Understanding Negative Scalars in Physics

[korters]

I don't understand how a scalar can be negative. Like work for example, how can this be - or + yet still be a scalar. I've read that scalars only have magnitude, while vectors have both magnitude and direction. As vague as these definitions seem, I reason that scalars wouldn't have any +/- sign attached to it since that would indicate somesort of direction. This is what doesn't make any sense to me. Also, this might be a little off topic, but in relation to speed and velocity, I've heard people say that speed is just the magnitude of velocity without any direction. This can be misleading to me because what if someone traveled all over yet started and ended in the same location, this would mean the speed would be some large value and velocity would be zero - but in regards to some people saying that speed is just the magnitude of velocity, wouldn't you then get two values for speed then (that large value and the absolute value of velocity)? This doesn't make any sense to me.

Not to be an *******, but please try to break this down in a way that makes sense to someone with only a mathematics education up to calc. 2 (I really hate it when "brainiacs" try to explain something using terms and mathematical ideas that are way beyond my education level in such a way that doesn't even coherently make any sense).

Thanks a ton for your help!

 

[symbolipoint]

Could a scalar be a number which can be represented on a one-dimensional number line? This is then a Real number, and Real numbers contain both positive, and negative numbers, and zero. This is not a characteristic of having direction; but just the characteristic of being greater than, less than, or equal to zero.

 

[Hurkyl]

Any 'number' can be a scalar -- it's all a matter of what structure you're working with. There is nothing deep to it; it's simply a matter of definitions. And in the settings you're studying, letting "scalar" refer to any real number is the most useful definition.

Other common situations are:
. In number theory, it is often useful to let "scalar" refer only to integers, or only to rational numbers.
. In physics, it is often useful to let "scalar" refer to complex numbers
. When working with fields (not fields), it is often useful to let "scalar" refer to any scalar field.

 

[mathman]

The simplest way to look at it is to consider sets of real numbers. If there is only one number, it is a scalar (+ or -). If it is a pair of numbers it is a vector in the plane. If it is a triplet, it is a vector in three space. Mathematicians also consider vectors in higher dimensions, but you can ignore this for a while.

When talking about velocity, it has a magnitude and a direction. When you take a round trip, the AVERAGE velocity is 0, although the average speed is obviously not.

Part of the confusion you may have is not distinguishing between magnitude (always non-negative) and scalar (any real number).

Advanced warning: if you go on further in mathematics, you will find the concept of scalar being extended to include complex numbers, while vector to mean multiples of complex numbers.

 

[Tac-Tics]

I don't understand how a scalar can be negative.

Because it's defined that way.

Math is different from physics because you can choose what kind of rules you want to use as long as your reasoning is logical. Since math is so abstract, names we give things are pretty arbitrary. Real numbers are called real because they were originally considered to be the values that lengths could take in the real world.

A scalars is a scientist's term for a real number. The two are (in most contexts) completely synonymous. That's the reason they can take on positive, negative, and zero values. If you ask 'why', the answer is because that's how we've defined it.

It seems you are confused about the relation between scalars and vectors. Vectors are funny things. You can represent them in a few different ways:

Cartesian Coordinates: (x, y) is a point given two real numbers x and y. Move x units to the right then y units up.

Polar Coordinates: (r, theta) is a point in space given by a real number r, representing the distance from the origin ("r" for radius), and a real number theta, representing the angle by the triangle between the point, the origin and the point at (1, 0) on the x-axis.

Cartesian coordinates are very simple, and each point has exactly one representation. If x != x' or y != y', then (x, y) != (x', y'). Very simple.

But polar coordinates are a bit more confusing. First, r can be positive or negative. When it's negative, it's the same as doing a 180º change in direction. That means that (r, theta) = (-r, theta + 180º).

It actually gets worse. Angles are not unique either. If you spin 360º, you wind up looking the same direction. So (r, theta) = (r, theta+360º).

For this reason, it's sometimes convenient to put restrictions on r and/or theta. In this situation, r would be restricted to a non-negative real and theta would be restricted to a real between 0 (inclusively) and 2pi (exclusively). And you still have to be careful with the 0 vector! (0, theta) = (0, theta') for all theta and theta', so we might also want to put a third restriction that says if r = 0, theta must also be 0.

These restrictions are a pain sometimes, though, so depending on the problem, we might just let r and theta be whatever, and correct for it whenever we do a comparison. It's pretty easy to convert any vector (r, theta) to a "normalized" one which obeys the restrictions above.

But the take home lesson is, in math, you can define things however you want as long as you keep the definition. You'll run into this in other areas as well, such as the infamous 0^0 issue. Everyone agrees on what x^y means when x > 0, but what about for x=0 and y=0? Does 0^0 = 0? 1? Or is it undefined entirely?

The truth is that ^ means whatever the hell you want it to mean! If you want it to equal 1, polynomial equations come out nicely. But then the function f(x) = 0^x is not continuous. If you want 0^0 to mean 0, then g(x) = x^0 is not continuous. If you want to make it so ^ is not defined for 0^0, then both f and g are continuous at all points in their domain, but their domain is now R - {0}, making ^ a partial function. It's a case of pick your poison.

(I really hate it when "brainiacs"try to explain something using terms and mathematical ideas that are way beyond my education level in such a way that doesn't even coherently make any sense).

Don't make such negative assumptions about people when you ask them for help. It doesn't reflect well on you to say things like this. Sometimes the help you get isn't helpful, but you just ask for a more appropriate explanation, and hopefully someone on the other end will help guide you.

 

[HallsofIvy]

You seem to have the wrong definition of a "scalar". A scalar is simply a number and can be either positive or negative.

You may be confusing this with the "length" of a vector which is never negative. The length is a number and so is a scalar. But not every scalar is a length.

 

[mgb_phys]

Even if physics a scaler can have a 'direction' in the sense that it is a change. So a temperature change is a scaler but can be signed.
I think the confusion is introducing scalers and vectors together and using the magnitude of a vector as the example of a scaler.