# Math Error in National Geographic Provides Some Teachable Moments

A few weeks ago I was at a very pleasant book store on Bainbridge Island across Puget Sound from Seattle. I started browsing through a special issue of The National Geographic that described many of the world’s most fascinating places, among which was the Gobi Desert.

I saw the following description of how the temperature in the Gobi can vary tremendously between day and night. I nodded in approval that the author had taken pains to give the temperatures not only in Fahrenheit but also in Celsius.

But soon my nodding in approval turned to shaking my head in disapproval. Why was that?

Spoiler alert: you may want to think about this for a while before reading any more. When you are ready, read on.

Let’s assume, for the purpose of this discussion, that The National Geographic is *correct* in its claim that the temperature can vary by up to 95 degrees Fahrenheit in one day. That is a huge amount, but let’s assume they are correct.

Now, the formula for converting Fahrenheit to Celsius is: C = (5/9)*(F — 32) so if you let F = 95 and apply the formula, you get C = 35.

In the above screen shot we see that The National Geographic puts 95F with 35C in parentheses. It is certainly true that a temperature *reading *of 95F is equivalent to a temperature *reading* of 35C (as it was on a few of our uncharacteristically-hot Seattle summer days).

However, it is **not** true that a temperature *change* of 95F is equivalent to a temperature *change* of 35C, and **therein lies the error**!

Why is that?

Well, it turns out that a *change* of 1 degree Fahrenheit is equivalent to a *change* of 5/9 degree Celsius. You can see this either by studying the conversion formula or, alternatively, by noting that when we go from the standard water-freezing temperature of 32F to the water-boiling temperature of 212F we have gone up by 180 degrees F. But in Celsius, water freezes at 0C and boils at 100C. So a change of 100C is equivalent to a change of 180F, and the fraction 100/180 is equivalent to 5/9.

This means that what The National Geographic **should** have written is that a temperature change of 95F is equivalent to a change of (5/9)*(95)C = 52.777…C. Of course they could have rounded it to 53C.

I emailed my findings to several departments at the National Geographic but got no reply.

What can we learn from this example? Well, several things.

- One has to be careful when applying math formulas. In particular, in this case you can’t directly apply the formula for converting F to C temperature
*readings*when what you*really*are after is a conversion from F to C temperature*changes*. For instance, even though 32F = 0C we would certainly not claim that a*change*of 32F equaled a change of 0C! - It was interesting to discuss this example with different people. Some immediately saw the error. Some questioned whether the temperature could really vary 95F in one day (that does seem remarkable, but that is not the main point here). Some were surprised that -40F = -40C (a temperature that few of us experience).
- If I were teaching mathematics, I would include this example in every class that was at a level such that they knew how to convert between Fahrenheit and Celsius. I wouldn’t tell them what the error was, but I would ask them to look at the example and talk about what they noticed.
- Examples like this still baffle Siri, Apple’s intelligent personal assistant, at least when I posed the question using a variety of formulations. Siri seemed quite insistent in telling me that a change of 95F was equivalent to a change of 35C.

In conclusion, I have a few requests.

- If you are able to pose the question to Siri in a way where Siri gives the right answer, please let me know.
- If someone has a personal contact who works at the National Geographic, please share this article with them.
- If you are teaching a math class at the appropriate level, please discuss this example with your class.