I also want to know! Does it have to do with different pipes? Or is the water coming from a different source?

Upon further thought I think the idea I'm trying to convey has to do with something such as, "phase states."

Ice is very cold, and it can only get warmer. So by adding salt to ice we make ice begin to melt quicker, which facilitates heat transfer in the case of making ice cream.

Water is not very cold. It can only get colder before it becomes ice. Adding salt to it will lower the freezing point, but this really doesn't matter because cooling water takes a lot of energy, and we aren't trying to cool the water, we are trying to cool the ice cream, which is most efficiently done by using ice, and adding salt.

The reason salt makes the ice melt faster is complex, and that it lowers the freezing point of water is relevant, it really isn't the actual answer here as far as I know, but I don't know much.

This is a mechanical and chemical physical question, and is fairly complex I suspect, but as far as I know from Feynman (what did he know besides winning a Nobel prize?) the reason is that the salt adds weight to the ice, and then that creates salt water, which further adds weight, and then you have this process almost like burning a candle where an ice cube will melt into water.

The maximum coldness of that ice cube is not increased by adding salt, but by adding salt it does increase the maximum coldness of the inner shell of the ice cream machine, because it helps the ice melt faster, which helps transfer the heat more efficiently. The 'grit' of this salt as it rubs into the salt is a major factor here, but more generally speaking if you were to simply drop salt onto ice in a very cold environment it will melt the ice a bit because simply dropping the salt on the ice is going to melt it.

https://www.youtube.com/watch?v=MO0r930Sn_8

Okay, do

i*i

using addition.

Well, how would you describe pi*e in terms of iterative addition?

If I were to try, I'd say e+e+e+(pi-3)e, but that's just shifting the multiplication somewhere else, no?

There seems to be disagreement on this, and not just by me (see my sources).

That was just an error. I intended it to say 4 x 8 + 9 - 2 x 7 + 9 ÷ 3 + 5

Can you elaborate? I feel like the logic should apply to all numbers. Hell even in the complex sphere you should be able to do multiplication through iterations of addition.

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Consider this formula:

4 x 8 + 9 - 2 x 7 + 9 ÷ 3

BUT, the +4 and -2 and +5 Where did the 5 came about?

How is there an XKCD for just about everything... that's an absolute 10/10.

Again, you can have a system that works perfectly well with multiplication prior to addition. There is no "inherent rule in nature" as OP phrased it guiding this.

This is the answer

Convention.

We could fully parenthesize everything, but it gets annoying to write polynomials if you have to put parens around everything, and it turns out that polynomials are super useful in math, so we set up a convention to make them easier to write purely for convenience.

If it happened to be that we most often wanted to do addition first instead of multiplication, then we would have set up the convention so that addition has a higher precedence so we wouldn't have to write a lot of parens for that thing we do all the time.

That's literally it. Mathematical notation simply exists to be a concise way of writing what we do most often to save us writing. It's nothing to do with math.

This is why it's super annoying when people insist that some equations like 6/2(1+2) are ambiguous. Literally the entire reason we set up these rules is to make sure that there are no ambiguities, so unless you're willing to accept that these super smart people that set up these rules just did a crap job of it and you found a flaw, the situation here is simply that you just don't know the rules they set up.

I think a lot of these answers are losing the forest for the trees.

What are the two most pointless symbols in mathematics education?

The symbols for multiplication (×) and division (÷).

Why? Because they disappear when you get to algebra.

Multiplication instead becomes the default of two values are next to each other without an operator, then they are multiplied:

Instead of 5 × x, we just write 5x.

All division turns into fractions.

Now think about something like a polynomial. For example,

5x^2 + 4x + 3 = 0

Is much easier to write than

5 × x^2 + 4 × x + 3 = 0

Which is also easier to write than

5 × (x^2 ) + 4 × x + 3 = 0

Now consider how the order of operations makes expressing the terms of a polynomial straightforward:

exponent first, so the variable can be raised to the appropriate power given for each term.

Multiplication next, so each term can be multiplied by a constant.

Addition last, for the final combination of fully calculated terms.

PEMDAS makes writing a polynomial straightforward and helps the notation capture meaning.

Obviously, there is more to math than polynomials and as other answers have touched on, there are lots of reasons why multiplication comes before addition.

I would also like to add that subtraction doesn't exist and it would be better if we stopped teaching it once negative numbers are introduced, but I do think that ship might have sailed.

So why not just have it be like SDPAEM? (subtract, divide, parentheses, add, exponent, multiply)? It doesn't make sense that the order is entirely arbitrary.

It seems to me that some arbitrary decisions were made, like to have addition before subtraction, or whether to have division before multiplication, but it seems clear the choice (for example) to have multiplication and division before addition and subtraction is not merely arbitrary and rather, is based on multiplication and division having a greater order of magnitude in their effect compared to addition and subtraction. Same with exponents being before multiplication and division.

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It's due to the amount of force over an area.

When the balls start slapping, they are typically covering a wide surface area with relatively little force.

A nut tap is concentrated to one area with more force.

Physics.

Probably because at its core, multiplication is just short form addition. Instead of writing 2+2+2 we can just write 3×2, and similarly instead of 3+3 we can write 2×3.

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It'd be less convenient to do it addition-first, but the system would still work and be consistent.

Hell there's even Polish notation, which you'd write

(1+2) x 4

as

x+124

This comment provides a similar explanation to what I'm saying:

https://www.quora.com/Is-there-a-logical-explanation-for-the-mathematical-order-of-operations/answer/Stephen-McBride-13?ch=10&oid=300169584&share=4e8656ca&srid=YAyd&target_type=answer

See also here: https://www.reddit.com/r/mathematics/comments/k2nfui/comment/gdvfjla/?utm_source=share&utm_medium=web2x&context=3

This comment provides a similar explanation to what I'm saying:

https://www.quora.com/Is-there-a-logical-explanation-for-the-mathematical-order-of-operations/answer/Stephen-McBride-13?ch=10&oid=300169584&share=4e8656ca&srid=YAyd&target_type=answer

See also here: https://www.reddit.com/r/mathematics/comments/k2nfui/comment/gdvfjla/?utm_source=share&utm_medium=web2x&context=3

Okay. So this is what I just figured out by trying to do a couple equations like these.

4 + 3 x 8 + 6 - 8 / 2

We can make sense of how the order works by trying to look at numbers separately. So 4 alone is 4. It’s absolutely 4. But if you look at the next number, it’s 3, but accompanied with x (3 times) So it’s 3x8. So the actual number there would be 24. So we put 24 instead of 3x8. Next, there is 6. Which again is a complete number by itself, so we leave it alone. Next comes 8, but it is accompanied with 8/2, that means it’s actually 4, not 8 alone. Once we’re done changing all the numbers to complete values, we do the addition and subtraction

It's actually GREAT practice! How else can you write this little gem in beginners all-purpose symbolic instruction code: 10 Print"Hello lovely!" 20 Goto10

It's mostly a matter of preference. While the word 'country' is often used interchangeably with 'sovereign state', it's not like the word is reserved only for that use. In the UK, country is just the word that they use for the major administrative divisions, similar to states, provinces, prefectures, and the like.

It is true that that preference reflects the historical status of those regions as independent kingdoms, but that is not something that is unique to the UK (Hawaii, Bavaria, Sicily, Saxony, etc)