0

## A Trick For Temperatures

Published on Thursday, July 04, 2013 in , , ,

Last month, I discussed how to convert between Celsius and Fahrenheit, including how to remember the correct steps.

There was one major fly in the ointment. When going from °F to °C, you have to multiply by 109, or 1.111..., which seems difficult to do in your head. In today's post, however, you'll learn how to perform this calculation easily!

MULTIPLES OF 9: When you think about it, multiplying by 109 is really just multiplying by 10 and then dividing by 9. When dealing with multiples of 9, this is very easy. What is 109 of 9? 109 × 9 = 909 - 10. Working through all the 1- and 2-digit multiples of 9, we get:

• 109 × 9 = 10
• 109 × 18 = 20
• 109 × 27 = 30
• 109 × 36 = 40
• 109 × 45 = 50
• 109 × 54 = 60
• 109 × 63 = 70
• 109 × 72 = 80
• 109 × 81 = 90
• 109 × 90 = 100
• 109 × 99 = 110
When dealing with tempertures of 158 °F or less, which is well over the hottest air temperature ever recorded on Earth, you're only going to be dealing with 2-digit numbers, so I'll be limiting this tutorial to 2-digit numbers.

If you're lucky enough to be working with a multiple of 9, you already have your answer at this point!

Besides being easy to remember, the multiples of 9 are also helpful in another way. If you're calculating with a number from, say, 50 to 59, is the answer going to be in the 50s or the 60s? Anything in the 50s below 54 would have to give an answer in the 50s, and any number in the 50s equal to or greater than 54 would have to give an answer in the 60s.

Using this technique, you can already say the first half of the answer without calculation! What is 109 of 73? Since we already know 72 would take us to 80, then so must 73, so we can start by saying, "Eighty...". How about 33? 36 takes you to 40, so 109 × 33 must still be in the 30s somewhere. If you say, "Thirty..." at this point, you already have the first half of your answer!

Now that we've narrowed down the answer to a specific range of 10 numbers, how do we get the rest of the answer?

DIGITAL ROOTS: The original number itself will clue you in to the rest of the answer in a surprisingly easy manner. To get the second half of the answer, you need to use what is known as the digital root.

The digital root of a number is always a 1-digit number from 1 to 9, and is calculated by adding the digits of the number together. For example, the digital root of 43 is 7 because 4 + 3 (the digits in the original number) = 7. Sometimes, you may have to repeat this process to get a 1-digit number. To find the digital root of 75, you would add 7 + 5 = 12, and then repeat the process with the 12, 1 + 2 = 3. So, the digital root of 75 would be 3.

Another way to get the digital root of a number is to subtract the nearest multiple of 9 that is less than or equal to your number. With 43, the nearest multiple of 9 less than or equal to it is 36, and 43 - 36 = 7. Similarly, with 75, 75 - 72 = 3.

You can learn about digital roots in more detail in this thonky.com post, as well as by watching this video.

Working out the digital root will tell you the final digits of the answer, in this way:
• A digital root of 1 will end in 1.111..., or 119
• A digital root of 2 will end in 2.222..., or 229
• A digital root of 3 will end in 3.333..., or 339, or 313
• A digital root of 4 will end in 4.444..., or 449
• A digital root of 5 will end in 5.555..., or 559
• A digital root of 6 will end in 6.666..., or 669, or 623
• A digital root of 7 will end in 7.777..., or 779
• A digital root of 8 will end in 8.888..., or 889
• A digital root of 9 will end in 0 (as discussed above)
This is a very easy pattern to remember, of course. This also gives you the precise second half of your number!

Let's work through the whole process now with some examples. What is 109 × 43? 43 is less than 45, so we know our answer will stay in the 40s, so we can already say, "Forty...". The digital root of 43, as explained above, is 7, so now we know that it will end in 7.777..., and we can finish by saying, "...seven point seven seven." If we make the same calculation with Wolfram|Alpha, you see that this is correct!

What about 75? 75 is greater than 72, so we already know our answer will be in the 80s, and say, "Eighty...". The digital root of 75 is 3, so we can finish by saying, "...three point three three." Sure enough, 109 × 75 is 83.333...!

Once you become comfortable with the process, it almost seems like someone else is making the calculation, and you're just listening to the answer as it comes out of your own mouth! What is 109 × 92? It's more than 90, so you can already say, "One hundred and...", and the digital root is 2 (9 + 2 = 11, 1 + 1 = 2), so you say, "...two point two two." Being able to give such decimal precision so quickly makes this seem impressive, as well.

NUMBERS ENDING IN .5: Since this calculation is used for temperatures, and dividing by 2 is so important when going from °F to °C, you're occasionally going to deal with numbers ending in .5, such as 62.5. You handle these in almost exactly the same way, but with one important change.

Just as before, start by asking yourself where the number is in relation to the multiple of 9. With 62.5, it's below 63, so the answer will be in the 60s. You say, "Sixty..." and then work out the digital root of 62.5: 6 + 2 + 5 = 13, and 1 + 3 = 4, so the number has a digital root of 4. However, instead of using the same numbers as we did previously, a number ending in .5 requires that you use these different final digits:
• A digital root of 1 will end in 6.111..., or 619
• A digital root of 2 will end in 7.222..., or 729
• A digital root of 3 will end in 8.333..., or 839, or 813
• A digital root of 4 will end in 9.444..., or 949
• A digital root of 5 will end in 0.555..., or 059
• A digital root of 6 will end in 1.666..., or 169, or 123
• A digital root of 7 will end in 2.777..., or 279
• A digital root of 8 will end in 3.888..., or 389
• A digital root of 9 will end in 5
Note that the digits after the decimal point are still the same as the digital root, but the number immediately preceding the decimal point always has a difference of 5 from the digital root. A calculation involving .5 should remind you that you need to add or subtract 5 (whichever is easier) from the digital root for the number in the ones place (the number immediately preceding the decimal point).

Finishing up with our 62.5 example, we've already said, "Sixty..." and know the digital root is 4, so we finish by saying, "...nine point four four." Yes, 109 × 62.5 is 69.44.

How about 87.5? It's above 81, so the answer will be in the 90s, it involves a .5, and the digital root is 2 (8 + 7 + 5 = 20, 2 + 0 = 2), so we give our answer as, "Ninety...seven point two two." Do you understand how we worked through that? If not, go back and read through the process before moving on. Also, feel free to ask questions in the comments below.

TEMPERATURE CALCULATIONS: Let's use our newfound ability to precisely calculate 109 of any 2-digit number by working through the full original process from °F to °C.

What is the Celsius equivalent of 82°F? First, we add 40, so 82 + 40 = 122. Next, we divide by 2, so 122 ÷ 2 = 61. 109 of 61 is, "Sixty...seven point seven seven" (67.777..., do you understand how we got that number?). The final step is to subtract 40, so 67.777... - 40 = 27.777..., so 82°F should be about 27.77°C, and Wolfram|Alpha verifies that this is correct!

How about 75°F? 75 + 40 = 115, and 115 ÷ 2 = 57.5. Make sure that you remember the rules for .5 here! 57.5 is more than 54, and has a digital root of 8 (5 + 7 + 5 = 17, 1 + 7 = 8), so 109 of 57.5 is, "Sixty...three point eight eight." Subtracting 40 in the last step, we get 63.88 - 40 = 23.88. Sure enough, 75°F is 23.88°C!

TIPS: Here are a few ideas that can help improve this process.

If you're dealing with a 1-digit number, it's digital root is itself. The digital root of 1 is 1, the digital root of 2 is 2, and so on. With 2-digit numbers, if one of the digits is either a 0 or a 9, you can ignore those, and use the remaining digit. Ignoring the 0 in 20 gives you 2, which is the digital root of 20. Ignoring the 9 in 49 gives you a digital root of 4, and ignoring the 9 in 96 gives you a digital root of 6.

You can also ignore pairs of numbers that sum up to 9, which is especially handy when dealing with numbers ending in .5. What's the digital root of 63.5? Well, since 6 and 3 equal 9, we can ignore those, and see that the remaining 5 gives us a digital root of 5!

Again, if you have any questions about clarifying this process, feel free to ask them in the comments!