As I write this, I'm in Vegas and it's hot. It's so brain-meltingly hot, I forgot to put up a post on Thursday!

Since the only thing my mind can conceive of is temperature right now, today's post will be about temperature conversion between Fahrenheit (°F) and Celsius (°C).

BASICS: The standard formulas taught for conversion are based on the fact that 0°C and 32°F are the same temperature. 32 is already a rough number to handle.

There's another formula that's based on the fact that -40°C and -40°F are the same temperature, and 40 is a much simpler number to deal with than 32.

To get a sense of both scales, there are some rhymes to help. For Celsius, remember: 30's warm, 20's nice, 10 is cold. 0's ice. For Fahrenheit, remember: 90's warm, 70's nice, 50's cold, 30's ice. Note that, for any temperature above -40°, Fahrenheit will be a bigger number than its equivalent in Celsius. The fact that the word Fahrenheit has a bigger number of letters than the word Celsius is an easy way to recall this.

THE FORMULAS: When converting from Fahrenheit to Celsius, follow these steps:

1) Add 40.

2) Divide by 2.

3) Multiply by ¹⁰⁄₉.

4) Subtract 40.

When converting from Celsius to Fahrenheit, use these steps:

1) Add 40.

2) Multiply by 2.

3) Multiply by ⁹⁄₁₀.

4) Subtract 40.

REMEMBERING THE FORMULAS: The easiest part of these formulas to remember is that, regardless of direction, the first step is always to add 40, and the last step is always to subtract 40. Already, you have half of the steps memorized!

Step 2 involves a 2, which is easy enough, but how do you remember when to multiply and when to divide? F, for Fahrenheit, is the 6th letter of the alphabet and C, for Celsius, is the 3rd letter of the alphabet. If you're going from F to C, then you can also think of that as going from 6 to 3. How do you do that? You divide by 2! Conversely, if you're going from C to F, you're going from 3 to 6, and you do that by multiplying by 2! The letters themselves practically tell you the exact step to take.

In step 3, you're always multiplying, but how do you remember whether it's ¹⁰⁄₉ or ⁹⁄₁₀? Remember the idea of Fahrenheit as bigger and Celsius as smaller?

When going from F to C, you're going from bigger to smaller, so the fraction should have the bigger number on top and the smaller number on the bottom: ¹⁰⁄₉. When going from C to F, you're going from smaller to bigger, so the formula should have the smaller number on top and the bigger number on the bottom: ⁹⁄₁₀.

Once you're able to recall the correct steps for each conversion, it's time to learn how to handle the math itself.

MENTAL MATH: The steps involving adding 40, multiplying or dividing by 2, and subtracting 40, are all easy to handle themselves. The only place you'll probably need help is mentally multiplying by the fractions.

For going from C to F, you have to multiply by ⁹⁄₁₀, but how do you actually do that in your head? What ⁹⁄₁₀ really means is to multiply by 9 and then divide by 10. Since 9 is equal to 3 × 3, you can break that step down farther. In short, take whatever number you have, triple it, triple it one more time, and then divide by 10. When you do that, you've effectively multiplied by ⁹⁄₁₀!

Assuming you started with a whole number for C, the corresponding F temperature will either be a whole number, or will end with a single decimal place from .1 through .9, because you're only dividing by 10.

When going from F to C, you're multiplying by ¹⁰⁄₉, which is the same as multiplying by 10 and then dividing by 9. Multiplying by 10 is easy enough, but you may or may not be comfortable dividing by 9. Often, you'll be dividing 3-digit numbers by 9, and some people can handle that with few problems.

For those who aren't comfortable dividing large numbers by 9, Dr. James Tanton shows you how to make this step ridiculously easy:



Notice that this method, when applied properly, also gives you the remainder. Even better, the remainder, when converted into a decimal, is just that number repeated endlessly. For example, a remainder of 8 means the number will end in .8888888..., as mentioned in the Grey Matters decimal division tutorial.

Personally, I usually give repeating decimals to 3 places, as that's enough to give people the idea.

EXAMPLES: Let's try this out with a few examples, shall we?

We'll start by converting 20°C to F. We're going from the 3rd letter to the 6th letter, which means multiplying by 2, and smaller to bigger, which means we multiply by ⁹⁄₁₀, so we should already know the steps involved from memory.

1) 20 + 40 = 60

2) 60 × 2 = 120

3) 120 × ⁹⁄₁₀ = (120 × 3 × 3) ÷ 10 = (360 × 3) ÷ 10 = 1080 ÷ 10 = 108

4) 108 - 40 = 68

So, if we did this correctly, 20°C should equal exactly 68°F, and double checking with Wolfram|Alpha confirms this.

How about figuring out what 87°F is in °C? We're going from the 6th letter to the 3rd letter, so step 2 is to divide by 2, and we're going from larger to smaller, so step 3 will involve multiplying by ¹⁰⁄₉. Let's work through this:

1) 87 + 40 = 127

2) 127 ÷ 2 = 63.5

3) 63.5 × ¹⁰⁄₉ = (63.5 × 10) ÷ 9 = 635 ÷ 9 = 70 remainder 5 = 70.555

4) 70.555 - 40 = 30.555

How close did we come to the correct answer? Wolfram|Alpha says we're exactly right!

FINAL THOUGHTS: These answers are exact, because the formulas above aren't estimates, they're the exact formulas used to convert. Many approaches taught use the idea of adding or subtracting 10% to get close, but as Numericana.com points out, adding 10% actually falls short of what needs to be added. Of course, some methods are even rougher!

Obviously, if you don't need an exact answer, the approximate methods work well. If you're going to need exact figures, however, you can have the confidence of being able to do them in your head.