## Convert between Fahrenheit and Celsius the easy way

Posted by Diego Assencio on 2014.11.26 under Physics (Thermodynamics)

Suppose you are calling someone from another part of the world and the person asks you "how is the weather there?". You answer promptly: "it's really nice today: $75^\circ\textrm{F}$", only to hear the confused reply "wait, what's that in Celsius?". Oops, what should you do now?

If you ever learned to convert temperature values between the Fahrenheit and Celsius scales, you probably learned to do it using the equation below: $$\displaystyle\frac{{}^\circ\textrm{F} - 32}{9} = \frac{{}^\circ\textrm{C}}{5}$$ where ${}^\circ\textrm{F}$ is the temperature in the Fahrenheit scale and ${}^\circ\textrm{C}$ is the temperature in the Celsius scale. More concise ways to express the equation above are: $${}^\circ\textrm{F} = {}^\circ\textrm{C} \times 1.8 + 32 \label{post_8a4390f653cdd7dca0e05db14bd8f760_C_to_F_exact}$$ and its inverse: $${}^\circ\textrm{C} = ({}^\circ\textrm{F} - 32)/1.8 \label{post_8a4390f653cdd7dca0e05db14bd8f760_F_to_C_exact}$$ However, both equations are unnecessarily complicated to be used for temperatures which one experiences on a daily basis because dividing or multiplying by 1.8 and subtracting or adding 32 are not trivially easy to do. Consider, for instance, the following equation: $$\boxed{ {}^\circ\textrm{F} = {}^\circ\textrm{C}\times 2 + 30 } \label{post_8a4390f653cdd7dca0e05db14bd8f760_C_to_F_approx}$$ and its inverse (just memorize the form which you find easier): $$\boxed{ {}^\circ\textrm{C} = ({}^\circ\textrm{F} - 30)/2 } \label{post_8a4390f653cdd7dca0e05db14bd8f760_F_to_C_approx}$$ Much less daunting, aren't they? Dividing or multiplying by $2$ is much easier than by $1.8$, and subtracting or adding $30$ is much easier than it is with $32$.

During the vast majority of the year, and on most regions where the Fahrenheit scale is used, temperatures are in the range $[-10^\circ\textrm{C}, 35^\circ\textrm{C}]$ = $[14^\circ\textrm{F}, 95^\circ\textrm{F}]$. Interestingly, equation \eqref{post_8a4390f653cdd7dca0e05db14bd8f760_C_to_F_approx} (or, equivalently, equation \eqref{post_8a4390f653cdd7dca0e05db14bd8f760_F_to_C_approx}) works very well for converting between ${}^\circ\textrm{C}$ and ${}^\circ\textrm{F}$ over this temperature range (see figure 1). In fact, when converting from ${}^\circ\textrm{C}$ to ${}^\circ\textrm{F}$, the largest difference (in magnitude) between the values computed using equations \eqref{post_8a4390f653cdd7dca0e05db14bd8f760_C_to_F_exact} and \eqref{post_8a4390f653cdd7dca0e05db14bd8f760_C_to_F_approx} is only $5^\circ\textrm{F}$ at $35^\circ\textrm{C}$: the exact value is $95^\circ\textrm{F}$ but equation \eqref{post_8a4390f653cdd7dca0e05db14bd8f760_C_to_F_approx} yields $100^\circ\textrm{F}$. A difference of only $4^\circ\textrm{F}$ occurs at the other end of the temperature range: equation \eqref{post_8a4390f653cdd7dca0e05db14bd8f760_C_to_F_approx} yields $10^\circ\textrm{F}$ at $-10^\circ\textrm{C}$ but the exact value is $14^\circ\textrm{F}$.

Before you start thinking "well, $5^\circ\textrm{F}$ is not negligible", consider that as you walk over your house, you might already experience a difference of a few ${}^\circ\textrm{F}$ (or ${}^\circ\textrm{C}$ if you prefer). Some rooms will be warmer than others and you might not even notice the difference. Additionally, temperatures you see reported on the Internet, TV, radio etc. are just the values measured at some location near you and often differ from what you would experience in your garden by a few ${}^\circ\textrm{F}$ (${}^\circ\textrm{C}$). Finally, notice that the errors discussed above happen in the extremes of the given temperature range: "very hot" ($35^\circ\textrm{C}$ or $95^\circ\textrm{F}$) and "very cold" ($-10^\circ\textrm{C}$ or $14^\circ\textrm{F}$). For temperatures in between, the errors are smaller. For instance, $20^\circ\textrm{C}$ is exactly $68^\circ\textrm{F}$ but equation \eqref{post_8a4390f653cdd7dca0e05db14bd8f760_C_to_F_approx} yields $70^\circ\textrm{F}$. Not so bad, right?

Not surprisingly, equation \eqref{post_8a4390f653cdd7dca0e05db14bd8f760_F_to_C_approx} converts from ${}^\circ\textrm{F}$ to ${}^\circ\textrm{C}$ within an error of $2-3{}^\circ\textrm{C}$ on the temperature range we chose, with the largest errors happening at the extremes (very hot and very cold). In other words, you can use equations \eqref{post_8a4390f653cdd7dca0e05db14bd8f760_C_to_F_approx} and \eqref{post_8a4390f653cdd7dca0e05db14bd8f760_F_to_C_approx} to convert between ${}^\circ\textrm{F}$ and ${}^\circ\textrm{C}$ and obtain good approximate answers with little effort.

 Fig. 1: Exact and approximate conversions from ${}^\circ\textrm{C}$ to ${}^\circ\textrm{F}$. These curves are described by equations \eqref{post_8a4390f653cdd7dca0e05db14bd8f760_C_to_F_exact} and \eqref{post_8a4390f653cdd7dca0e05db14bd8f760_C_to_F_approx} respectively.