Posts tagged with linear transformation

## What’s half of 100 degrees Fahrenheit?

Hint: it’s not 50 degrees Fahrenheit.

100 ℉ = 311 K, half of which is 105.5 K = −180℉

$\large \dpi{200} \bg_white \begin{matrix} 100 \, ^{\circ} \rm{F} & \longrightarrow & 311 \, \rm{K} \\ \\ && \downarrow \\ \\ -180 \, ^{\circ} \rm{F} & \longleftarrow & 155 \, ^1\!\!/\!_2 \, \rm{K} \end{matrix}$

Yup — half of 100℉ is −180℉.

The difference between the Kelvin scale ℝ⁺ and the Fahrenheit scale is like the difference between a linear scale and an affine scale.

You were taught in 6th form that `y = mx + b` is a “linear” equation, but it’s technically affine. The `+b` makes a huge difference when the mapping is iterated (like a Mandelbrot fractal) or even when it’s not, like in the temperature example above.

(The difference between affine and linear is more important in higher dimensions where `y = Mx` means `M` is a matrix and `y` & `x` vectors.)

Abstract algebraists conceive of affine algebra and manifolds like projective geometry — “relaxing the assumption” of the existence of an origin.

(Technically Fahrenheit does have a bottom just like Celsius does. But I think estadounidenses conceive of Fahrenheit being “just out there” while they conceive of Celsius being anchored by its Kelvin sea-floor. This conceptual difference is what makes Fahrenheit : Celsius :: affine : linear.)

It’s completely surprising and rad that mere linear equations can describe so many relevant, real things (examples in another post). Affine equations — that barely noticeable `+b` — do even more, without reaching into nonlinear chaos or anything trendy sounding like that.