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.

Shuffling Digits

Let's consider all those numbers in base 10 that retain their digits when changed into another number base. In other words, the digits of the number in base 10 are simply shuffled about. Let's start with base 9. In the range up to 40000 and ignoring single digit numbers, Table 1 shows the numbers with this property.

Table 1 showing
conversions to base 9

The base 10 numbers are 158, 227, 445, 1236, 1380, 2027, 2315, 2534, 5270, 5567, 5637, 5783, 10235, 10453, 11750, 13260, 13402, 13620, 21322, 21763, 21835, 23568, 26804, 32348, 34582, 35001, 35081, 35228, 37465 (permalink).

In base 11, the numbers and their conversions are shown in Table 2.

Table 2 showing
conversions to base 11

The numbers are 196, 283, 370, 1723, 4063, 7587, 8665, 15680, 16121, 18291, 19463, 19730, 23146, 26931, 32321, 35024, 38276 (permalink).

I won't show any further tables as they take up a fair amount of space but I'll list the results for other bases. Let's start with base 2 and work our way up (both only considering the range between 10 and 40000):

• base 2: none
• base 3: none
• base 4: 13
• base 5: none
• base 6: 1045, 1135
• base 7: 23, 46, 265, 316, 1030, 1234, 1366, 1431, 1454, 2060, 2116, 10144, 10342, 10542, 11425, 12415, 12450, 12564, 12651, 13045, 13245, 13534, 14610, 15226, 15643, 16255, 16546, 16633
• base 8: 1273, 1653, 2154, 3160, 3161, 3162, 3163, 3164, 3165, 3166, 3167, 3226, 16273, 21753, 30576, 31457
• base 9: 158, 227, 445, 1236, 1380, 2027, 2315, 2534, 5270, 5567, 5637, 5783, 10235, 10453, 11750, 13260, 13402, 13620, 21322, 21763, 21835, 23568, 26804, 32348, 34582, 35001, 35081, 35228, 37465
• base 11:196, 283, 370, 1723, 4063, 7587, 8665, 15680, 16121, 18291, 19463, 19730, 23146, 26931, 32321, 35024, 38276
• base 12: 2193, 6053, 7140, 7141, 7142, 7143, 7144, 7145, 7146, 7147, 7148, 7149, 7243, 7941, 9825, 24871, 25061, 25169, 34179
• base 13: 43, 86, 191, 774, 958, 4621, 5272, 5812, 5920, 7364, 7834, 7873, 9304, 9343, 9413, 29103, 29610, 30189, 31112, 31481, 35731, 36417
• base 14: 834, 6572, 9143, 9730, 9731, 9732, 9733, 9734, 9735, 9736, 9737, 9738, 9739]
• base 15: 261, 5180
• base 16: 53, 371, 913, 4100, 5141, 5412, 6182, 8200, 9241

We'll stop at base 16 but we could go on of course. Let's put all those numbers above into one ordered list:

13, 23, 43, 46, 53, 86, 158, 191, 196, 227, 261, 265, 283, 316, 370, 371, 445, 774, 834, 913, 958, 1030, 1045, 1135, 1234, 1236, 1273, 1366, 1380, 1431, 1454, 1653, 1723, 2027, 2060, 2116, 2154, 2193, 2315, 2534, 3160, 3161, 3162, 3163, 3164, 3165, 3166, 3167, 3226, 4063, 4100, 4621, 5141, 5180, 5270, 5272, 5412, 5567, 5637, 5783, 5812, 5920, 6053, 6182, 6572, 7140, 7141, 7142, 7143, 7144, 7145, 7146, 7147, 7148, 7149, 7243, 7364, 7587, 7834, 7873, 7941, 8200, 8665, 9143, 9241, 9304, 9343, 9413, 9730, 9731, 9732, 9733, 9734, 9735, 9736, 9737, 9738, 9739, 9825, 10144, 10235, 10342, 10453, 10542, 11425, 11750, 12415, 12450, 12564, 12651, 13045, 13245, 13260, 13402, 13534, 13620, 14610, 15226, 15643, 15680, 16121, 16255, 16273, 16546, 16633, 18291, 19463, 19730, 21322, 21753, 21763, 21835, 23146, 23568, 24871, 25061, 25169, 26804, 26931, 29103, 29610, 30189, 30576, 31112, 31457, 31481, 32321, 32348, 34179, 34582, 35001, 35024, 35081, 35228, 35731, 36417, 37465, 38276

There are no repeated numbers and there 158 numbers in total. Of these numbers, 31 are prime:

13, 23, 43, 53, 191, 227, 283, 1723, 2027, 3163, 3167, 4621, 5783, 6053, 7243, 7873, 9241, 9343, 9413, 9733, 9739, 10453, 15643, 16273, 16633, 19463, 25169, 31481, 32321, 35081, 35731

What stands out are the runs shown in blue above:

• 3160, 3161, 3162, 3163, 3164, 3165, 3166, 3167 (base 8: permalink)
• 7140, 7141, 7142, 7143, 7144, 7145, 7146, 7147, 7148, 7149 (base 12: permalink)
• 9731, 9732, 9733, 9734, 9735, 9736, 9737, 9738, 9739 (base 14: permalink)