Let’s say you have a division. You’ve got 3/2, where 3 might be acres and 2 might be brothers so from there you could deduce that the division just means that the 3 acres of land will be distributed between two brothers. Summation and multiplication are quite straight forward, but what the hell does one do with multiplication and exponentiation? If I got 20 N-m of torque, wtf does the multiplication tells me there? If something follows a square root?
Please feel free to add any other operations to these. Thanks!
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“Order of operations” just means “when a problem is written down, you solve it in *this* order”. There is no “physical meaning”.
Sure, 3/2 could be used for “3 acres of land are divided up equally between 2 brothers; how much land does each brother get?” and the answer is “3/2 acres”, or “1.5 acres”. None of your other examples mean anything though.
There is no physical meaning, in the sense you seem to be using, for mathematical operations. They’re just a way to use two concepts to indicate a third; as such, they only apply to ideas. Those ideas can be mapped onto physical situations in turn.
multiplication simply is counting individuals in equal groups say i had 4 boxes each with 5 balls you would be adding 4 to 4 5 times. i’d say exponential would be the pattern the same except the boxes are in separate houses but you want to know how many balls there are but you know the balls and boxes are the same for however many houses you need
Multiplication: Each of x brothers brings y beers to the family reunion. You have x*y beers total.
Square root: A field x square feet in area has 4 equal sides. Each side is sqrt(x) long.
Exponentiation: You are putting together an outfit that needs x types of clothing (pants, shirt, hat, etc.) You have y options for each type of clothing. There are y^x unique outfits you can wear.
The trick is that we often find ourselves in situations that require various mathematical operations to properly express. Just work backwards from there.
Multiplication is just repeated addition. In fact that’s how computers do multiplication at a circuit level.
> 20 N-m of torque
That’s not multiplication. Newton-metre is a unit of measure. It’s just twenty of them. Like saying 20 dollars.
A newton metre is just the equivalent of putting one newton of force on a moment arm 1 metre from the point of rotation.
And a newton is how much force it takes to accelerate 1 kg by 1 metre per second per second.
Numbers are invented not discovered, so after a certain point of applicability it’s all abstract. That’s how people have come up with complex maths that are later able to be applied to another practical field.
Physics provides a lot of insight into representation of mathematical operations.
For multiplication, you can imagine a car going a certain rate of speed, e.g. 60 km/h. If it travels for 2 hours, how far will it go? Well, 60km/h * 2hrs = 120Km.
Taking your N-m of Torque example, you can imagine that as applying 1 newton of force over 1m of distance. The distance in this case is rotational instead of linear, but the idea is the same.
Exponentiation is similar, if you have a cube shaped tank with a side length of 2m, then how much water will fit inside it? (2m)^3 = 8 cubic meters. Cubic meters might seem a weird measurement, but it just means the volume taken up by a cube with a side length of 1m, so you could fill 8 1m side length cubes with the water from one cube with a side length 2m.
Square roots are just the opposite of squaring, in case you wanted to know the side length of a square given the area that it covers.
I disagree with u/Melenduwir but it may come down to semantics: I think mathematical operations *do* have physical meaning. Take Einstein’s equation E=mc^(2) for example. We can read this as a sentence “Energy is Mass and they are related by a factor of c^(2)”. That equals sign is our human attempt to write down what the universe just does naturally (convert matter into energy like in stars).
The beauty of math is that it works the same with or without any physical meaning. Its logically self consistent as abstract symbol manipulation.
Now when it comes to applying math, of course you need to match some parts up with physical meanings as needed, but you can treat a lot of it as black box internals of which you know are logical and self consistent.
As for some examples for practical use of math concepts.
Multiplication is when something such as torque is proportional to several things such as force applied and length of the leverage.
Division is often inverse proportionality, such as speed is proportional to distance traveled and inversely proportional to time taken.
Exponentation describes growth very well, processes with feedback. If you reinvest your earnings hour earnings grow even faster.
Many things can have polynomial relations, reflected radar signal is proportional to 4th power of distance for example.
Complex numbers line up very well with oscillations where energy is coupled between two different things such as tension and inertia or electric and magnetic fields.
Matrix operations and linear algebra lines up well with coordinate transformations, inherently pure mathematics, but describes physical space and motion very well.
But just because a mathematical concept is often used in one physical meaning, doesn’t mean it can’t be used to describe completely different physical meanings. Mathematics is just abstracted logic, with whatever type of logic you understand some physical phenomena, you can transform it into a mathematical model and for there manipulate it as such.
The concept is called Dimensional Analysis. You probably learned about it in HS physics. Just as the operation applies to the numbers, the operation applies to the dimensions.
I used to apply this through calculations to check if the answer made sense – if I lost the way through the problem, the dimensions would not agree with what i was trying to solve.
Miles / hour = velocity or speed
acres / brother
velocity or speed = distance/time
mass * (distance/time)*(distance/time) = Force
velocity^(2) = acceleration
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