How to Derive -40°F = -40°C
Kirt's Cogitations™ #264

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Some of what you and I consider common knowledge is largely unrealized by most people. Call me a geek, but I take pleasure in pointing out to people that the Fahrenheit and Centigrade scales are equal at -40°, and I especially enjoy working out the simple proof for them. Most people appreciated the effort and are amazed, claiming to have never seen that before. When I read the following in Smithsonian magazine, "Winter temperatures here, some 250 miles northeast of St. Petersburg, sometimes plunge to minus 40 degrees Fahrenheit," I wondered whether the author knew that -40°F = -40°C. Maybe he just didn't want to confuse his readers by omitting the redundant superfluous 'F' or 'C,' and it couldn't be 'K' because there are no negative Kelvin degrees. It could also be that he knew but figured most people do not, so he chose his favorite temperature unit. The other possibility is that he simply did not know, but it really doesn't matter.

Since every once in a while people land on the RF Cafe website who do not have a background in science or engineering, I'll take this opportunity to present the aforementioned proof:

The conversions back and forth between Fahrenheit and Celsius are

1. F = (C * 1.8) + 32

2. C = (F - 32) / 1.8

As a quick test, almost everyone knows that 0°C = 32°F, so

C = (32 - 32) / 1.8 = 0 / 1.8 = 0

In the other direction,

F = (C * 1.8) + 32 = (0 * 1.8) + 32 = 0 + 32 = 32

Additionally, 100°C = 212°F, so

C = (212 - 32) / 1.8 = 180 / 1.8 = 100

Now for the proof. Substitute equation 1. into equation 2. to find where C = F,

C = (C * 1.8) + 32 = 1.8C + 32

Subtract 1.8C from both sides of the equation,

C - 1.8C = 32

-0.8C = 32

Divide both sides by -0.8,

C = 32 / (-0.8)

C = -40

So, C = F at -40

It works the same for substituting equation 2. into equation 1.,

F = (F - 32) / 1.8

Multiply both sides of the equation by 1.8,

1.8F = F - 32

1.8F - F = -32

0.8F = -32

F = -32 / 0.8

F = -40

So, F = C at -40

QED

The entire exercise takes less than a minute with pencil and paper. Enlighten a friend or relative the next time an opportunity arises.

Posted April 24, 2013

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