0:00:00.750,0:00:02.610
We have seen multiple times in our life

0:00:02.610,0:00:08.055
that distance can be viewed as rate times[br]time.

0:00:08.055,0:00:10.990
Now what I wanna do in this video is use[br]this fairly

0:00:10.990,0:00:15.730
simple formula right over here, this[br]fairly simple equation, to understand that

0:00:15.730,0:00:20.140
units can really be viewed as algebraic[br]objects, that you can kind

0:00:20.140,0:00:24.360
of treat them like variables as we work[br]through a formula or equation.

0:00:24.360,0:00:27.480
Which could be really, really helpful to[br]make sure that

0:00:27.480,0:00:33.000
our, that we're getting results in units[br]that actually make sense.

0:00:33.000,0:00:38.030
So for example, if someone were to give[br]you a rate, if they were to say a

0:00:38.030,0:00:43.360
rate of, let's say, 5 meters

0:00:43.360,0:00:48.960
per second, and they were to give you a[br]time, a time of 10

0:00:48.960,0:00:53.220
seconds, then we can pretty, in a pretty[br]straight forward way, apply this formula.

0:00:53.220,0:00:57.770
We say well distance is equal to our rate,[br]5 meters per

0:00:57.770,0:01:03.440
second, times our time, times our time[br]which is 10 seconds.

0:01:03.440,0:01:07.050
And what's neat here is we can treat the[br]units,

0:01:07.050,0:01:11.160
as I just said, like algebraic constructs,[br]kind of like variables.

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So this would be equal to, well

0:01:13.640,0:01:16.010
multiplication doesn't matter what order[br]we multiply in.

0:01:16.010,0:01:17.760
So we can change them, the order.

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This is the same thing as 5 times 10.

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5 times 10 times meters per second, times[br]meters per seconds, times seconds.

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And if we were to treat our units as these

0:01:32.593,0:01:35.141
kinds of algebraic objects, and we could[br]say, hey look,

0:01:35.141,0:01:37.273
we have seconds divided by seconds, or you[br]have a

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seconds in the denominator multiplied by[br]seconds in the numerator.

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Those are going to cancel out.

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And 5 times 10, of course is, 5 times 10,[br]of course is 50.

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So we would be left with 50, and the

0:01:50.900,0:01:54.540
units that we're left with are the meters,[br]50 meters.

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So that's pretty neat.

0:01:55.750,0:01:56.690
The units worked out.

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When we treated the units out like[br]algebraic objects, they worked out so that

0:02:00.900,0:02:06.430
our end units for distance were in meters,[br]which is a unit of distance.

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Now you're saying, okay, that's, that's[br]cute and

0:02:07.910,0:02:09.310
everything, but this seems like a little[br]bit

0:02:09.310,0:02:14.290
of too much overhead to worry about when[br]I'm just doing a simple formula like this,

0:02:14.290,0:02:18.020
but what I wanna show is that even with a[br]simple, with a simple formula like

0:02:18.020,0:02:22.490
distance is equal to rate times time, what[br]I just did could actually be quite useful.

0:02:22.490,0:02:25.570
And this thing that I'm doing is actually[br]called dimensional analysis.

0:02:25.570,0:02:27.150
And it's useful for something as simple as

0:02:27.150,0:02:29.370
distance equals rate times time, but as[br]you go

0:02:29.370,0:02:32.340
into physics and chemistry and[br]engineering, you'll see

0:02:32.340,0:02:35.880
much, much, much more I would say hairy[br]formulas.

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And when you do the dimensional analysis,[br]it makes

0:02:38.090,0:02:40.620
sure that your, that the math is working[br]out right.

0:02:40.620,0:02:42.660
It makes sure that you're getting the[br]right units.

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But even with this, let's try a slightly[br]more complicated example.

0:02:45.960,0:02:49.170
Let's say that our rate is, let's say,

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let's keep our rate at 5 meters per[br]second.

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But let's say that someone gave us the[br]time.

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Instead of giving it in seconds, they give[br]it in hours.

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So they say the time is equal to 1 hour.

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So now let's try to apply this formula.

0:03:04.250,0:03:09.596
So we're gonna get distance is equal to 5[br]meters per second, 5

0:03:09.596,0:03:15.820
meters per second, times time, which is 1[br]hour, times 1 hour.

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But what's that gonna give us?

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Well the 5 times the 1, so we multiply the[br]5 times the 1.

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That's just going to give us 5.

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Well then we have to, remember we have to,[br]the units algebraically.

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Where we're going to do our dimensional[br]analysis.

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So it's 5, that we have meters per second,[br]times hours, times hours.

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Or you could say 5 meter hours per second.

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Well this doesn't look like a, this isn't[br]a, a,

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a set of units that we know, that, that[br]makes

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sense to us, this doesn't feel like our[br]traditional units

0:03:45.490,0:03:48.350
of distance, so we wanna cancel this out[br]in some way.

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And it might jump out of you, well if, if[br]we can

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get rid of this hours, if we could express[br]it in terms

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of seconds, then that would cancel here[br]and we'd be left with

0:03:55.240,0:03:58.902
just the meters, which is a unit of[br]distance that we're familiar with.

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So how do we do that?

0:04:00.410,0:04:04.710
Well we'd wanna multiply this thing by[br]something that has hours in

0:04:04.710,0:04:07.150
the denominator, and seconds in the

0:04:07.150,0:04:10.330
numerator, times essentially, seconds per[br]hour.

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Well how many seconds are there per hour?

0:04:13.160,0:04:19.540
Well, there are 3600, let me do this in a,[br]I'll do this color, there are 3600 seconds

0:04:19.540,0:04:27.510
per hour, or you could even say that there[br]are 3600 seconds for every 1 hour.

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So when you, now when you multiply, these[br]hours will

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cancel with these hours, these seconds[br]will cancel with those seconds,

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and we are left with, we are left with 5[br]times

0:04:38.385,0:04:43.232
3600, what is that, that's 5 times 3000[br]would be 15000.

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5 times 600 is another 3000.

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So that is, it's equal to 18000, and the[br]only units that

0:04:51.956,0:04:57.786
we're left with, we just have the meters[br]there 18 oh, it's

0:04:57.786,0:05:03.310
18000, 18000, 18000 meters, right?

0:05:03.310,0:05:04.540
And so, this is, we're done.

0:05:04.540,0:05:08.300
We've now expressed our distance in terms[br]of units that we recognize.

0:05:08.300,0:05:13.960
If you go 5 meters per second, for 1 hour,[br]you will go 18000 meters.

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But let's just use our little dimensional[br]analysis muscles a little bit more.

0:05:17.020,0:05:18.540
What if, what if we didn't want the answer

0:05:18.540,0:05:20.550
in meters, but we wanted the answer in[br]kilometers?

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What could we do?

0:05:22.000,0:05:27.570
Well we could take that 18000 meters,[br]18000 meters, and if

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we could multiply it by something that has[br]meters in the

0:05:30.700,0:05:36.190
denominator, meters in the denominator,[br]and kilometers in the numerator, then

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these meters would cancel out and we'd be[br]left with the kilometers.

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So what could we multiply it so we're not[br]really changing the value?

0:05:42.760,0:05:45.354
Well we want to multiply it by essentially[br]1, so

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we wanna write equivalent things in the[br]numerator and the denominator.

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So 1 kilometer is equivalent to 1000[br]meters.

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So one way to think about it, we're just[br]multiplying this thing by 1.

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One kilometer over 1000 meters.

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Well, one kilometer is 1000 meters.

0:06:03.260,0:06:05.670
So this thing is equivalent to 1.

0:06:05.670,0:06:07.970
But what's in need is when you multiply,[br]we

0:06:07.970,0:06:12.446
have meters cancelling with meters, until[br]you're left with

0:06:12.446,0:06:16.570
18000 divided by 1000 is equal to 18, and

0:06:16.570,0:06:20.010
then the only units we're left with is the[br]kilometers.

0:06:21.020,0:06:23.980
And we are done, we have re-expressed our

0:06:23.980,0:06:27.560
distance instead of in meters, in terms of[br]kilometers.