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- Thread starter Ayesha_Sadiq
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UltrafastPED

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If you started with two unit vectors, that would be the end of it. To include general vectors we then multiply this "projection" by the magnitudes of the two vectors.

See http://mathworld.wolfram.com/DotProduct.html

And http://betterexplained.com/articles/vector-calculus-understanding-the-dot-product/

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A.T.

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Start with something simple like: velocity * time = distanceBut I still couldn't get what is implied when we multiply two physical quantities.

50 candies per packet * 2 packets = 100 candies

50 miles per hour * 2 hours = 100 miles

No, because it's 5N * 3m not 5N/m * 3m.Then wouldn't a 5N times 3m be like 15 newtons total force applied in 3 unit distances of 1m each.

5 meters * 3 meters = 15 square meters

5 Newton * 3 meters = 15 Newton meters

For a nice visal introduction into these concepts I recommend this book:

Thinking Physics by Lewis Carroll Epstein

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In physics there is no thing as multiplication,there is only dot product or cross product.

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In physics there is no thing as multiplication,there is only dot product or cross product.

Oh. But then what would division be in physics? I am guessing this is how relationships are developed between physical quantities. How physical quantities are sort of quantified relative to one another, and many factors considered at once. And it just all somehow works out.

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A.T.

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Your example with candy is also a "rate" multiplication: candies per person, candies per package.Thank you for the reference of the book. I'll try to get my hands on it as soon as possible. But then, in this case, wouldn't it be a somewhat 'rate' multiplication. Like, I am covering 2 metres per second, so how many metres I cover in 10 seconds would be a sort of 'rate' multiplication.

Velocity and time are both physical quantities.Not of two physical quantities, would it?

100 candies / 50 candies per packet = 2 packetsOh. But then what would division be in physics?

100 miles / 50 miles per hour = 2 hours

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Hmm. I think I see it. This is how it works then. Thank you. =)

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sophiecentaur

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That is a bit of a sweeping statement. If you multiply heat capacity by temperature change, to get heat input, you are just multiplying scalars and there is no 'dot' or 'cross' involved.In physics there is no thing as multiplication,there is only dot product or cross product.

The original concept of Multiplication would have been repeated integer addition. e.g. ten sheep with four legs gives you forty legs altogether. sheep plus sheep plus sheep etc.

Once the mechanics of this was established, it was found that you got the right answer if non-integers were involved. The history of Maths is full of instances of taking a basic operation and then finding it works in complicated situations.

The use of Maths to model the real world is an interesting point of philosophical discussion.

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The original concept of Multiplication would have been repeated integer addition. e.g. ten sheep with four legs gives you forty legs altogether. sheep plus sheep plus sheep etc.

Once the mechanics of this was established, it was found that you got the right answer if non-integers were involved. The history of Maths is full of instances of taking a basic operation and then finding it works in complicated situations.

The use of Maths to model the real world is an interesting point of philosophical discussion.

Thank you, I think this is what I was looking for. So, it's just Mathematics can describe relationships between physical quantities too, without it necessarily being a 'multiplication' of some sort.

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sophiecentaur

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There seems (to me) to be a dichotomy between discrete (integer) operations and continuous operations. Take the exponential operator, for instance. You can square, cube etc an integer number and it is easy to grasp what goes on. One step further - into fractional indices and logarithms - and, despite the fact that the same rules apply, it is hard to believe it's the same thing.Thank you, I think this is what I was looking for. So, it's just Mathematics can describe relationships between physical quantities too, without it necessarily being a 'multiplication' of some sort.

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BruceW

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Ayesha - in your first post, (2 people with 3 candies each), you can think of this in terms of sets and elements. Consider each candy to be a unique element (they are all different from each other). Then each person represents a disjoint set that contains 3 elements each. So the union of these two sets contains 6 elements. So your interesting 'intuitive' definition of multiplication can be closely related to the concept of the cardinality of sets. (cardinality means how many elements).

So, is it fairly easy to extend this to multiplication of continuous numbers? No, I don't think so. And so maybe this is the reason why multiplication of continuous numbers is much less intuitive than multiplication of integers, which can be thought of in terms of unions of sets.

edit: I should also have said, the reason why this works is because each person has the same number of candies. But I'm guessing you know that already. So the main idea is that the cardinality of the union of disjoint sets of the same size is equivalent to the cardinality of each set, multiplied by the number of sets. Which is a very intuitive concept.

So, is it fairly easy to extend this to multiplication of continuous numbers? No, I don't think so. And so maybe this is the reason why multiplication of continuous numbers is much less intuitive than multiplication of integers, which can be thought of in terms of unions of sets.

edit: I should also have said, the reason why this works is because each person has the same number of candies. But I'm guessing you know that already. So the main idea is that the cardinality of the union of disjoint sets of the same size is equivalent to the cardinality of each set, multiplied by the number of sets. Which is a very intuitive concept.

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A.T.

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Everything about integers is more intuitive than continuous numbers. But multiplication of continuous numbers is not specific to physics.And so maybe this is the reason why multiplication of continuous numbers is much less intuitive than multiplication of integers, which can be thought of in terms of unions of sets.

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BruceW

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neither is multiplication of the integers... I don't understand what you mean here. I didn't really mention physics because I thought that the OP's issue was more with the mathematical intuition.A.T. said:But multiplication of continuous numbers is not specific to physics.

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A.T.

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Maybe, but she asks specifically about multiplication of physical quantities in the physics forumI didn't really mention physics because I thought that the OP's issue was more with the mathematical intuition.

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BruceW

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sophiecentaur

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There are some things which we can 'count', however and that can give complete accuracy.

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BruceW

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I agree that multiplication of the reals does not have a direct link to your 'candy intuition'. I think most mathematicians would say the 'candy intuition' is one specific example of multiplication, and multiplication of continuous numbers is another specific example. I think 'multiplication' in maths is used as a very loose term. For example, multiplication of square matrices would be another specific example of a kind of multiplication.So then, real, continuous numbers are multiplied that way, and it doesn't really mean what multiplication means generally.

So the word 'multiplication' can mean a lot of things. I think the most general definition of 'multiplication' is the definition that you find when people talk about a ring. So we have a set with some 'addition' and 'multiplication' defined. The set is a monoid under multiplication and multiplication distributes over addition. This defines the most general kind of multiplication (I think).

Surely our ability to measure to arbitrary precision is a different issue? It looks like you are implying that if we could measure to arbitrary precision, then physical quantities would not take real value...Ayesha_Sadiq said:And I think physical quantities do take a real value, as we cannot measure them more than a specific degree of precision. And their actual value is really not what we measure, only a very, very close approximation, which works. I hope I am getting this right...?

edit: I think what SophieCentaur said is right. If we have a physical quantity that takes on real value, and we use some arbitrary units, then we will generally get a number with an arbitrarily large number of decimal places. So we can't even write down our measurement, because we have only a finite amount of pen and paper.

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sophiecentaur

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I think the best idea is to separate the thing into two areas of worry. First of all, get to grip with the Maths and 'make it work' for you. Then, when you have time to spare, get into Mathematical Analysis - or whatever they call it these days and see how everything that (legit) Mathematicians do is justifiable, going back to simple stuff like 0+1 = 0 and 1X1 = 1. Trying to tie those two things together at every step can really spoil your day.

PS You are quite happy to sit and watch TV and use your computer and you probably have very little idea of the working details of those things. We do it all the time. Chill.

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Yes. Physical quantities do take real values, but we can only measure them to a specific degree of precision. Got no tools or instruments for that. =P No division on an instrument could be the smallest, you could go on making the smallest divisions smaller and smaller, and increasing it's precision. So then, a real value like that, would be a continuous number? And we don't perceive multiplications of continuous numbers like we do for integers?Surely our ability to measure to arbitrary precision is a different issue? It looks like you are implying that if we could measure to arbitrary precision, then physical quantities would not take real value...

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Indivisible pebble, eh? =P

I think the best idea is to separate the thing into two areas of worry. First of all, get to grip with the Maths and 'make it work' for you. Then, when you have time to spare, get into Mathematical Analysis - or whatever they call it these days and see how everything that (legit) Mathematicians do is justifiable, going back to simple stuff like 0+1 = 0 and 1X1 = 1. Trying to tie those two things together at every step can really spoil your day.

PS You are quite happy to sit and watch TV and use your computer and you probably have very little idea of the working details of those things. We do it all the time. Chill.

Well, the quantisation of energy for electrons certainly amazed people.

I am still in college, and working towards the day I can tell to a youngie that I study the complicated working details of things all the time too. But the working of algebra in a physics context, I did not understand, and I asked.

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BruceW

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when I said 'real number' I mean the mathematical definition 'real number'. They have certain nice properties. In a sense, they 'fill up' the number line very tightly. So, for example, the set of all possible fractions is also infinite, but they don't fill up the number line as tightly as the real numbers do. So anyway, the 'real numbers' is the strict mathematical name for what we would usually call 'continuous numbers' in everyday talk.Yes. Physical quantities do take real values, but we can only measure them to a specific degree of precision. Got no tools or instruments for that. =P No division on an instrument could be the smallest, you could go on making the smallest divisions smaller and smaller, and increasing it's precision. So then, a real value like that, would be a continuous number?

I agree, we don't perceive multiplication of continuous numbers like we do for integers. For integer multiplication, we have the 'candy intuition' that you mentioned. But for continuous numbers, I don't think we have any kind of intuition like that.Ayesha_Sadiq said:And we don't perceive multiplications of continuous numbers like we do for integers?

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