Negative Numbers in Math & Physics

Question

We say that $-4 < -2$ and that $-3 < 0$ and that $-192 < 24$. I'm aware that there are simple, easily understandable definitions for less than / greater than / equal to e.g. $a < b$ iff there is some positive number $c$ such that $a + c = b$.

This is in mathematics where, to further emphasize the point, a negative number is less than a positive number.

Now come to physics. I've seen--done some myself--calculation in motion physics that yield negative velocities/accelerations. So, $-20\space m/s$ is to be interpreted as $20 \space m/s$ in a direction opposite to a velocity of $20 \space m/s$. It doesn't seem to make sense to say that $-20\space m/s < 20 \space m/s$.

How do I reconcile these two usages of negative numbers?

Mathematically $-20 < 20$ but in physics we can't say that $-20 \space m/s < 20 \space m/s$.

Answer 1

Often times in physics, you work with “quantities” which can be thought of as vectors $v\in\mathbb R^n$. As such, you cannot compare them. After all, what would $v>w$ mean for $v,w\in\mathbb R^n$? To compare such elements, you usually are interested in their magnitudes. Physical magnitudes are often measured using mathematical norms $\|\cdot\|:\mathbb R^n\to\mathbb R_{\geq 0}$, usually the Euclidean norm. In that case, we would say that $v$ is greater than $w$ (or better: $v$ has a greater magnitude than $w$) if $\|v\|>\|w\|$. Note that we might have $\|v\|=\|w\|$ without $v$ and $w$ being equal to another.

Answer 2

In both mathematics and physics, negative numbers have various but related functions. Negative numbers in mathematics extend the number line by introducing values lower than zero and making it easier to solve equations involving losses, debts, or opposites. They serve as a foundation for ideas like absolute value and inequality and are essential for operations like subtraction.

Negative numbers have more contextual relevance in physics. They represent vector quantities with direction opposite to a selected reference as well as numbers less than zero. In this area, a change in a certain quantity, such as displacement, velocity, or acceleration in the opposite direction, is frequently indicated by a negative value.