Why can scalars have a sign?

The sign of a scalar depends upon the scale with which it is measured.

In the case of temperature, a Fahrenheit scale arbitrarily says 32 degrees is the freezing point of water, and zero degrees is a mark on a scale of numbers. Any temperature less than zero has a negative sign.

The Celsius scale says zero is the freezing point of water, so temperatures below zero have negative signs.

But a Kelvin scale says zero is the coldest possible temperature, so there are no negative numbers in the Kelvin scale.

The sign of a scalar is a matter of convention determined by how the scale is defined and numbered. Most scales do not include negative numbers because it would make computation confusing.

However, scalars also exist as the dot product of vectors. For example, work is the dot product (or scalar product) of force and displacement, both of which are vectors. In these cases, the negative sign is a useful convention, though work does not have a direction.

Think of a vector as having direction in space (north, south, east, west). Scalars may or may not be capable of having negative values. It just depends on the nature of the quantity.

The statement that negative values for scalars are just convention is rather misleading. Some "conventions" just naturally make a whole lot of sense, and changing them would be illogical. (and btw, despite another answer, it is possible to have negative temperature, even with the Kelvin scale, or with any scale. Just look at any college level text on thermodynamics.)

A negative value for a scalar does not imply a direction in space. Temperature can be negative, but temperature doesn't have a direction in space. A circuit can have a negative amount of voltage at a given point, but the voltage isn't pointing in any direction. Work can be negative, meaning you're taking energy from something. (think of friction slowing a car down.)

The "one-dimensional axis" that you're referring to isn't something in real space. Does this axis point towards the north, the south, the east, or the west? Or does it point into the sky? It's just a concept that you (or your teacher, or someone) has invented to help you understand things.

The modern notions that separate "scalars" and "vectors" goes as follows:

• Scalars are elements of fields. Examples of fields include the rational numbers, the real numbers, and the complex numbers. Scalars can be added and multiplied and divided.

• Vectors are spaces over fields. These are basically just lists of elements of fields (You can get fancier than that, but let's not). Velocity vectors, for example, are triples of real numbers. Vectors can be added and subtracted, but not multiplied nor divided.

That's it! Here are some examples.

1. Real scalars that can have a minus sign include the coulomb and gravitational potentials, as well as any other potential (like $mathrm{Pe}=mgh$, $h$ can be negative).

2. Complex numbers are scalars with no notion of positive or negative (you cannot say $i<0$ or $i>0$). They do have a notion of "direction", but in quantum mechanics for example, the "direction" of a complex number is meaningless (we say "the wavefunction is unique up to a phase"), so you really do have scalars with no possible meaning of "direction", but which still have $+1$ and $-1$ as scalars.

3. Temperature in kelvin, and mass, are both [real] scalars almost always positive. But it's incorrect to say "that's the case because mass and temperature are scalars". There are other reasons that's the case.

Middle school teachers might tell you that "$-3$ cannot be a scalar, because it has a direction", but my example with the gravitational potential is a good counter-argument, and my "wavefunction" example seals the deal. If your teacher insists "$-3$ cannot be a scalar", memorize their sentence, remember it on their test, and forget it immediately afterwards :)

I've read all the answers here and believe there is still more to be said that could be useful. I've been obsessing over this lately because I'm currently writing a physics book. The concept of distance, a scalar concept, (as opposed to displacement, a vector concept), knows no direction and is often mistakenly referred to as a signed quantity. The OP is correct in pointing out that a signed quantity can imply direction.

An object moves, on whatever shaped path and accumulates distance as a positive quantity. Distance cannot be meaningfully negative. Consider throwing a ball vertically upward into the air. It travels a certain distance $h$ upwards and then falls the same distance back again having travelled a total distance $2h$.

If you tried to make the return distance negative, to somehow capture that it's come back to where it started from, you'd then have to explain how that worked for an object travelling in a circular path of circumference $2h$. Does it get half way around the circle a distance $h$ then travel the other half a distance $-h$? This is not a tenable idea at least because it's artificially introducing a one dimensional direction and then forcing it into a two dimensional problem.

So, the distance concept, properly understood, can only be positive and distances cannot be used to cancel each other. A properly drawn distance graph can only have zero or positive gradient. So, distance cannot be used to model uniformly accelerated motion in one dimension - displacement is required for this.

Because distance is always increasing, change in distance is also, always positive, so instantaneous speed is always positive. Change in distance is still not a vector concept and cannot be used to define velocity, which is a vector concept. Instantaneous speed in turn can be used to define a stunted form of acceleration, which is a signed quantity, because speed can decrease, unlike distance. But speed still cannot be negative, otherwise an implicit one dimensional direction would sneak in again. In the thrown ball example above, this stunted acceleration would be a negative constant going up to slow the ball down and at the top of the ball's ascent would instantaneously and discontinuously change to a positive constant to speed the ball up going down.

So that's an example of a scalar concept with a very limited application. Now consider a vector space example, say $mathbb{R}^2$ over $mathbb{R}$, where the scalars have a very powerful and certainly not limited application. Here the scalars are real numbers and so can be negative as well as positive. If you multiply $(1,1)$ by $2$ it becomes $(2,2)$, that is, it gets bigger but does not change direction. But if you multiply $(1,1)$ by $-2$ it becomes $(-2,-2)$. It gets bigger and changes direction. So, scalar multiplication can change the direction of a vector as well as its size and can be used to define inverse vectors. The scalar real numbers themselves have no meaningful direction as such in the way that vectors do.

The set $mathbb{R}$ itself can be a vector space over $mathbb{R}$. On one side playing the role of vectors and on the other playing the role of scalars. So, its not surprising that the real numbers can be used to model one dimensional displacements and can therefore be used to model uniformly accelerated motion in a straight line.

So the use of the word scalar is varied. There are cases where they must strictly only be positive and there are cases where being a signed quantity is essential and there are cases where a single set can play both the role of vectors and scalars. Hopefully this discussion has highlighted further the difference between a scalar quantity and a vector quantity.