### Video Transcript

In this video, we’re talking about
converting between the Celsius and Fahrenheit temperature scales. Although these are the only two
temperature scales there are, they are two of the most commonly used. By the end of this lesson, we’ll be
able to take a temperature given in one of these two scales and convert it to the
other. To understand how to do that, it
will help to know where the Fahrenheit and Celsius temperature scales came from in
the first place.

Now, when we talk about temperature
scales in general, we know that one end of the scale corresponds to hotter
temperatures and the other end to cold ones. And any particular temperature
scale is defined by the values it gives to specific temperatures. For example, let’s say that we
wanted to make up our own temperature scale. We could call it the Nagwa
Scale. Let’s say that we define this scale
by saying that the temperature of the sun is 1000 and the temperature of a snowflake
is zero. So then, if someone asked us,
what’s the temperature of the sun? We could tell them that it’s 1000
degrees Nagwa. Or similarly, a snowflake is zero
degrees Nagwa. That’s the temperature scale we’ve
come up with.

Now, this may seem a little silly,
but actually not much more than this goes into defining a temperature scale. Partly for this reason, throughout
history, many different scales have been developed. Two of the most common are the ones
we’re talking about here, Celsius and Fahrenheit. Of these two, it was the Fahrenheit
temperature scale that was developed first, and it’s named after its inventor, the
German physicist Daniel Gabriel Fahrenheit. The Fahrenheit scale is oriented
around the temperature at which water boils. And for historical reasons, the
number on this scale at which that event occurs is 212. Then, if we go down to a colder
temperature, the temperature at which water freezes and become a solid, on this
scale that’s at 32.

Now, on some temperature scales,
and the Fahrenheit scale is one of them, when we name a value on that scale, we
include the word degrees in the description of it. For example, we would say that
water boils at 212 degrees Fahrenheit. That where degrees points us back
to the fact that the original form of this temperature scale was developed by an
astronomer. And the idea was that temperatures
could be subdivided in the same way that angles could be divided, into degrees. So anyway, water boils at 212
degrees Fahrenheit, and it freezes at 32 degrees Fahrenheit.

Over time, thanks in part to Daniel
Fahrenheit’s ability to make very accurate thermometers, the Fahrenheit temperature
scale grew in popularity. Sometime later though, a Swedish
astronomer named Anders Celsius came up with an idea for a slightly different
scale. For his scale, Celsius used the
same kinds of physical events, water boiling and water freezing, but he assigned
different numbers to them. Celsius wanted to make a
temperature scale that could easily be divided into 100 equal parts. In other words, he wanted to make a
centigrade scale. In this word, we can see the word
cent, which has to do with 100 or one 100th. For example, a century of years or
a cent of money.

Anyway, with that goal in mind,
Celsius assigned the value of 100 to the temperature at which water boils and zero
to the temperature at which it freezes. So then, on the Celsius scale,
there’s a 100-degree difference between these two temperatures. But notice that that’s not the case
on the Fahrenheit scale. If we subtract 32 from 212, we find
the difference between these values is 180 degrees Fahrenheit. This helps us see that there are
really two differences between the Fahrenheit and the Celsius scales. One difference, as we’ve already
seen, is that they assign different temperature values to the same events.

For example, when water boils in
the Fahrenheit world, we say that happens at 212 degrees. But in the Celsius world, we say it
happens at 100 degrees. So, different numbers there, that’s
one difference between the scales. But another difference is that
changing one degree in one scale is not the same as changing one degree in the other
scale. Let’s say that we take these two
temperature differences on the two different scales, 180 degrees Fahrenheit and 100
degrees Celsius, and we divide them both by 100. If we do that, we find that a
temperature difference of 1.8 degrees Fahrenheit is equal to a temperature change of
one degree Celsius.

In other words, if we increase the
temperature of something by one degree on the Celsius scale, then we’ve increased
the temperature of that same object by almost two degrees on the Fahrenheit
scale. If we express this difference as a
ratio, we would say it’s 1.8 to one or, written as a fraction, 1.8 divided by
one. Now, if we multiply top and bottom
of this fraction by five, then we get this result of nine-fifths. That’s the ratio of how much a
given temperature change in degrees Fahrenheit is to how much that same temperature
change is in degrees Celsius. And we’ll keep this fraction in
mind because it will show up again later.

Now, knowing that we have these two
temperature scales, Fahrenheit and Celsius, we want to find a way to convert from
one to the other. That is, to take a temperature in
one of the scales and change it over to the equivalent temperature on the other
scale. Now, there is a mathematical
relation that shows us how to do this conversion. And knowing what we know so far
about these two scales, we can begin to derive that equation. So far, we have these two
temperature correspondences, that 100 degrees Celsius is equal to 212 degrees
Fahrenheit and that zero degrees Celsius is equal to 32 degrees Fahrenheit.

Using these correspondences, we
want to come up with an equation that relates the temperature in one scale — say the
temperature in degrees Celsius, we’ll call it 𝑇 sub C— to an equivalent temperature
in degrees Fahrenheit, we’ll call it 𝑇 sub F. Now, the way it’s written right
now, we know that this relationship is inaccurate. But we want to figure out what
mathematical steps to apply to either side of it so that it is. To start to see what those steps
may be, let’s consider this first correspondence, zero degrees Celsius corresponds
to 32 degrees Fahrenheit.

This tells us that in a correct
equation, if we substitute in zero for our temperature in degrees Celsius, then a
correctly defined equation will output 32 in degrees Fahrenheit. That must be the case because we
know that zero degrees Celsius corresponds to that temperature in degrees
Fahrenheit. So then, what can we do to this
equation to make that happen? Well, perhaps the simplest thing is
to add 32 to the left-hand side. This way, when we input zero for
our temperature in degrees Celsius, then we add 32 to it, we get the result we
wanted, 32 in degrees Fahrenheit.

So then, as it’s written, this
temperature conversion formula agrees with this first temperature correspondence we
have, that zero degrees Celsius corresponds to 32 degrees Fahrenheit. But now, let’s check it for the
second correspondence, that 100 degrees Celsius is 212 degrees Fahrenheit. If we substitute in 100 for our
temperature in degrees Celsius and then we add 32 to it, we get 132, which is not
the 212 we know we should get. This tells us that, in its current
form, our conversion formula is still missing something. As it turns out, that something is
the conversion ratio we found earlier between degrees Fahrenheit and degrees
Celsius. If we multiply that ratio by our
input temperature in degrees Celsius and then add 32 to that, we’ll get a different
result.

Let’s try it out using 100 degrees
Celsius as our input. Nine-fifths times 100 plus 32 is
equal to 900 divided by five plus 32, which is equal to 180 plus 32, which adds up
to 212. So, we’ve input 100 degrees Celsius
and as our output we’ve gotten 212 degrees Fahrenheit. That means that this form of our
conversion formula agrees with this correspondence of 100 degrees Celsius
corresponding to 212 degrees Fahrenheit. But now, let’s check and see if it
still works for the other correspondence. If we insert zero for our
temperature in degrees Celsius, then nine-fifths times zero is itself equal to zero,
and then when we add 32 to that, we get a result of 32. So, by inputting zero degrees
Celsius, as an output we’ve got 32 degrees Fahrenheit, just like we wanted.

What we’ve developed is a general
equation for taking a temperature in degrees Celsius and calculating its
corresponding value in degrees Fahrenheit. One important point about this
equation is that its unitless. Even though there are temperatures
in degrees Celsius and degrees Fahrenheit, when we plug them into the equation, we
leave off the units. For example, when we substituted in
zero degrees Celsius, we did that just plugging in a zero. And then, we calculated out 32,
which we knew, as the output was 32 degrees Fahrenheit. This is one of the only cases where
we leave off units in the calculation. Now, at this point, given any
temperature in degrees Celsius, we could solve for the corresponding temperature in
degrees Fahrenheit.

But what about in the opposite
direction? To see how to make that kind of
conversion, all we’ll need to do is rearrange this equation we already have
algebraically. We could think of it this way. Right now, the subject of this
equation is 𝑇 sub F, a temperature in degrees Fahrenheit. We want to change this to the new
subject, it’s 𝑇 sub C. Our first step we could take
towards getting there is to subtract 32 from both sides of this equation. If we do that, the positive 32 and
the negative 32 on the left-hand side cancel one another out.

Then, as a next step, we can
multiply both sides of the equation by the inverse of nine-fifths, in other words
five-ninths. When we take this step, looking on
the left-hand side, multiplying and dividing by five and multiplying and dividing by
nine gives a result of one. And that leaves us with this
result, 𝑇 sub C is equal to five-ninths times the quantity 𝑇 sub F minus 32. Using this new form of our
equation, we can now take a temperature in degrees Fahrenheit and convert it to its
equivalent value in degrees Celsius. So then, here are the two
equivalent versions of this temperature conversion. Let’s now get some practice with
these relationships through an example.

What is 67 degrees Celsius in
degrees Fahrenheit? Give your answer to one decimal
place.

All right, so here we want to
take this temperature in degrees Celsius, it’s 67 degrees, and convert it to
degrees Fahrenheit. That is, given this temperature
we can call it 𝑇 sub C in degrees Celsius, we wanna convert it to the
Fahrenheit temperature scale. To do this, we can recall the
mathematical relationship that connects these two scales, Celsius to
Fahrenheit. Given a temperature in degrees
Celsius, if we multiply that by nine-fifths and add 32 to that result, we get
the corresponding value in degrees Fahrenheit. And by the way, this ratio,
nine-fifths, is not a random number. This is equal to the ratio of
how much a temperature change on the Fahrenheit scale corresponds to on the
Celsius scale.

So then, to solve for 𝑇 sub F,
the temperature in Fahrenheit that corresponds to 67 degrees Celsius, we’ll take
our input temperature written as a pure number with no units, multiply it by
nine-fifths, and then add 32 to that. When we do this, the result we
find is 152.6. Now, because everything here
has no units, our answer also has no units in its current form. But because we’re solving for a
temperature in degrees Fahrenheit, we know that really 152.6 is 152.6 degrees
Fahrenheit. In our problem statement, we’re
told to give our answer to one decimal place, which it is, 152.6. So then, this is our final
answer. 67 degrees Celsius is 152.6
degrees Fahrenheit.

Let’s look now at a second
example.

What is negative 40 degrees
Celsius in degrees Fahrenheit?

In this exercise then, we’re
doing a temperature conversion from a temperature in degrees Celsius to one in
degrees Fahrenheit. Whenever we do that, it’s
helpful to recall the conversion relationship between these two separate
temperature scales. Given a temperature in degrees
Celsius, we can call it 𝑇 sub C, if we multiply that temperature by nine-fifths
and then add 32 to that result, we get the corresponding temperature in degrees
Fahrenheit, we’ll call it 𝑇 sub F.

So, what we want to do is solve
for 𝑇 sub F, given an input temperature of negative 40 degrees Celsius. To do that, we’ll use this
conversion relationship, and we’ll insert negative 40 for our temperature in
degrees Celsius. Multiplying negative 40 by
nine-fifths, we get negative 360 divided by five. And that fraction is equal to
negative 72. When we add positive 32 to
negative 72, we get a result of negative 40. And it’s now that we put the
correct units onto this result. That’s degrees Fahrenheit.

Now, at first this may seem
strange. We started out with negative 40
in degrees Celsius, and we found that, apparently, that’s negative 40 in degrees
Fahrenheit. In fact, there is nothing wrong
with what we’ve done. Rather, what we’ve discovered
is the one single temperature at which that temperature value in degrees Celsius
is equal to the same temperature value in degrees Fahrenheit. There is no other temperature
for which this is true, but it is true for negative 40. So, negative 40 degrees Celsius
is negative 40 degrees Fahrenheit.

Let’s now summarize what we’ve
learned about converting between Celsius and Fahrenheit scales. First off, we saw that many
different temperature scales exist. The Celsius and the Fahrenheit
scales are two of the most common. Furthermore, these scales are
defined by the boiling and freezing points of water. Water boils at 100 degrees Celsius,
or 212 degrees Fahrenheit, and it freezes at zero degrees Celsius, or 32 degrees
Fahrenheit.

And then, lastly, we learned the
conversion formula for going from degrees Celsius to degrees Fahrenheit and
back. Written one way, it tells us that a
temperature in degrees Fahrenheit is equal to nine-fifths times that equivalent
temperature in degrees Celsius plus 32. And written another way, it tells
us that a temperature in degrees Celsius is equal to five-ninths times the quantity
temperature in degrees Fahrenheit minus 32.