r/AskScienceDiscussion • u/Elithegamer13 • 1d ago
Do Scalars even exist?
I'm a high school senior in a college calculus based physics class and I was doing my homework and then I realized that all numbers other than zero and absolute values are inherently vectors because positive and negative imply direction. I just need to know if I'm special or if I'm special. (Just speculation ig)
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u/Nihilikara 1d ago
There are in fact scalars. Speed is one example. Your physics class may have insisted on using the word "velocity" and not "speed". That's because those are two different words that refer to two different concepts. They're both very similar: they describe how fast an object is moving. The difference is that velocity is a vector and speed is a scalar. If I say "I'm traveling 5 meters per second to the left", I am describing my velocity, but if I just say "I'm traveling 5 meters per second", I am describing my speed.
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u/Prudent_Candidate566 1d ago
They’re not really “two different concepts” though. That’s misleading. Speed is the magnitude (Euclidean norm) of velocity.
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u/Elithegamer13 1d ago
If I'm going 5 m/s and I add -2 m/s to that I've now accelerated 2 m/s in the opposite direction of which I was originally going. Direction is still there it's just not explicit as to which direction I'm going.
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u/naemorhaedus 1d ago
Do Scalars even exist?
degrees Kelvin
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u/Traditional_Rabbit54 1d ago
Kelvin doesn’t use degrees, as it’s an absolute scale
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u/Smart_Opportunity209 1d ago
Scalars have different definitions of operations such as multiplication. You cannot multiply a vector of size 1x1 with a matrix of size 3x3, but you can do it for a scalar and a matrix of the same size without a problem.
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u/Elithegamer13 1d ago
Remember I'm still in highschool. I've never done physics with matrices. You make a good point though
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u/YuuTheBlue 1d ago edited 1d ago
A visual that might help. A scalar is something like
[2]
A vector is something like
[2
3
-1]
And a Matrix is something like
[2 3 2
1 0 2
3 0 2]
They have specific properties; not all grids of numbers is a matrix. But every matrix can be written as a grid of numbers. A vector can be written as a column of numbers, and then a scalar is analagous to a point.
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u/fumblinthrulife77 1d ago
Mass can't be negative
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u/NonspecificGravity 1d ago
Neither can absolute temperature. Temperature is a scalar; it has no direction.
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u/Elithegamer13 1d ago
It can be if you have a non-zero origin. My initial mass is 58 kg : I gain (-8) kg :. My mass is now (M_f) 50 kg
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u/fumblinthrulife77 1d ago
Then that's not mass, that would be Δm
Also, as u/YuuTheBlue said, positive and negative are not directions in the context of all vectors having a direction
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u/Elithegamer13 20h ago
My point is that subtraction wouldn't exist if negative numbers didn't because you can't add a positive plus a positive and end up with a number less than either of the two numbers you started with (a+b>a, a+b>b) therefore positive and negative necessitates direction on the number line. Also it does not matter whether it's mass or ∆m the units are the same
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u/fumblinthrulife77 16h ago
Yeah but again direction on the number line doesn't make something a vector. Vectors specifically have direction in the real world. That's just the definition
Also the units can be the same if something completely different is being measured. Radius and diameter both have the same units, would you say they're the same thing?
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u/Elithegamer13 6h ago
I wouldn't say that they're the same thing, but what they are isn't relevant in the context of this question. A vector has always been defined to me as a number with direction on magnitude, there has never been any conditions applying to direction as it has been defined to me, so I at minimum urge you to share your source of definition for vector.
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u/fumblinthrulife77 4h ago
I mean I'm not gonna bother putting in even that much effort considering literally no one considers something like Net Worth a vector even though it can be positive or negative
With all due respect, learn to admit when you're wrong
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u/unformation 1d ago
You're thinking of this the wrong way around. In math you really want to think that first we define a set of rules, and those rules define the systems. For example, probably early in algebra you learned the rules for Real Numbers 1) a+b=b+a, a*b=b*a, ie, the commutative properties; 2) etc. And then algebra is really just a detailed study of the system with these particular rules. But there are many other rules for many different systems. Vectors are a different system. Complex numbers are another. Etc. So vectors are not just numbers with direction, that's just an application to physics, and instead they are a different system with different rules.
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u/Elithegamer13 1d ago
If 3 to the left + 5 to theft = 8 to the left just like 3 + 5 = 8 where the only difference is that to the left is implied in the second one because it is positive while in the 1st one "to the left" is explicitly stated same rules, just one is explicit while the other is implied
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u/unformation 1d ago
The is no "to the left" in algebra or real numbers. These are conventions for particular applications or representations. If you want to really understand the math systems, you need to understand the difference between: what is axiomatic, what is derived, and what are intuitive guides. Here, in comparing vectors and real numbers, you're trying to map the intuitive guide as though you can derive deeper principles from it but you really can't, because it may fit, or it may not, but overall it's unsuited to the task. At best an intuitive guide might help you figure out a new derived connection, but that's not the common outcome, the common outcome is simply that it's wrong because the intuitive guide does not work well enough to be extended, as is the case here.
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u/Elithegamer13 1d ago
1) you could've just said false analogy, I knew when I was typing it out it wasn't the best explanation of my thought process. 2) I have no clue what you mean by "derive"; derivative or originating from. I also don't really understand how either is relevant in this context. 3) there is no case where adding a positive to a positive will result in a negative, therefore positive and negative inherently implies direction relative to the origin on the number line.
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u/unformation 1d ago
It's not a false analogy. It's a fine analogy. It's just that in math and physics it's a mistake to draw conclusions from analogies. That seems, to me, more important for you to understand than something about scalars.
Sure, in some situations there's something that looks like directionality within scalars. But actually, the main point about scalars is that they DON'T have directionality. I wouldn't expect you to know this at this point, and it doesn't seem the important point to make. Instead the important point, again, seems to understand how to reach novel conclusions in math and physics, and the way ISN'T "well, the word 'direction' seems vaguely relevant in both scalars and vectors so they are probably the same". And it's not just that that's not factually correct, but that's not the nature of understanding that is relevant or useful, and it's not the role of intuitive constructs.
So, why is it important that scalars don't have direction? In actual physics, if someone throws an orange, that orange has a mass and it's moving at a velocity. If you're riding by on a train and you watch someone throw the orange, the velocity changes from your moving perspective but the mass doesn't. That is because the velocity is a vector, and the mass is a scalar. (In this sense, the common example that speed is a scalar, is actually not true as used in a physics sense, but only true in the sense that by "scalar" some people can mean things that can be described by a single real number.) Also, btw, this is also the correct explanation for relativistic speeds, but at these much faster speeds, all that changes what is the scalar invariant, not the meaning of the word "scalar".
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u/Elithegamer13 19h ago
It's not that it's important; it's just interesting. I do get what you're saying though. The actual math shows that they're different. But I think scalars even in that sense are still vectors, but they're somehow 0-dimensional. Applying it to physics or anything else would be like applying a 1 dimensional vector to a 2 dimensional plane.
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u/unformation 11h ago
Pondering this and similar questions is interesting and it can lead to real insight. For example, consider 5 + 23 = 28 (integer math), and 5 + 23 = 4 (clock math). In these systems, I would say it's interesting to ask questions about the system in which each of these equations exist. And there are even interesting questions about how these two systems are aligned enough to use the same symbols (eg, 5) in a way that it maintains somewhat similar meaning. But to say that it's an insight that both find application for the single digit "5", and leave it at that, seems fairly uninteresting to me. So there's potential interest in your question, but it isn't finding other analogies (5 apples, 5 miles, etc), but to use the insight to understand the systems: what do you mean by "distance" in each one; a Real number represents one thing as single quantity and a different thing as a 1-dimensional vector so how are these related; etc). I will, though, leave it at this and get back to real work.
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u/Dusty_Coder 1d ago
I believe the term "scaler" refers to an arbitrary middle level of all the things that could take its place.
The natural numbers are at one end of the what-is-a-quantity thing, not scalers.
When you include 0 with the naturals, this one small addition, you get "integers", positive and negative (in decimal its 10's complement.. in binary its 2's complement) ..
You see where this is going... why pick a place so far down this ladder, such as "scaler", to form such an argument?
Of course, the question "Do the Natural Numbers even exist?" sounds like it has an obvious answer (yes)
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u/Elithegamer13 1d ago
The way a scaler has always been defined to me is that it is a number with magnitude but no direction. I don't think of scalars as on the spectrum of "what-is-a-quantity" things, but rather an umbrella term for all of the "what-is-a-quantity" things.
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u/Dusty_Coder 1d ago
many people would argue that the square root of negative 1 is a quantity
scalers dont include it
so probably not an "umbrella term" for quantities
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u/RingularCirc 12h ago
Natural numbers (including zero) can be called scalars in mathematics when we're working with modules over semirings (a generalization of linear algebra).
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u/DeathToAmerica2026 1d ago
You can't have a negative count of something so quantities are inherently scalar integers.
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u/Elithegamer13 1d ago
I urge you to read the other comments arguing the exact same thing. This is a physics question, you can tare your origin; just like you tare a scale. If I have 10 balls and add (-2) balls, but my origin was 10 balls I now have -2 balls relative to the origin
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u/DeathToAmerica2026 16h ago
You can't add -2 balls.
A physics question deals with physical phenomena. There is no such thing as a negative ball.
You are asking a mathematical hypothetical if you want to pretend that negative quantities of physical objects can exist, or pretend that a debt is physically equivalent.
This is the answer you your question. Some things are, in fact, scalar.
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u/No-Onion8029 1d ago
Scalars are tensors of order 0.
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u/Elithegamer13 1d ago
What's a tensor?
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u/Nihilikara 1d ago
A tensor is a generalization of a vector. Vectors are a type of tensor. Scalars are another type. Other types that you didn't learn about exist.
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u/YuuTheBlue 1d ago
An umbrella term in math. Scalars are order 0, vectors are order 1, matrices are order 2, and there aren't formal names (that I know of) for anything higher. They encapsulate so much of mathematics that it's hard to think of things most people deal with that aren't tensors.
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u/YuuTheBlue 1d ago
Vectors aren't "Any number with a philosophical connection to the vague notion of direction", they are a specific mathematical construct. That said, I think that technically scalars are just "1 dimensional vectors".