Explain me a deep derivation of Celsius and Fahrenheit equation. 🌡️🤒

 The conversion between Celsius (°C) and Fahrenheit (°F) temperatures relies on establishing a relationship between the freezing and boiling points of water on these two scales. Here's a breakdown of the derivation:

**1. Defining the Scales:**

* **Celsius:** Assigns 0°C to the freezing point of water and 100°C to the boiling point of water.

* **Fahrenheit:** Assigns 32°F to the freezing point of water and 212°F to the boiling point of water.

**2. Establishing the Relationship:**

The key concept is that both scales measure the same change in temperature. In other words, if the temperature increases by 10 degrees on the Celsius scale, it should also correspond to a 10-degree increase on the Fahrenheit scale (ignoring potential non-linearities at extreme temperatures).

**3. Setting Up the Proportion:**

Let's denote the unknown temperature in Celsius as "°C" and the corresponding temperature in Fahrenheit as "°F". We can set up a proportion based on the defined freezing and boiling points:

```

Change in °C / Change in °F = (100°C - 0°C) / (212°F - 32°F)

```

This proportion represents that the change in temperature on the Celsius scale (100°C - 0°C) is proportional to the change in temperature on the Fahrenheit scale (212°F - 32°F).

**4. Simplifying and Solving:**

The proportion simplifies to:

```

°C / (°F - 32) = 100 / 180

```

We can multiply both sides by (°F - 32) to isolate °C:

```

°C = (100 / 180) * (°F - 32)

```

**5. Conversion Formula:**

To convert from Fahrenheit to Celsius, we can rearrange the equation:

```

°C = (5/9) * (°F - 32) // This is the standard Celsius to Fahrenheit conversion formula

```

**Additional Notes:**

* The factor (5/9) arises from the ratio of the temperature changes (100°C / 180°F).

* This derivation assumes a linear relationship between the two scales. In reality, the relationship is slightly non-linear at extreme temperatures. However, for most practical applications, this linear approximation is sufficient.

**Converting Celsius to Fahrenheit:**

Following the same logic, we can set up a proportion based on the defined freezing and boiling points for converting Celsius to Fahrenheit:

```

Change in °F / Change in °C = (212°F - 32°F) / (100°C - 0°C)

```

This simplifies to:

```

°F = (9/5) * °C + 32 // This is the standard Fahrenheit to Celsius conversion formula

```

I hope this deep derivation clarifies the relationship between Celsius and Fahrenheit and the reasoning behind the conversion formulas.