When Linear Relationships Make –40 the Same in Celsius and Fahrenheit

Think of how you convert between Celsius and Fahrenheit or how you figure out if a hill is steep enough for a ramp. At the heart of these tasks is something deceptively simple: the linear relationship. These are functions of the form y = mx + b—straight lines on a graph—but they show up everywhere from science labs to bank accounts. Today, we’ll dive into one of the coolest examples of linear relationships: why –40° ends up being the same point on both the Celsius and Fahrenheit scales, and how this idea of a constant rate of change powers real-world feats.

Where did this come from?

The idea of plotting equations as straight lines came from René Descartes in the 1600s when he invented coordinate geometry—linking algebra to geometry. Fast-forward to the 1700s, and Daniel Gabriel Fahrenheit built the first reliable mercury thermometer, picking 32°F as water’s freezing point and 212°F as its boiling point (a 180° gap). Shortly after, Anders Celsius set up a 0–100 scale but flipped freezing and boiling; later scientists reversed it to the familiar 0° for freezing and 100° for boiling. When you match these two scales, you get a neat linear formula: °F = 1.8×°C + 32.

Where you'll see this in real life

Temperature conversions (°C, °F and even Kelvin) rely on linear formulas so your weather app makes sense in any country. Currency exchange also works linearly: if one Australian dollar buys you 0.70 US dollars, two will buy you 1.40, ten will buy you 7.00, and so on—simple multiplication and an intercept of zero. Calculating pay is the same: hourly wage (m) times hours worked (x), plus any fixed allowances (b), gives your total earnings. Even reading maps or blueprints uses a linear scale: 1 cm on paper might equal 1 km on the ground, so every centimetre you measure becomes a fixed real-world distance.

A common misconception

Many students think a ‘linear relationship’ must pass through the origin—that is, a direct proportion y = mx. But general linear functions have the form y = mx + b, and b can be nonzero. If a straight-line graph doesn’t hit (0,0), it’s still linear. Also, remember vertical lines (x = constant) look straight but aren’t functions, since they don’t give you exactly one y-value for each x.