Algorithms for Mental Conversion between the Fahrenheit and Celsius Scales

Robert A. Warren

This is a slightly revised and fully corrected update to an earlier article. It's well worth reading. Editor.

Yes, quick mental conversions can be made between the Fahrenheit and Celsius scales, using the algorithms presented in this paper. But before these algorithms are presented, some background material is needed.

Figure 1. Fahrenheit-Celsius conversion scale. National Institute of Standards and Technology.

The standard equations that are used for conversions between the Fahrenheit (F) and Celsius (C) scales are the following:

F = 9/5C + 32

C = 5/9{F –32}

The problem with these equations is that they are hard to remember, because it is easy to confuse the form of the one equation with the form of the other.

I have devised conversion equations and algorithms that are easy to remember and easy to keep straight. I refer to these as the AMS equations and algorithms. AMS stands for, “add, multiply, subtract.” These letters could also be thought to represent the words, “a mistake-proof system.” The words “a mistake-proof system” help us to remember the initials (and thus the order of operations) “AMS.”

The AMS algorithms are based on the fact that - 40 degrees Celsius = - 40 degrees Fahrenheit. When 40 degrees is added to a temperature (Fahrenheit or Celsius), we obtain the distance in degrees from the top of the mercury or alcohol line in the thermometer (assuming that it is a liquid thermometer) to the common point on the Fahrenheit and Celsius scales. By using the appropriate multiplier (5/9 or 9/5) we move from one scale to the other. By then subtracting 40 degrees, we obtain the difference in degrees between 0 degrees on the new scale and the top of the line of liquid in the thermometer, i.e. the equivalent temperature on the new scale.

The AMS algorithm for conversion from Fahrenheit to Celsius is the following:

1. Add 40 to the Fahrenheit Temperature.

2. Multiply this sum by 5/9.

3. Subtract 40 to obtain the equivalent Celsius temperature.

The AMS algorithm for conversion from Celsius to Fahrenheit is the following:

1. Add 40 to the Celsius Temperature.

2. Multiply this sum by 9/5.

3. Subtract 40 to obtain the equivalent Fahrenheit temperature.

As you can see, the algorithm for converting from Fahrenheit to Celsius is almost identical to the algorithm for converting from Celsius to Fahrenheit. The only difference between the two algorithms is the multiplier. We multiply by 5/9 when going from Fahrenheit to Celsius and by 9/5 when going from Celsius to Fahrenheit.

One way to remember which multiplier to use is to remember that Fahrenheit temperature values are almost always larger in magnitude than the equivalent Celsius temperatures. Thus, to obtain larger values (Fahrenheit values) we multiply by a number (the number 9/5) larger than one. To obtain smaller values (Celsius values) we multiply by a number (the number 5/9) smaller than one.

Mental Conversion Algorithms

If we use as multipliers 2 (instead of 9/5) and ½ (instead of 5/9) in the AMS conversion algorithms, we will only be introducing about 10% error, and this change of multipliers will make the calculations simple enough that the conversions can be made mentally.

Although the amount of error introduced by the use of these approximate multipliers is only about 10%, the amount of error in the final answer will sometimes be a much greater percentage. Thus it is desirable to include a correction term into the calculations. The mental AMS conversion algorithms involve the use of a correction term to obtain a highly accurate final answer.

The mental AMS algorithm for conversion from Fahrenheit to Celsius is the following:

1. Add 40 to the Fahrenheit Temperature.

2. Multiply this sum by ½.

3. Add 10%

4. Subtract 40 to obtain the equivalent Celsius temperature.

The mental AMS algorithm for conversion from Celsius to Fahrenheit is the following:

1. Add 40 to the Celsius Temperature.

2. Multiply this sum by 2.

3. Subtract 10%

4. Subtract 40 to obtain the equivalent Fahrenheit temperature.

Sample calculations

The examples that follow will help you learn to do quick conversions between Celsius and Fahrenheit.

Example 1.

Let's run through the steps required to convert 100 degrees Celsius, the boiling point of water, to the Fahrenheit equivalent.

We know that this answer is correct.

Example 2.

Let's convert 0 degrees Celsius to the Fahrenheit equivalent using the appropriate mental AMS algorithm.

We know that this answer is correct.

Example 3.

Let's convert 212 degrees Fahrenheit to a Celsius temperature with the appropriate mental AMS algorithm.

We know that this answer is a very close approximation.

Example 4.

Let's go through the required steps to convert 32 degrees Fahrenheit to the Celsius equivalent.

We know that this answer is a very close approximation.

Example 5.

Finally, let's run through the steps required to convert - 40 degrees Fahrenheit to the Celsius equivalent.

We know that this answer is correct.

From these sample conversions, we can see that the mental AMS conversions from Celsius to Fahrenheit give exactly correct answers. The mental AMS conversions from Fahrenheit to Celsius give answers that are very close approximations for every temperature except – 40 degrees. The further one moves from - 40 degrees in either direction, the greater the degree of error in these conversions. For most purposes, the slight degree of error in the mental AMS algorithms when going from Celsius to Fahrenheit should be inconsequential

The AMS temperature conversion equations

The AMS temperature conversion equations are the equation equivalents of the AMS algorithms. These equations will now be derived from the standard temperature conversion equations.

The standard equation for conversion from Celsius to Fahrenheit is:

F = 9/5C + 32

Performing some manipulations,

F = 9/5C + 72 - 40

We obtain the AMS equation for conversions from Celsius to Fahrenheit

F = 9/5(C + 40) – 40

The standard equation for conversion from Fahrenheit to Celsius is the following:

C = 5/9(F – 32)

Performing some manipulations,

C = 5/9{F - 72 + 40)

We obtain the AMS equation for conversions from Fahrenheit to Celsius

C = 5/9(F + 40) – 40

These AMS temperature conversion equations are easier to remember than the standard temperature conversion equations.

Putting these two AMS temperature conversion equations side-by-side, we can see that they both have the same basic form.

F = 9/5{C + 40} – 40

C = 5/9{F + 40} – 40