Temperature Conversion

I came across this interesting snippet of information today:

I was interested in why this interchange should be possible and so I asked Gemini. Here was its response:

The trick works because the conversion slope between Celsius and Fahrenheit:$$\frac{9}{5} = 1.8$$closely mirrors how reversing a two-digit number changes its numerical value in base-10 arithmetic, combined with simple integer rounding.

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The Exact Arithmetic

The standard conversion formula from Celsius ($C$) to Fahrenheit ($F$) is:

$$F = \frac{9}{5}C + 32 = 1.8C + 32$$

Every time $C$ increases by 12°C, $F$ increases by:

$$\Delta F = 1.8 \times 12 = 21.6$$

which is approximately 22°F.

Evaluating each step from the image reveals how closely the exact arithmetic matches the digit reversal:

• 04°C: $1.8(4) + 32 = 7.2 + 32 = 39.2$ → 39.2°F ≈ 40°F 
  (reversed digits: 04 → 40)
• 16°C: $1.8(16) + 32 = 28.8 + 32 = 60.8$ → 60.8°F ≈ 61°F
  (reversed digits: 16 → 61)
• 28°C: $1.8(28) + 32 = 50.4 + 32 = 82.4$ → 82.4°F ≈ 82°F
  (reversed digits: 28 → 82)
• 40°C: $1.8(40) + 32 = 72.0 + 32 = 104.0$ → 104°F
  (reversed digits: 40 → 04, plus a leading 1)
• 52°C: $1.8(52) + 32 = 93.6 + 32 = 125.6$ → 125.6°F ≈ 126°F ≈ 125°F
  (reversed digits: 52 → 25, plus a leading 1)

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Why Digits Invert in Steps of 12

Let a two-digit Celsius value be represented by digits $a$ and $b$:

$$C = 10a + b$$

Its digit reversal $R(C)$ is:

$$R(C) = 10b + a$$

If you add $12$ to $C$, the tens digit increases by $1$ and the units digit increases by $2$ (assuming no carryover):

$$C' = 10(a+1) + (b+2)$$

The reversal of this new value is:

$$R(C') = 10(b+2) + (a+1) = (10b + a) + 21 = R(C) + 21$$

Because reversing digits after adding $12$ increases the reversed value by $21$, and the physical temperature conversion rate is $1.8 \times 12 = 21.6$, the rounded whole-number sequences track together with remarkable accuracy across the everyday range from 0°C to 50°C.