00:00:47.700 --> 00:00:52.560
45 degree line as there are above it. Okay, good.

00:00:52.679 --> 00:00:55.539
Now let's look at those same lines but in a different

00:00:55.539 --> 00:00:59.210
way. Think about their slopes. Every line between

00:00:59.210 --> 00:01:02.909
the x -axis and the 45 -degree line has a slope

00:01:02.909 --> 00:01:07.250
between 0 and 1. And as the lines rotate upward,

00:01:07.489 --> 00:01:11.269
their slopes increase until we reach the 45 -degree

00:01:11.269 --> 00:01:15.150
line, which has a slope of exactly 1. But keep

00:01:15.150 --> 00:01:18.890
rotating towards the y -axis. Now the slopes

00:01:18.890 --> 00:01:22.609
exceed 1. We've got a slope of 2, and then 10,

00:01:22.930 --> 00:01:25.890
and then eventually 100, a million. And as we

00:01:25.890 --> 00:01:29.370
get closer and closer to the y -axis, the slopes

00:01:29.370 --> 00:01:34.349
increase without bound. So, here's our puzzle.

00:01:34.890 --> 00:01:38.250
The lines in the bottom half correspond to all

00:01:38.250 --> 00:01:42.230
the real numbers between 0 and 1. The lines in

00:01:42.230 --> 00:01:44.989
the top half correspond to all the real numbers

00:01:44.989 --> 00:01:50.290
exceeding 1. And yet, geometrically, the two

00:01:50.290 --> 00:01:56.179
groups of lines seem to be the same size. So,

00:01:56.179 --> 00:01:59.700
are there really just as many real numbers between

00:01:59.700 --> 00:02:04.400
0 and 1 as there are between 1 and infinity?

00:02:06.019 --> 00:02:09.580
Hmm, think about that for a while. Maybe draw

00:02:09.580 --> 00:02:12.819
a picture, try some ideas. And since we've spent

00:02:12.819 --> 00:02:15.620
this week talking about community, here's another

00:02:15.620 --> 00:02:18.900
idea. Find someone else and ask them the question.

00:02:19.300 --> 00:02:21.699
And maybe they'll see something that you don't,

00:02:21.699 --> 00:02:24.979
or maybe you'll see something they don't. And

00:02:24.979 --> 00:02:28.080
then maybe one of you will say, wait, do you

00:02:28.080 --> 00:02:32.539
see what I see? I think that's a pretty good

00:02:32.539 --> 00:02:36.580
way to do math. This is Dave from Intersecting

00:02:36.580 --> 00:02:36.759
Us.