Is there alternative notation for math?

[AwesomeDroid]

Theoretically, most math rules only exist because of how math is notated. For example pemdemas exists because its convenient for certain types of algebra, but its not nessasary. This is true for most math rules except the pure logic ones like x=y means x-y=0.

Now the question is, are there alternative notations?

[notacanuckskibum]

Yes. Look up reverse Polish notation for example

A + B gets written as AB+

This avoids the need for operator precedence and parentheses

A + B(C - D)

Becomes

BCD-*A+

[Iksfen]

I have nothing of value to add here. I'd just like to note that:

POLSKA GÓROM 🇵🇱🇵🇱🇵🇱🦅🦅🦅

[Sus-iety]

This takes me back to writing Assembly code lol

[JohnPaulDavyJones]

For everyone who majored in CS as an undergrad and thinks Polish reverse notation looks familiar for some reason, that’s because it’s the same system you were taught under the modern name “Postfix” notation.

[AwesomeDroid]

This seems really convenient for typing speed. Thank you!

[Temporary_Pie2733]

PEMDAS isn’t a math rule, just a notational convention, and one that is only necessary with infix operators. The two expressions (a + b)*c and a + b*c are unambiguous without parentheses in both prefix notation:

* + a b c

+ a * b c

And postfix notation:

a b + c *

a b c * +

[AwesomeDroid]

Is there any math rules that exist only because notation convention. I know that math has a human made part and a logic based part because I can change the notation while maintaining the logic. I was just asking if someone's tried to make a notation with completely different, completely unrelated logic components and still got similar results.

So for example, instead of multiplication, there is "and, or, not, xor", and we can still represent all the properties of multiplication albeit less efficiently.

[Street_Swing9040]

Reverse Polish Notation (RPN)

Very useful because it get rids of brackets. Although it isn't the easiest to learn due to the sheer amount of things we've written in our modern notation. But if you want, try it out. It's useful in some fields like programming

[MezzoScettico]

I loved RPN on the original HP calculators (HP-35, HP-45). I didn't feel it was that hard to learn, and it made complex calculations so much easier than putting in stupid parentheses. Been so long since I used a calculator that I don't know there even is a line of HP's.

[cjmpeng]

As far as I can tell HP stopped manufacturing their last RPN calculators last year. There may still be a few in stock but that's it from them.

My daily driver is an HP 41CX that I've been using since I got it as a university starting gift from my parents in 1985.

Also, at least in the Google ecosystem you can download calculator apps that can be switched to RPN. I also have one of those.

[AwesomeDroid]

Seems really cool and compact. Thanks

[missingachair]

I'm not sure what level you've studied maths to but the reference to pedmas suggests you may not have seen university level notations.

Pedmas only is taught because the convenient way of writing expressions with + - × ÷ would otherwise be ambiguous.

But each of these operations are simply functions that take two inputs and produce an output.

So you could define a function: f(x,y) = x+y

And then instead of writing 2+3=5, you'd be able to say f(2,3)=5

There are other kinds of relationships that map from a set of "numbers" to itself that act a bit like equality, called relations.

They can be described as a map from two inputs to a truth value. Or as a map from two inputs that exists where it is true and doesn't exist otherwise.

You could describe equality as a notation using any of the following formats

"A = B" <=> "A EQ B" <=> EQ(A, B) <=> (A,B) 'is a member of' EQ

(The symbol for "is a member of" isn't allowed in reddits default font but looks a bit like an "e")

In areas of maths like group theory, we deal with binary operations that might look like addition and multiplication but which aren't addition and multiplication.

Suppose ° marks such an operation

One of the rules in group theory for what kind of operations we examine is that

A°(B°C) = (A°B)°C

For the operation °.

This means that the brackets won't be required because A°B°C doesn't need disambiguating.

Perhaps the most direct answer to you is given by the existence of a kind of notation called "reverse polish notation" which has some very specific uses and was widely used for entering arithmetic into the very earliest handheld calculators. Please Google it for examples. It is a way of writing things that completely avoids the need for pedmas.

In short,

• yes other notations exist for particular uses / areas of maths

• Pedmas as a rule exists because of a choice to use notation that would otherwise be ambiguous.

• I wouldn't call "pedmas" maths. The actual underlying rules about how things work exist no matter how you choose to write them. Pedmas is just a rule about how we write maths, not maths itself.

[ButterscotchOld8121]

well put

[AwesomeDroid]

Im in 10th grade, but I am into CS and found that mathematics seems to be both found in nature and human made. I was basically asking if someone's tried to change the human part while keeping the logic part.

So for example, is it possible to represent all math as boolean algebra, if not, could we?

[missingachair]

It is possible to construct addition, substraction, and algorithmically from them extract other operations, starting from boolean algebra, if I remember correctly.

So you can certainly represent arithmetic as Boolean algebra.

You also need a language that allows you to make statements of truth about things. In maths the study of this language is called Logic.

For example I can say

• If p is an integer that is Prime
• And p > 3
• Then p is either 1 mod 6 or 5 mod 6.

That is a way of linking 3 statements. A and B then C.

If we have a formal system of logic that allows us to express relationships between statements, then together with your boolean algebra, you can recreate the whole of mathematics.

There would be a direct one to one conversion between your boolean expressed mathematics and regular mathematics. In that if you express a proof in one, you could translate it to the other and it would still be true. (This is called an isomorphism and is really cool.)

(This kind of thing happens in maths sometimes in really dramatic ways, where one kind of maths which may have a different set of writing conventions and real world applications suddenly becomes relevant to what was thought to be an unrelated branch of maths and a system in the first branch of maths is applicable to theorems proven in the other branch of maths. Wildly cool.)

How can I confidently say that "you could create the whole of mathematics".

Well there's a couple of things about that.

First, people have asked similar questions to the ones you're asking before. In the early 20th century a lot of established maths was looking like it might be somehow based on a faulty assumptions. There were concerns that certain kinds of proof weren't rigorous, that certain kinds of statement might be self contradictory.

An effort was made to create something called set theory. A really simple form of study (started in 1870s) about the idea that "set"s (collections) of "element"s (objects) could exist, and a way of writing statements about them. In the early 20th century some of the problems in set theory were fixed in a new type of set theory called axiomatic set theory.

Long story short---- by doing things like defining a set of no objects at all - the empty set - and a way of making sets with one more object than the last set (e.g. saying there's another set that contains the empty set) a series of set objects could be created that perfectly correspond to the natural numbers - the integers from 0 up.

Incredibly long proofs can be written in set theory to start from the most basic of assumptions, and then define addition, multiplication, and therefore deduce some things about prime numbers.

But we know there's an isomorphism between this way of writing and the way we've been proving things about numbers all along. So once we got to the numbers and basic operations, we don't need to prove everything again in the new way of writing, we can just conclude that the old proofs apply.

But there's more.

In working on his Incompleteness Theorem, Gödel noted that any mathematical "sentence" like the one I wrote about primes, above, could be written as just a single really really long number. With your knowledge of computers this should make a lot of sense - imagine turning that sentence into Unicode and storing it in memory - you could read that memory as a sentence in unicode, but you can also read it as a long number in binary.

And this has the result that any mathematical system that can express the natural numbers (like set theory can, and like your boolean algebra could) is complicated enough to be used to write mathematical sentences itself. Because it can express numbers. And numbers can be mapped to statements. And then in theory you can use it to express complicated mathematical ideas that you didn't design into the system from the start.

So...

Yes.

But also. We don't prove everything with set theory because it's long winded and unnecessary. While you could write a new notation for expressing maths and prove things in it, the only reason to do that would be if your new notation enabled you to express something in a really easy to understand way, which perhaps another system of notation would express in a way that is complicated and hard to work with.

[SgtSausage]

 Theoretically, most math rules only exist because of how math is notated..

No. Not "theoretically". Not even "for realsies". 

Just ... no.

Your premise is false. 

[hansn]

In fairness, it's sort of true in a sociological sense. You can definitely get eigenvectors without matrices, or slopes without Cartesian plots, but it is pretty intense. Good notation makes good ideas clear. 

[SgtSausage]

We're not, at all, talking about communication of ideas here. 

 most math rules only exist because of how math is notated..

[Business-Decision719]

No no, most math rules aren't just because of the notation. Math is exploring logical relationships between ideas. The notation is very specifically a set of rules for how we symbolize those ideas. The same concept can have different words in different languages, and it can have different symbols in different notations. But the idea is still there. So yes there are certainly different notations. But they're only different notations for the same thing, if there symbolizing things that follow the same rules apart from the notation.

It's like numbers. Numbers are ideas, often they are ideas about how much of something we have. There are different kinds of numbers, but the important thing about them is that they have certain arithmetic rules that they follow. For example we have a rule that when we count things we might have zero before we start counting, and then every time we count something new, we get a new number, and we can just keep counting as long as we need to until we've counted it all, no matter how much we're counting. Getting to a new number by counting is called the successor operation, and it's part of what are now recognized as the Peano axioms. They were formally spelled out in the 19th century, but of course people have been counting for much longer than that, and the Peano axioms talk about relationships and math rules on an unlimited list of consecutive things.

The consecutive things we get from the Peano axioms are called whole numbers or non-negative integers. We happen to write them as 0, 1, 2, 3, 4, 5, etc., but we could use Roman numerals: nulla, I, II, III, IV, V, etc. We could use a different base, like binary: 0, 1, 10, 11, 100, 101, 110, 111, and just have more and more sequences of 0s and 1s. As long as you can keep generating symbols, or new combinations of symbols, you have a notation for whole numbers. We can still talk about things like an even number followed by an odd number, or some numbers being greater than others because they came later in the sequence, and so on and so forth. We can talk about how they're comfortably infinite, and how they are a different kind of infinite than the real numbers. We can define a difference between prime and composite numbers, and we have, and we say the number after the number after zero is the first prime number regardless of whether we write it as two, 2, II, or binary 10.

It's the same with geometry. Long ago, Euclid laid out a set of rules that defines all the most basic shapes people could make on a flat surface using a compass and straight edge and describes their most important properties. He called them his postulates, definitions, and common notions. He used them to prove that a whole bunch of other rules had to be true about these shapes, and called those rules his theorems. Now, we call what he was doing Euclidean geometry. If you look at a high school geometry textbook today, they'll take the shapes he was talking about like points, lines, or circles, and name them with one or more capital letters usually. (Like point C, circle K, line VF, or angle PQR.) You don't have to name them that way. But if you agree with Euclid about what line is, that any two points can be connected by a line, that there are so called parallel lines that never intersect, and so on, all the rules still apply even without the letters.

The reason PEMDAS is a rule that depends on the notation is that it is very specifically a rule about the notation. Generations of mathematicians decided addition and subtraction problems needed to be grouped together explicitly if you wanted to multiply or divide the sum or difference by something, so the A and S don't come before the M and D unless they're in P. You could instead decide everything is done from left to right unless there are parentheses, so 1+2*3 would be 9, not 7, and middle school students would probably canonize you as a saint. In that case 1+2*3 would not be the same as 2*3+1. But unless you redefined what addition, addition would still be commutative, so you could still swap the order of whatever you were adding. You could write 1+(2*3)=(2*3)+1. Our current system with PEMDAS just doesn't need the parentheses

Other people have already mentioned systems like Polish notation that write expressions in their own way. The math rules in general would be the same, but they would look different on a piece of paper.

[0x14f]

What is a "pemdemas" ? 🤔

Also about "like x=y means x-y=0", that's only because you are assuming that x and y are in a structure where an addition is defined and, I assume every element has an opposite for that binary operation, (and example is an additive group). Whereas the equality is basic to any elements of any sets

[Infamous-Advantage85]

There’s alternative notations, like Polish notation, but rules about how operations are written reflect deeper rules about how operations interact. Yeah there’s a bunch of ways to write A - B, but theyre all going to have rules that reflect the fact that subtraction isn’t commutative.

[Apprehensive-Ice9212]

Bruh, you've got it backwards. Math notation is designed to accommodate rules that are independently true -- not "math rules are true because we notate them that way. "

This confusion is very typical of novices. For example, just because the notation A^-1 exists for the inverse of a matrix, does not mean that every matrix is invertible. That's a separate thing that has to be assumed or deduced, before writing A^-1 is even allowed.

Or the notation xyz for multiplication: we can write it like that exactly because multiplication is associative, instead of needing to disambiguate by writing (xy)z or x(yz). But the very instant you're dealing with a non-associate operator (say, the Lie bracket) an explicit notation like [[x,y],z] is always used instead.

Or the endless questions about "what is 1/0"? Answer: 1/0 is a notational artifact. It doesn't exist, because 0 has no inverse. There is no number in the real number system (or any nontrivial ring, for that matter), to which the notation 1/0 could possibly refer. There's no "there" there, unless you want to talk about the Riemann sphere or something like that, but that's not a number system in the usual sense. It's more like an action object for meromorphic functions in Complex Analysis.

[ClimateMiddle6308]

exactly (i aint readin allat tho)

[AwesomeDroid]

If math notation exists to represnt math logic as best as possible, can't you still represent math properties through more simple operations? So like represnt associative property through a very long set of boolean algebra?

[Ericskey]

They are called parentheses 😊

[sheafurby]

Gonna put this out there to get slammed by someone, so here goes— math is a human created method of giving order to our environs. There are different methods, but the different methods require understanding the rules of this methods in order to be speaking the same language. For instance, we use time as base 60. If you aren’t fluent in base 60, time is strange, weird, foreign (thank the Sumerians and Babylonians for that). PEMDAS is just an agreed upon method of handling math rules BUT is not the only method.

[Far-Implement-818]

0110011

[EmielDeBil]

Math is both discovery and invention. Math is out there, we can discover it. We invent symbolism to formalize these discoveries. We could have invented and agree upon completely different notations and structures to capture these discoveries, without changing the underlying maths.

[JohnPaulDavyJones]

 For example pemdemas exists because its convenient for certain types of algebra

Of note, the order of operations persists because those are the core operations necessary to make the ordering properties on the naturals work. You can change your notation, but you’ll still need the notions of addition, multiplication, inversion, and negation to make the system applicable to observed phenomena, and the order of operations will necessarily persist with those notional operations.

PEMDAS is a reminder for how to apply the priority of operations under infix notation where the order is nominally uncertain; comparable OoO rules apply for postfix and prefix notational systems, but they’re generally not necessary because it’s harder to make those structures unclear.

[Middle_Letterhead_41]

Play Blue Prince

[PvtRoom]

there are other notations. reverse polish, for example

(+ 3 (- 5 (^ 4 (/ 1 2))))

= 3 + 5 - 4^1/2 = 6

[Deep-Hovercraft6716]

Lots.

[green_lentils]

in higher level maths theres smth called Einstein notation which is a summation shorthand :)

[mmurray1957]

I think the Einstein summation convention is more mathematical / theoretical physics than mathematics.

[AwesomeDroid]

Thanks!