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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 $-20 \space m/s < 20 \space m/s$.

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2 Answers 2

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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.

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    $\begingroup$ I think that this lifting to higher dimensions is indeed the correct illustration of the issue. It points at a subtle difference between the way that math and physics understand vectors: for physicists a vector is an object with magnitude and direction whereas for mathematicians a vector (in $\Bbb{R}^n$; no being annoying with algebra please) is a tuple of numbers. Of course we can translate between these, but the mathematician's essential building blocks are $\Bbb{R}$ and $\times$, whereas the physicist's are $\Bbb{R}^+$ and $O(n)$. $\endgroup$
    – Eric Nathan Stucky
    Commented Aug 28, 2023 at 8:24
  • $\begingroup$ Merci beaucoup for the answer. It appears that the notion of less than/greater than/equal to is undefined for vectors. So am I right to conclude that negative numbers shouldn't be considered vectors but ... they look so pretty as 1 dimensional vectors with only an $\hat i$ component. $\endgroup$
    – Hudjefa
    Commented Aug 28, 2023 at 11:00
  • $\begingroup$ You hit the point home as far as I'm concerned Zuy. Vectors have magnitude & direction and it doesn't make sense to say $\vec a > \vec b$ because the concept of greater/less than doesn't apply to direction (I'm ignoring the fact that the direction of a vector is given as an angle, of which it may be said that one is greater/less than the other). So if you want to make a greater/less than comparison of vectors, we must extract their magnitudes and check how they stack up to each other. $\endgroup$
    – Hudjefa
    Commented Aug 28, 2023 at 11:06
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    $\begingroup$ @AgentSmith In 2 dimensions, we can represent a vector's direction with an angle measure number, yes. But we need to be careful when using $<$ and $>$ comparisons on these, since adding to this angle number can bring us back to where we started, which breaks some usual ordering assumptions. In 3 dimensions (and higher), we can't even represent a vector's direction with a single number in a continuous way. $\endgroup$
    – aschepler
    Commented Aug 28, 2023 at 15:31
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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.

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  • $\begingroup$ I think I see the problem. There's a pattern and mathematicians hate to mess up a pattern. For example, if we define less than as $a < b$ IFF $a + c = b$, where c is a positive integer then this definition of less than will have to abandoned (it feels like a pretty good definition) if we refuse to accept that, say, $-4 < 4$ because $-4 + 8 = 4$ and 8 is a positive integer. $\endgroup$
    – Hudjefa
    Commented Aug 28, 2023 at 7:15
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    $\begingroup$ @AgentSmith, your sentence doesn't quite make sense ... if you define "less than" like that, there's no question around "accepting" the latter statement. It IS true ... because we've defined is as such. $\endgroup$
    – Brondahl
    Commented Aug 28, 2023 at 16:51
  • $\begingroup$ @Brondahl, thanks. I believe negative & positive numbers are a special case of vectors in 1 dimensional space (a line with only two directions, left & right), one in which less than/greater than has been defined for vectors like so: $\vec a < \vec b$ iff there is a rightward pointing vector (positive number) $\vec c$ such that $\vec a + \vec c = \vec b$ $\endgroup$
    – Hudjefa
    Commented Aug 29, 2023 at 1:14

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