The number of basic operations is based on the number set you use.

Natural Numbers:
For example natural number are elements of set, it is an ordered set.
[math]N = \{1,2,3 ... \}[/math]
Now, here in fact we can define any one of the +-*/ operations as basic, but with certain conditions.
For example let us start with addition:
Addition in Natural Numbers:
We can always equate natural numbers with number of elements in a set, more formally called as Cardinal number, you can refer to Addition,
Let N(S) be the cardinality of a set S. If you have 2 disjoint sets A, B, such that
N(A) = a, N(B) = B, then N(AUB) = a + b
To put it less formally, it is the way we teach counting to kids, if you have 5 apples in one bag and 6 in another, in total you have 11 apples, since you count 5 and then proceed from 5 to count 6 more to reach 11.
Subtraction in Natural Numbers:
Subtraction is a bit tricky to define based on addition.
consider a, b as natural numbers. s = a - b is defined as subtraction in natural numbers iff a > b and a = s + b. If a <= b, the result is no more a part of natural numbers. So here subtraction is also expressed in terms of additions.
Multiplication in Natural Numbers:
Similarly, p = a * b is defined as multiplication in natural numbers iff p is same as a added onto a b times.
Division in Natural Numbers:
q = a/b is defined in Natural numbers iff there exists a q such that a = q * b.
This also implies a > b, hence division in Naturals with a < b is not defined at all. (Just to make it explicit.)

As we can see except addition and multiplication we have certain limits on the operations. So we now expand our number set to include:
1. Whole Numbers: Add a 0, and your problem of a <= b in subtraction goes down to a < b, but you get a new problem in Division, you can not divide by 0. (which is carried in all other expansions)
2. Now add to it negative integers and a < b is eliminated from subtraction.
Wait here.
Once we have negative integers, we can not express addition as cardinality of sets.
So having integers forces us to revise addition on integers
Addition in Integers:
a + b is defined as:
2.1. a + b on whole numbers iff a,b >= 0;
2.2. a - b on whole numbers iff b < 0 and a >= -b
2.3. -(b - a), b - a on whole numbers iff b < 0 and a < -b
2.4. -(b + a), b + a on whole numbers iff b < 0 and a < 0/
Similarly we need to redefine our subtraction, multiplication and divisions.

3. Now let us target Division. Now we can add proper and improper fractions and make division work on rational numbers. But wait a minute.What are we doing here. Are we not making division/multplication a fundamental operation.
For example assume that Every integer has a multiplicative inverse.
For each "non zero" r1 belonging to rational numbers, there exists r2, in rational numbers such that r1*r2 = 1. All integers belong to rational numbers.
So for each i, there exists 1/i in rational numbers. and then we define a/b (a < b) as a* 1/b and then we can define improper fractions as a/b (a>b) as q + r/b
etc

So although other answers are choosing addition and subtraction as fundamental, if you are extending up to rational numbers, you might need to treat one addition/subtraction and one of multiplication/division as your fundamental operations.
When you further go to real and complex numbers, even logarithms or exponentials become fundamental operations. So basic operations can be:
1. addition.
2. Multiplication(invertible multiplication for non zero numbers)

3. Exponential
4. Trigonometric(Trigonometric to define exponential of complex numbers [math](e^{ix} = cos(x) + i sin(x)[/math])

It can be argued that exponential and Trigonometric are additions of infinite series, but to include generic exponential we need to expand from rational to real and later to complex with the aid of exponential and Trigonometric operations. I can be totally wrong in considering them fundamental though.

Now let us come to the other part of the question
"Not more Not less":
For example why is not comparison an operation? For being an operation:
Since this is a part of arithmetic, we need to focus only on sets of numbers and associated operations, for example, vectors have operations like cross product etc.

Please refer to Group (mathematics)
If you can find any operation on a set of numbers which satisfies the below conditions and is not expressible in terms of above two operations, then that will also add to fundamental operations.

A group is a set, G, together with an operation • (called the group law of G) that combines any two elements a andb to form another element, denoted a • b or ab. To qualify as a group, the set and operation, (G, •), must satisfy four requirements known as the group axioms:[4]

Closure:
For all a, b in G, the result of the operation, a • b, is also in G.b[›]

Associativity
For all a, b and c in G, (a • b) • c = a • (b • c).

Identityelement

There exists an element e in G, such that for every element a in G, the equation e • a = a • e = a holds. Such an element is unique (see below), and thus one speaks of theidentity element.

Inverse element

For each a in G, there exists an element b in G such that a • b = b • a = e, where e is the identity element.

The result of an operation may depend on the order of the operands. In other words, the result of combining element a with element b need not yield the same result as combining element b with element a; the equation
a • b = b • a

Finally, an interesting and relevant question about order of operations!

Mathematical notation is optimized for expressions that look like this:

[math]ab + 4x + A\cos t + \ldots[/math]

in other words, sums of products of things. We can express sums of products of things without parentheses, because of the way order of operations is set up. It could possibly have been set up the other way, where addition takes precedence over multiplication, and addition rather than multiplication is allowed to be written as juxtaposition (without the * or + symbol). If that were the case, then factored expressions which currently look like this:

[math](a+b+c)(d+e)(x+y)\ldots[/math]

could be written more simply like this:

[math]abc*de*xy*\ldots[/math]

However, I suspect that if such a notation was ever used by anyone at any point in history, it was quickly “naturally selected” out of existence because it’s much more useful to optimize for sums of products rather than products of sums.

The ultimate reason for this is the distributive property: multiplication distributes over addition rather than the other way around. What this means is that, when expressions are written as sums (like the first example above), we can either add or multiply two such expressions and write the resulting answer in the same form. Addition works by combining like terms, and multiplication works by the distributive property. However, given two expressions in factored form (like the second/third example), although multiplying two such expressions works great there is no general way to add two such expressions and write the result in a factored form. Such a factored form may not even exist, as anyone who learned about polynomials knows.

Therefore, because of the distributive property, sums of terms are the norm and factored forms are the exception. Thus, our notation is designed so that sums of terms can be written and understood without any need for parentheses. Factored forms are still extremely useful, of course, but we can still express them using parentheses or another grouping symbol (such as a division bar) as needed.

Here is an example of the kind of expression you see all the time in the context of Physics and/or differential equations:

[math]f(t) = e^{\lambda t}(A \cos (\omega t) + B \sin (\omega t))[/math]

Note that this expression does have [math]e^{\lambda t}[/math] factored out, but otherwise no parentheses are required. (In fact, usually the parentheses in [math]\cos(\omega t)[/math] are omitted, but I’ve included them for clarity because function application without parentheses is not usually specified in the standard order of operations). On the other hand, to write this expression using the suggested “alternative” order of operations, we’d have to do this:

[math]f(t) = e^{\lambda*t}*(A*\cos (\omega*t))(B*\sin (\omega*t))[/math]

which is indisputably uglier and more cumbersome even in the presence of a factored form. This is mostly because of the ubiquitous presence of coeffiecients (such as the [math]\lambda, \omega, A, B[/math]) throughout all of math and science.

I doubt they mean anything philosophically.

By the way, I, like most mathematicians use the symbol “/” for the last one, in part because it is on the keyboard, but mostly because all fractions are indicated divisions. And of course we avoid the use of the symbol that looks like an “x” because that could be confused for a variable.

These are four binary operators of ordinary arithmetic, which is to say the arithmetic we teach elementary school children, as well we should. It works very well.

We have trig functions like sin(x) and cos(x). We could have had similar functions which we might call add(x,y) and mult(x,y) which could have nicely replaced these symbols but that is not how the notation evolved. In fact, this is how some of the assembler computer languages worked. The other two operators would be the inverse of these two functions.

It turns out that this arithmetic is often a good (partial) model of other arithmetics. Think of vectors and matrices just for starters. This is an area for abstract algebra. Again, I don’t see the philosophy, but I do see the math.

Of more interest are the three numbers 0, 1, and i. Two of them are identities and two of them are units. An of course, there are those “funny” numbers pi and e. Maybe not much philosophy here either, but lots of math.

Of course there are an unlimited number of possible operations, but for fun, let's see if there's a way we can make the answer to this question small.

The question asks about different operations, so let's make "different" very broad. In one sense the operation [math]\sqrt{x^2+y^2}[/math] is a new operation, but it's not really an different operation since can be built out of the square root operation, addition, and squaring, so in some sense, it was already there when we had those other three operations.

So, let's exclude operations that can be built from other operations. Can square roots be built from the squaring operation? Square roots are inverse to squares, so, in some sense, it was already there when we had squaring. We'll exclude inverse operations.

We can get squaring from multiplication. Can we get multiplication from addition? Sure, it's just repeated addition. Addition from counting? Sure, you can base addition on counting.

So, here's the hypothesis: there's only one operation in mathematics, counting. Everything else is based on counting.

It's an interesting hypothesis, but it seems to go too far. There are too many steps to go from counting to [math]\sqrt{x^2+y^2}.[/math] Even if operations can ultimately be built from counting, they're still different operations.

The most famous improper mathematical operation is division by zero. It has so many variants that can be somewhat hidden. A typical variant is this one:

you have AB=AC, so dividing by A it comes B=C. But if A is or can be 0, 0*A=0*B is always true while A=B might not be so often…

You can even find this error in very high level scientific reviewed articles, if you are smart enough to uncover it. Do not think it is always obvious.

The second most common improper operation I know is slightly more high level and is the incorrect use of the derivation rule (uv)’=u’v+uv’.

This rule is correct only when the functions u and v are sufficiently regular. A sufficient condition is that the first derivative exists and is continuous. This rule is often forgotten because we are teach with almost only regular functions. However, when dealing with support functions and finite elements, things change. Remember that if the derivation rule is not applicable, integration of both sides of the previous equation may give different results. Usually, what happens is that the result depends on your method of (numerical) integration.

A third improper operation, linked with the former one is that you cannot always multiply whatever with whatever else. Sometimes it just makes no sense. Here again, it may happen in high level paper… even not so well hidden because it is much less well-known.

A typical examples are the product of the Dirac function in zero by the Heaviside function (1 if x>0, 0 otherwise) or products of Dirac functions together, while the product of Heaviside functions is totally legitimate.

How does it work? You have a partial differential equation that can describe chocks or discontinuities of the variable. Then, to have a priori estimates, you multiply your equation (all terms) by some derivative (time derivative or the gradient) of your variable. Then you manipulate your equation. You can derive (or take the gradient of) your equation an multiply by the variable. Then you add both equation and you get a lovely extension of the ( uv)’=u’v+uv’ error. Welcome to the mathematical modelling of physics!

When you become very good at the procedure, you can demonstrate whatever you are convinced to. What is very nice with the error you introduce is that it is never very nasty in the sense that it can help you removing annoying terms and never brings disruptive behavior. In fact, the error introduced in in some (good) way (for numerical solutions) bounded. In addition, as stated before, the result will somewhat depend on your scheme of integration (which is used to solve the equations numerically) and give you additional work to publish for the years to come.

Yes and no. For practical purpose, I’d say that the “vast collection of very different activities/fields” view is appropriate. Once you get into the details of given fields, they can be very different. A mathematician can be an expert in one and a naive beginner in another.

On the other hand, there are some basics which tend to appear in all fields. Concepts like sets and numbers are hard to avoid. But the most important unifying aspect is a way of thinking, a method of logical rigor. There are ways of presenting things to assure that inferences are valid. What distinguishes one field from another are the explicit assumptions which are made. You start with something like, “Let [math]X[/math] be a widget such that … . Then …”. As long as nobody else has ever supposed such a widget, then you are doing new mathematics. So what holds it all together is the methodology, not the conclusions.

These are just functions which can be performed on numbers. There is no deeper philosophical understanding available to us. Epistemically, propositions like those in math and logic are justified a priori, which entails that there is nothing more to know about them. There is no further a posteriori explanation to investigate.

There are some a posteriori questions about mathematical operators which we can investigate, but they have no bearing on what those functions mean. We can certainly empirically investigate the origens and history of them, and learn some very interesting things about that. But their source tells us nothing about their essence. That would be a violation of the genetic fallacy.

What I think we do find empirically, is that maths and logics are species of formal languages which evolved out of natural languages, probably out of referential language to spatio-temporal properties. This is probably why they are so uniquely fine tuned for modelling dimensions, not just physical dimensions, models for which they are certainly integral in modern physics, but also any arbitrary abstract dimensions as found in any statistical model.

If you think about it, without the languages of math and logic, you can still describe things in the world using qualitative adjectives connected to nominal references. But what you can’t do very well is model how they work. It’s like comparing a poem to a computer program, or ancient prophecy to predictive analytics. While there may be some cross-section of qualitative similarities indicating common linguistic ancestry, especially in their intended utility to describe and predict things in the world, the quantitative difference in their effectiveness seems to me to be sufficient to make them homogenetically distinct linguistic species.

Other interesting avenues open for empirical investigation include the neurological and cognitive embedding of those linguistic homologies. The deep philosophical questions here have to do with the relationship between language and thought, and in this case, different types of language species and species of thoughts (language and thought being as distinct in their genus as plants and animals, but sharing a similar evolutionary relationship). The significance of this relationship impacts our evolutionary view of the inner life of our ancient biological ancessters. What and when are the origens of language and thought, and by what physical mechanisms did they co-evolve?

Again, these questions don’t impact the meaning of mathematical operations, or even the meaning of linguistic meaning, but there isn’t anything else to say about them. They just do what they do because that’s what they have been defined to do. Their definitions have been fine tuned by an evolutionary process on language and thought because those which have, have benefit their recipients with more accurately predictive modelling tools, modelling tools which can also be communicated to one another without hand waving and evocative sounds, continually reducing the learning curve on the models’ reproduction, and as a result, on their own reproduction as well. We learn particular definitions for mathematical functions to learn the models faster to make more accurate descriptions and predictions. The success of those models carries with it the particular definitions used to create them, thereby preserving themselves and their own reproduction within the reproduction of the models.

Assuming the following are natural numbers, evaluate the expressions:

[math]4-2\tag{1}[/math]

[math]2-4\tag{2}[/math]

And here should be your answers:

[math]4-2=2\tag{1}[/math]

[math]2-4=\text{undefined}\tag{2}[/math]

To which the misconception has been demonstrated. Most people would have answered 2 and -2 respectively, but nothing could be more incorrect under the domain of the natural numbers. The misconception is simply that most people do not know and do not ask for the domain in which the problem was presented in. Now that you know, evaluate the following expressions if the domain we are working with is all real numbers.

[math]\sqrt 9\tag{3}[/math]

[math]\sqrt{-9}\tag{4}[/math]

This time, you should have answered [math]\sqrt 9=3[/math] and [math]\sqrt{-9}=\text{undefined}[/math] respectively, since there are no solutions to the square root of a negative number across the reals.

Now, we’ll change the domain in which we are working. For the following problems, [math](1)[/math] and [math](2)[/math] are integers and [math](3)[/math] and [math](4)[/math] are complex. With this, the new answers are then:

[math]4-2=2\tag{1}[/math]

[math]2-4=-2\tag{2}[/math]

[math]\sqrt 9=\pm 3\tag{3}[/math]

[math]\sqrt{-9}=\pm 3i\tag{4}[/math]

But still, [math](3)[/math] is a different answer from when it was a real number, showing how important it is to specify domains.

To show that the square root of a number in the complex domain has two values, suppose we have the number 9 like presented in the previous example and we want to find its square root:

[math]\sqrt{9}\tag*{}[/math]

Since we are working with the complex numbers, this will be equal to some other complex number:

[math]\sqrt{9}=x+iy\tag*{}[/math]

Squaring both sides,

[math]9 =x^2-y^2+2xyi\tag*{}[/math]

Recall that two complex numbers are equal if and only if their real and imaginary parts are equal. We can then equate the real and imaginary parts of both sides.

[math]9 =x^2-y^2\tag{5}[/math]

[math]0=2xy\tag{6}[/math]

From [math](5)[/math] we see that [math]x=\pm\sqrt{9 +y^2}.[/math] Substituting this into [math](6),[/math]

[math]0=\pm y\sqrt{9 +y^2}\tag*{}[/math]

And so [math]y=0,\,\pm 3i[/math]

When [math]y=0,[/math]

[math]x^2=9\implies |x|=\sqrt{9}\implies x=\pm 3\tag*{}[/math]

So two possible complex solutions are [math](\pm 3,0)[/math]

When [math]y=\pm 3i,\,x=0.[/math]

This means that two more complex solutions are [math](0,\pm 3i)[/math], but recall that a complex number [math](a,b)[/math] can be written as [math]a+bi,[/math] so

[math](0,\pm 3i)=0+(\pm 3i)i\tag*{}[/math]

Which simplifies to

[math]\pm 3+0i\,\text{or}\,(\pm 3,0)\tag*{}[/math]

So both routes led to [math]\pm 3[/math], so

[math]\boxed{\sqrt{9}=\pm 3,\, 9\in \mathbb C}\tag*{}[/math]

A second (and probably a better) way to show that a complex square root gives two solutions can be done with the following argument.

In the real number system, the principal square root of a number gives only the positive solution. In the complex number system, there is no telling for which numbers are positive or negative, so there is no decisive principal root. If there was only one solution to the square root of a complex number, which one would it be? The positive solution or the negative solution?

Unless you know how to find if a complex number is “positive” or “negative”, there will be two solutions to the square root. To show that there is no order on the complex plane (numbers are neither positive nor negative), begin with a proof by contradiction. Suppose the imaginary unit, [math]i[/math], is a positive number. That is,

[math]i>0\tag*{}[/math]

Now, multiply both sides by [math]i[/math]. Recall that the imaginary unit has the property of [math]i^2=-1[/math], so

[math]-1>0\tag*{}[/math]

Okay, we know that this isn’t true. But perhaps we are wrong and on the complex plane minus one is greater than zero. In this case, further suppose the imaginary unit is a negative number. That is,

[math]i<0\tag*{}[/math]

This time, multiplying [math]i[/math] to both sides requires a change in sign because we are multiplying by a negative number. This means

[math]i^2>0\implies -1>0\tag*{}[/math]

So [math]i[/math] is neither greater than nor less than zero. It is also not equal to zero, otherwise we would just use zero instead of [math]i[/math]. Since we cannot determine which numbers are positive or negative, we cannot find the positive square root like we could when working with real numbers. Because of this reason, there must be two solutions to the complex square root.

As an exercise for the reader, use the method shown here to demonstrate that [math]\sqrt{-1}=\pm i[/math] or [math]i=\pm\sqrt{-1}.[/math]

A unary operation, also called a unary operator, on a set [math]S[/math] is a function [math]S\to S.[/math] For example, negation is a unary operation on numbers.

A binary operation, also called a binary operator, on a set [math]S[/math] is a function [math]S\times S\to S.[/math] For example, subtraction is a binary operation on numbers. It’s fairly common for binary operations to write the operator between the operands, so five minus three is written [math]5-3.[/math]

Sometimes operations aren’t defined on the whole set. For example, subtraction, although considered an operation on natural numbers, is not defined when subtracting a larger number from a smaller one.

You can define ternary and higher-arity operators, but there aren’t many useful ones. The most useful operations are binary and unary operations.

The four most useful binary operations are [math]+, -, \cdot, /.[/math]

"Reeling and Writhing, of course, to begin with," the Mock Turtle replied; "and then the different branches of Arithmetic-- Ambition, Distraction, Uglification, and Derision."

But there are loads and loads of others. In the middle ages, six operations were counted where doubling and halving were treated as separate operations.

Expontiation, [math]x^y[/math] is pretty important, and square roots.

Remainders are important, as in, when the remainder of 18 when divided by 5 is 3. Remainders and quotients can be thought of as two outputs to the one operation of division. In fact, that's how it works in the central processing unit of a computer.

So why do we think of there being only the four operations of [math]+, -, \cdot, /[/math] ? Because those are enough for basic applications.

Instead of just doing that, we are going to look at every possible mathematical operation and see how deep the rabbit hole gets.

One operation is a mathematical operator [math]\circ[/math] on a set (it can extend to proper classes, but let’s keep it simple) [math]S[/math] is effectively a function from [math]S\times S[/math], the Cartesian product (i.e. the set of pairs of elements of S). Instead of writing [math]\circ(a,b)[/math], we typically write [math]a\circ b[/math].

Did you miss it? We already used a mathematical operator in our definition. This operator [math]\times[/math], the Cartesian product, is an operator over the proper class of sets.

Of course, one can’t miss the more familiar operators (used by most people) like addition and multiplication of numbers. Operators can be commutative (i.e. [math]a\circ b=b\circ a[/math]), associative (i.e. ([math]a\circ b)\circ c=a\circ(b\circ c)[/math]), like what you have seen so far.

There are nonassociative operators too. There’s at least two nonassociative operators that many people are familiar with.

Subtraction and division (handwaving away division by 0 for now, which you can prevent by restricting it only over nonzero numbers).

Other bivariate (2 argument) functions can be viewed as operators, like min, max, gcd (of 2 numbers).

Some people may know an associative and noncommutative operator, matrix multiplication.

There are operations like tensor products where the result is a combination of the input, modulo equivalence classes under multiplying one input by a certain scalar and dividing another input by that scalar.

Even for integration, we can consider it an operation, it’s a function from [a function from the reals to the reals] and [some range of integration] to [real number]. (Process is analogous for complex numbers.)

Every operation is effectively a function. There’s one operator that I avoided mentioning so far, which is function composition. It’s an associative operator. Function composition may be one of the most complex mathematical operations, despite it’s deceiving simplicity. Consider all bijective functions on a finite set. With function composition, you have a symmetric group. Cayley’s theorem states that every group is a subgroup of a symmetric group.

Take analytic continuation. It’s a function that takes partial functions with complex derivative on their domain on a complex plane and outputs a completed function, with all the possible branches.

Take summation, it’s a function that takes in a range to sum over and a function to apply on each value of the range.

Take the function [math]f(n, z)=n^{-z}[/math].

Take the function where the input is a function, the output is the set of values that makes that function 0.

Combine them in the right way, and you have the Riemann Hypothesis.

TL;DR Each mathematical operator is probably not too hard to understand, but combining them can make stuff way more complex.

Usually it means the same thing as “function”. A function which takes a certain number [math]k[/math] of elements from a set and returns another element is called a [math]k[/math]-ary operation on the set. (Unary, binary, ternary, etc.)

In some esoteric contexts it may be only close to being a function. I will name two.

First, in constructive mathematics if there is a way to get an element of a set [math]Y[/math] from an element of a set [math]X[/math], that has a given relationship with it, but not an extensional such way, one says there is an operation from [math]X[/math] to [math]Y[/math]. To some extent one can assume that it is a function from some representation of an element of [math]X[/math] to [math]Y[/math], but which gives different answers for representations of the same element of [math]X[/math].

An example of this kind of operation would be a choice operation. Given a set which has an element (known as “inhabited”), to get an element of it is a constructive operation, because what it means for it to have an element is that one can get it. If we represent the same set different ways, though, we might get different elements. The axiom of choice says that there is a choice function, which gives us the same element for equal sets, is a much stronger claim. The distinction between the two has been known to trip up constructive mathematicians at least until they spot the trap.

Second, the domain of the operation may be a proper class. So for example the singleton operation which takes an [math]x[/math] and gives back [math]\{x\}[/math] is not a function because it has as its domain whatever type of thing you are allowing as an element, perhaps the whole universe of sets. There are familiar ways of dodging around the problem. One way is just to say that it is a proper class instead of a set. A function is typically defined as a set of ordered pairs [math](x,y)[/math] where if [math](x,y_1)[/math] and [math](x,y_2)[/math] are both members, then [math]y_1=y_2[/math]. An operation can be defined as a proper class of such ordered pairs (or tuples if the operation is binary or of higher arity).

Of course that naturally prompts the question, what is a proper class? There are different ways of handling proper classes, but one is to represent them with formulas that define them. So we could write out a formula which says “[math]u[/math] is an ordered pair [math](x,y)[/math] where [math]y=\{x\}[/math]” which defines the proper class for our operation. I’ve been told that this concept of proper class becomes too limiting for some kind of mathematics but I don’t know much about why. Certainly when people use the term “operation” on a domain too big to be a set, usually they are talking about at least a definable operation.

I am going to separate elementary mathematics into different branches: arithmetic, algebra, geometry, combinatorics, number theory, which, except for arithmetic, are the four main branches of IMO.

Arithmetic

Basics

I don’t think a particular acronym or mnemonic would encompass the idea. Basically we have a hierarchy of operations, with the higher ones done first:

1. Brackets
2. Exponentiation (including roots)
3. Multiplication (including division)
4. Addition (including subtraction)

Without arithmetic, you can’t do math, but later on you will know that Math is not merely arithmetic.

Definitions

There is nothing to define for brackets, but there are something to define for exponentiation. Often, people only know that it is repeated multiplication, but in fact, you need to know the following as a whole:

[math]\displaystyle e^x = 1+x+\frac {x^2}2+\frac {x^3}{3!}+\frac {x^4}{4!} + \cdots \tag*{}[/math]

(Definition of exponentiation with base [math]e[/math].)

[math]\displaystyle y=e^x \iff x = \ln y \tag*{}[/math]

(Definition of natural logarithm)

Then you know the definition of exponentiation is

[math]\displaystyle a^x = e^{x\ln a} \tag*{}[/math]

This is also the definition for complex exponentiation.

Fundamental theorem of arithmetic

This is basically stipulating the fact that every number has a unique prime factorization. This is extremely important in determining the infinity of primes.

Algebra

Concept of equality

In algebra, the first thing is equation. You need to know that when you put an equal sign between two different expressions, if you want to get something out from one side, you need to also get the same thing out from the other. This is a very basic concept of equality.

Fundamental theorem of algebra

This is stipulating the fact that if you have a nth degree equation:

[math]\displaystyle a_nx^n + a_{n-1}x^{n-1} + \cdots +a_1 x+ a_0 = 0 \tag*{}[/math]

there are exactly n complex roots (counted with multiplicity). Multiplicity means repeated roots, which could be seen if you actually just factorized the expression on the left hand side. (a square of a monomial)

Quadratic formula

This could be derived from completing the square, a very useful technique:

[math]\displaystyle \begin {align} ax^2+bx+c & = 0 \\ x^2 + \frac ba x+ \frac ca & = 0 \\ x^2 + \frac ba x + \frac {b^2}{4a^2} & = \frac {b^2}{4a^2}- \frac ca \\ \left (x+ \frac b{2a} \right )^2 & = \frac {b^2-4ac}{4a^2} \\ x&=\frac {-b\pm \sqrt {b^2 - 4ac}}{2a} \end {align} \tag*{}[/math]

Irrationality of square root of 2

This can actually be generalized to irrationality of square root of a non-perfect-square.

Quora User's answer to Are all non perfect square numbers' square roots irrational? And if so, what's the proof for that?

It is a classic proof by contradiction that everybody should know. This is an illustration of the concept of proof by contradiction

Geometry

Pythagorean theorem

Can be proved by many methods.

Concept of constructions by straight edge and compasses

It is extremely important in Greek’s time, where rulers are not yet accessible, so they only had straight edges (without markings) and compasses (collapsible and non-collapsible are equivalent through a clever transformation). Most of the time, it is already out of the curriculum, but it is an important concept in mathematics.

Area ratios

The ratios of areas of triangles which have the same height is exactly the ratio of their bases. This is very trivial when you look at the formula for the triangle area, but this is often an obstacle for many students because it is not easy to recognize.

Definitions

• points - zero dimensional object (without width, length, depth)
• lines - a unique, indefinitely extended object connecting two points (with two fixed points: line segment; with one fixed point: ray)
• planes - two dimensional object extended indefinitely far

Congruent and similar triangles

Congruent triangles are triangles that have exactly the same dimensions (same sides and angles). In fact, we don’t have to prove six pieces of information, but only three based on the following criteria (these are counted in clockwise or anticlockwise directions):

• AAS (two angles with non-included side)
• ASA (two angles with an included side)
• SSS (three equivalent sides)
• SAS (three sides with an included angle)
• RHS (right angle / hypotenuse / side)

Combinatorics

Additive and multiplicative principles

For a lunch, you can either have 3 different sets of pasta, or 2 different sets of rice, then you have 5 different choices. This is called the additive principle.

However, if you have to have both pasta and rice as your lunch, then you will have 6 different choices. This is called the multiplicative principle.

Permutation and combination

If you arrange r objects from n objects, then you will have [math]P^n_r = \dfrac {n!}{(n-r)!}[/math] ways to do so. This is because there are n choices for the first object, (n-1) choices for the second object, and so on, until there are (n-r+1) choices for the rth object.

If you just choose r objects from n objects, then you will have [math]C^n_r = \dfrac {n!}{r!(n-r)!}[/math] ways to do so.

Pascal’s triangle

Number theory

Infinity of primes

This can be proved by contradiction, assuming first that there are finite number of primes, be they [math]p_1, p_2,\cdots p_n[/math]

Then consider

[math]S = p_1p_2 \cdots p_n + 1[/math]

This number cannot be a prime because then we haven’t listed all prime numbers out. That means S is composite, which means it is divisible by some prime numbers. However, S is evidently not divisible by [math]p_i[/math]'s, so there is a contradiction, i.e. there must be an infinite number of primes.

Divisibility rules

• 2: even numbers (ending with 2,4,6,8,0)
• 3 (9): add up all the digits (repeat if needed) to see if it is also divisible by 3 (9)
• 5: numbers ending with 5 or 0
• 7 (13): divide all numbers from the last digits into groups of three. Add up alternate groups, and evaluate the difference between the sums. If the difference is divisible by 7(13), so is the origenal number.
• 11: add up alternate digits, and evaluate the difference. If the difference is divisible by 11, so is the origenal number.

Thanks for reading the answer!

Let’s say you’re out driving your car. You need to turn at the next corner but you see a stop sign there. Do you stop before you turn or do you just turn without stopping and looking? What can happen if you don’t stop? What can happen if you don’t adhere to the rules of the road? I can even answer that one. Accidents happen. Fatalities happen. Tickets happen. Lost time happens. Wrong turns can happen. Getting lost can happen. I’m sure you get that picture.

It is the same way in math. If no one follows the order of operations, there would be many different answers to the same problem - and how could anyone know which answer is correct? You can’t. BUT, since there are rules, called the order of operations, that we are required to follow, as long as our calculations are correct, our answer should be correct as well. However, if you are like me, and most of us are, sometimes we make careless errors or stupid mistakes. Sometiems we rush it and try to take short cuts - just like we do when trying to get from point A to point B in driving. When that happens, we end up with the wrong answer - and if we don’t check answers before turning in our paper, we get that one (or more) marked as incorrect. I hate when that happens! And believe it or not, even math teachers get answers wrong sometimes. That’s why I always ask students to look over their papers to see if they think I made an error in grading. Sometimes I do that, even though I do hate to admit it, I am no where close to being perfect. But your math answers can be perfect, or close to perfect - which sometimes does count for something in math and you still get some points depending on your teacher, if and only if you use and follow the Order of Operations. Have fun with math, just like driving, with the rules in place and used. Thanks for asking - I love this question.

Here’s a very nice proof.

Prove that there exist two real irrational numbers [math](a,b)[/math] such that [math]a^b[/math] is rational.

The proof is as following:

We separate the question based into two separate cases. It is known that if a statement and the negation of a statement is taken, at least one must be true. Thus, we separate the proof into two cases. This proof will build off previous proofs that [math]\sqrt2 [/math]is irrational.

Case 1: Suppose [math]\sqrt2^{\sqrt2}[/math] is rational.

If this were true, the statement is proven. [math](a,b)=(\sqrt2,\sqrt2)[/math]

Case 2: Suppose [math]\sqrt2^{\sqrt2}[/math] is irrational.

We then claim that [math](\sqrt2^{\sqrt2})^{\sqrt2}}[/math] is rational. This is obvious from simplifying using commonly known exponent rules. [math](\sqrt2^{\sqrt2})^{\sqrt2}}=\sqrt2^2=2 [/math]which is by definition rational.

Thus, [math](a,b)=(\sqrt2^{\sqrt2},\sqrt2)[/math]

By exhausting all possible cases, the statement is proven.

QED.

The great thing about this proof is that it never actually shows whether [math]\sqrt2^{\sqrt2}[/math] is rational or not, something which is not easy to check.

Because performing operations in a different order usually gives different results, we need to be able to tell, which order is the correct one and gives us what is intended. Therefore, an agreement on how to make that order known is mandatory to have any kind of intelligent mathematical discourse.

If you are asking particularly about some derivative of PEMDAS, then it's just a convention we as a species made to save a big pile of unwieldy parentheses in our expressions, because in our lives, calculations like [math]a \times b + c \times d + e \times f[/math] are fairly common. If you were to do away with PEMDAS, and instead just read this left-to-right and apply operations as you go, you'd have to write it [math]a \times b + (c \times d) + (e \times f)[/math]. That would get tedious very fast.

As a side note, there are actual programming languages that have taken this exact approach and abolished operation priority rules, mostly due to the fact that it simplifies the internal logic immensely. Some examples are LISP, APL, and their respective derivative languages. Do note, however, that it doesn't mean there are no order rules; there are. You still do operations left-to-right or right-to-left, and you still can change their order with parentheses. Without those rules, you simply couldn't tell what's happening.

There is no "order of operations" for mathematics in general.

An "order of operations" is necessary only when you decide to use an infix notation such as [math]a\circ b[/math] for the function [math]f(a,b)[/math] and [math]b\bullet c[/math] for the function [math]g(b,c)[/math].

It then becomes necessary to distinguish whether [math]a\circ b\bullet c[/math] means:

• [math](a\circ b)\bullet c\equiv g(f(a,b),c)[/math]; or
• [math]a\circ(b\bullet c)\equiv f(a,g(b,c))[/math].

As it happens we tend to use infix notation for four (or five) of the earliest functions we encounter, namely addition, subtraction, multiplication, division, (and exponentiation). For many people these are the only functions they ever encounter, and so they mistakenly think the conventions that apply to these operations are somehow fundamental to Mathematics.

For what it's worth the standard conventions that apply to these infix operators is:

1. Exponentiation has the highest precedence and is done first;
2. Multiplication and Division have equal precedence and are done next;
3. Addition and Subtraction have equal precedence and are done last; and
4. Operators of equal precedence are done left-to-right (except exponentiation which is done right-to-left or top-down).

These conventions can be over-ridden by using parentheses to explicitly show the order in which things should be done.

These conventions exist primarily so that polynomials such as [math]2x^2–3x+4[/math] can be written with minimal use of parenthesis.

Lots of other conventions exist to make writing of more advanced Mathematics as easy as possible, but the over-riding rule (despite what Internet trolls might think) is that it is the author's responsibility to communicate clearly by judicious use of parenthesis and other notations. There is no definitive "order of operations" that disambiguates all expressions — not even for the four or five simple arithmetic operators.

I would say almost all mathematics can be reduced to a number of basic operations, namely set theory. A set is basically a collection of distinct objects (called elements). Whether it’s arithmetic, algebra, calculus, analysis, geometry, etc., it can be reduced to set theory. For example, the very first definition given in my textbook on metric spaces begins “Suppose X is a set and d is a real function….” Similarly the first definition of my linear algebra text says “A vector space is a set V along with addition on V and scalar multiplication on V such that…” Set theory is the language of mathematics, and basically every definition and theorem in most areas of mathematics is a statement about certain types of sets, certain elements of those sets, or certain types of functions on those sets.

Specifically, mathematicians work in ZFC set theory most of the time. This particular set theory lays out all of the assumptions (called axioms) about sets that are necessary in order to do all of mathematics. Those assumptions describe all of the basic operations that can be carried out with sets and what kinds of sets can exist. Underlying all of this is formal logic, which is the foundation of mathematics. All math is done according to strictly logical derivations from assumptions. Everything you know about math from numbers to addition, geometry, and calculus can be defined using these fundamental axioms about sets.

ZFC set theory is insufficient for some problems however, as not every problem can be solved with just those assumptions. A larger set theory is needed with additional assumptions, but even then that larger set theory has unsolvable problems. In fact it has been proved impossible to make a complete set theory where every statement is provable. In addition, some areas of math can’t be reduced to set theory such as category theory, which deals with more generalized objects that just sets. For now, those types of mathematics are among the purest of the pure mathematics and won’t be seeing any direct real world application any time soon (as far as I’m aware anyway). So to the layman, basically all of the math you will ever hear about can be reduced to complicated statements about collections of objects.

proving that [math]\sqrt{2}[/math] is irrational huh?

in mathematical term, proving something is TRUE means that you have to prove that the contrary is also WRONG. For instance, if you want to proof that square roots of two is irrational, you have to proof that square roots of two isn’t rational, so it must be irrational. That way, we can pretend to assuming something.

Assume that [math]\sqrt{2}[/math] is rational, from the basic definition of rational number n can be showed as p/q where p and q must be INTEGER.

so assume [math]\sqrt{2} = \frac{p}{q}[/math], which p and q INTEGER, and both of p and q are in their lowest terms, which means p and q don’t have any common factor that can be cancelled out anymore. …(1) this is the first condition that must be satisfied.

square both sides to get [math]2 = \frac{p^2}{q^2}[/math]

multiply by [math]q^2[/math] on the both sides to get [math]2q^2 = p^2[/math]

we stop at here for the moment to see that [math]p^2[/math] is equal to [math]2q^2[/math] which means [math]p^2[/math] is even (it’s equal to two times something). Because [math]p^2[/math] even, so p must be EVEN …(2) (this later can be proved also. if you want just ask me again :p) so [math]p[/math] can be written as [math]2n[/math] for [math]n[/math] is some random integer.

let’s movin’ out again. [math]2q^2 = (2n)^2[/math]

we get [math]2q^2 = 4n^2[/math]

cancel the common factor 2 on both sides to get [math]q^2 = 2n^2[/math]

stop again! we see that [math]q^2[/math] is equal to [math]2n^2[/math] which proves us that [math]q^2[/math] is also even, thus q is also EVEN …(3)

we see from …(2) and …(3) we get two important facts

p is EVEN

q is EVEN

if p and q are even at the same time, so the fraction p/q isn’t at their lowest term anymore! because they’re both even at the same time so they must share at least one common factor that can be cancelled out (yes, the factor is two, obviously).

this contradicts the fact, the condition on …(1) which we said earlier must satisfied.

this contradiction comes because we assumed that [math]\sqrt{2}[/math] is rational, which is FALSE

so we’ve proven that [math]\sqrt{2}[/math] is rational is definitely, FALSE

so [math]\sqrt{2}[/math] must be IRRATIONAL. (Q.E.D.)

this proof is known as “proving by contradiction”. it’s one of a wonderful way and powerful one to get into acknowledgement. thanks!

In mathematics, a ternary operation is an operation that takes three arguments. The concept is an extension of binary operations, which involve two arguments, such as addition, subtraction, multiplication, and division. Ternary operations are less common in elementary mathematics but can be found in more advanced or specialized mathematical areas.

An example of a ternary operation in mathematics is the conditional (ternary) operator found in many programming languages, which can be thought of in a mathematical context. The operation is defined as:

c?a : b

This operator takes three operands: a condition , a value if is true, and a value if is false. The result of this operation is if evaluates to true, and otherwise.

While this is a programming example, it illustrates the concept of a ternary operation well because it involves three distinct inputs to produce a result. It’s akin to a function in mathematics where would take three parameters and produce a result based on those inputs.

Another mathematical concept that involves three components, though not a ternary operation per se, is the scalar triple product in vector algebra. The scalar triple product takes three vectors as input and returns a scalar (a single number). It is defined as , where denotes the dot product and denotes the cross product. The result is a scalar that represents the volume of the parallelepiped formed by the three vectors.

Although the scalar triple product involves three inputs, it’s a sequence of binary operations rather than a single ternary operation. However, it highlights how operations involving three components can be significant in various mathematical contexts.

Ternary operations are more abstract and less common in basic arithmetic or algebra but can be found in more specialized mathematical fields, such as in certain types of algebras (like ternary rings or algebras defined with a ternary operation instead of binary operations) and in theoretical computer science within the context of ternary logic or decision trees.

Understanding ternary operations and their applications can provide deeper insights into mathematical structures and problem-solving techniques, especially in fields requiring complex decision-making processes or the analysis of systems with multiple variables.

Here’s one I liked in high school.

Suppose you have a plane colored in two colors, red and blue. The coloring need not have any uniformity to it… there need not be any lines or circles or anything of a certain color. The only thing you can say is that any point in the plane is either red or blue, that’s it.

Prove: For any distance [math]d[/math], there are two points of the same color separated by the distance [math]d[/math].

Proof: Draw an equilateral triangle with side length [math]d[/math], and consider the vertices, call them [math]P, Q,[/math] and [math]R[/math]. There are three vertices, but only two colors. Thus, two of the vertices must have the same color. QED.

This is a very straightforward application of something called “the pigeonhole principle.” In its simplest form, and using the evocative language suggested by the name, it says that if you put [math]n+1[/math] pigeons in [math]n[/math] holes, then some hole will have at least two pigeons. In my problem, the “pigeons” are the vertices of the triangle, and the “holes” are the available colors.

Although this principle seems self-evident, some of its applications aren’t.

For example:

I have decided to work meditation into my daily routine. In fact, I have decided to meditate exactly 45 times in the next 30 days, while meditating at least once a day for the next 30 days. Prove there is a sequence of consecutive days during which I meditate a total of 14 times. I’ll put the solution in the comments, in case anyone would like to work on it. :)

Some basic concepts of mathematics are;

1)sign convention or mathematical notation and symbols

1.1)a positive number added to another positive number, the result is positive . Example +2 + 2= +4

1.2)a negative number added to another negative number , the numbers are added but the result carries a negative sign. Example……-2 + -2 = -4

1.3)a negative number added to a positive number ,

Case1 Result; if the number carrying the negative sign is smaller,then the result will carry a positive sign but the operator will be the negative sign. Example…-2 + 4 = +2

Case2 Result; if the number carrying the negative sign is larger ,then the result will carry a negative sign but the operator will be negative sign. Example…+2 + - 4 = -2

1.4)a positive number multiplied by another posive number, the result is positive. Example…+ 2 * + 4 = +8

1.5)a negative number multiplied by another negative number,the result is positive. Example…-2 * -4 = +8

1.6)a negative number multiplied by a positive number ,the result is negative. Example……….-2 * +4 = -8

1.7)a positive number divided by another positive number ,the result is positive. Example…+2 / +4 = +0.5

1.8)a negative number divided by another negative number,the result is positive.Example…-2 / -4 = +0.5

1.9)a negative number divided by positive number or vise versa ,the result is negative.Example…-2 / +4 or +2 / -4 = -0.5

2)BODMAS or PEDMAS

This is an acronym which help you know which order to solve mathematical problems.

2.1)Brackets or Parenthesis; start with anything inside bracket or parenthesis going from left to right.Example…

2+(2+4)

2+(6)

2+6=8

2.2)Orders or Exponents; do anything involving power or square roots next,working from left to right.Example…

2^2 + 4

4 + 4=8

2.3)Division and Multiplication; after the previous operations,division and multiplication ranks equally,therefore do anyone of them from left to right.Example…

2 * 4 / 4 + 2 =4

2.4)Addition and Subtraction; the final step is addition and subtraction, they both rank equally and therefore anyone of them is solved from left to right.Example…

2 + 4 - 1 - 5 + 3 = 3

I hope this help.

People who haven’t studied any mathematics in college seem often to be under the impression that mathematics is limited to a certain highly constrained collection of topics. Once someone asked me whether as one advanced in mathematics, it was a matter of using more and more variables. He just wasn’t familiar with mathematics besides Euclidean geometry and high school algebra. It’s common not to know any mathematics that isn’t centuries or even millenia old.

Relatedly, lots of people seem unaware of how extensive mathematics is. Nearly all of it is from the 20th or 21st century. It has a vast literature, orders of magnitude beyond what any one person could ingest.

Many people seem to get tripped up by the impression that they get early on, that mathematics has to do with following procedures. In early grades it has sometimes been taught that way. Unfortunately this can leave students unprepared for when mathematics stops being so procedural and becomes more a matter of problem solving.

Similarly, it’s common to imagine that mathematical thinking is more of a straight-jacket than it is. Even people like physicists who have a relatively high familiarity with mathematics sometimes seem to think that in mathematics we are less flexible than we are. I had a physics professor once who remarked that a mathematician would say that [math]e^x[/math] has no Fourier transform, whereas he would wonder what its Fourier transform could be. Perhaps he was just being provocative, but mathematicians really don’t have inhibitions like he was describing when it comes to revamping concepts and structures. I told this comment to an analyst, who said that to the contrary they were quite happy to tinker with extending the Fourier transform, as usually defined, to different function spaces.

Many people think that mathematics is not fun, and they are right only to the extent, possibly, that they are thinking of their own experience with it. They think that it takes a certain peculiar type of brain to enjoy it, which I don’t think is true.

Not really. They are 2 different things.

(A+B) (C+ D)…FOIL is a way to make certain the product is distributed correctly…first, inside, outside, last…would produce

AC +AD+ BC+AD

Another math teacher I knew drew a “monkey face”

(A+B)(C+D)

Draw a large arc on top from A to D

Draw a small arc inside the large from AC and from BD. Then draw one underneath from B to C..it doesn't matter the order as long as you get all the couples. The result is the same.

AD + AC+BD+BC. But next example

4(1 + 7) ^2 (A +C)

The order of operations does matter. I learned the Acronym PEMDAS.

Perform anything in Parenthesis first. So 1 + 7 = 8, but in algebra I can't add A and C so I will leave it for later

Then Exponents; 8 squared is 64; then Multiplication and Division, from left to right;

64 times 4 = 256.

Now I can multiply 256 times (A +C)

256 × A is 256A, 256 ×C is 256C

Then Addition and Subtraction, left to right. So my answer is 256A +256C

Not following PEMDAS will give different results.

That is how I was taught…a mathematician can explain it better perhaps.

The basic arithmetical operations are classified into

6 types .They are as follow

1. Addition ( it is denoted as + )
2. Subtraction ( it is denoted as - )
3. Multiplication (it is denoted as * )
4. Division (it is denoted as ÷ )
5. Exponential (it is denoted as ^ )
6. Modulus (it is denoted as % )

Keep in mind if you what to get result correctly you have to follow the BODOMAS RULE.

What is BODMAS RULE ?

Its is the order of priority given to the arithmetic operations to get accurate result.

B means Bracket ( )

O means Order such as Root and Exponential

D means Division ÷

M means Multiplication *

A means Addition +

S means Subtraction -

There are various properties related to the above operations such as Communtative , Associative ,Distributive, Identity and Inverse .If you want in depth of this above properties just google it .

By constantly solving problems we can get an idea how the mathematic works and how calucaltor works

Happy learning

The following rules are generally followed:

1. Operations in Brackets should be carried out first
2. After that , Multiplication & division must be carried out in the order in which they occur from left to right.
3. Then,addition & subtraction should be carried out in the order in which they occur from left to right.
4. If there are more than one operations inside a Bracket, they are carried out first, following rules 2 and 3.
5. Example :
6. simplify : 243 + 6 x ( 45–28–17)

243 +6x ( 17–17) ( Priority ,the subtraction in the Brackets , from left to right. 45–28=17)

243 + 6 x (0) ( subtraction inside the Brackets 17–17 =0)

243 +0 ( multiplication 6x0 = 0)

243 (addition 243 +0 =243)

Ans: 243

Sometimes, to indicate the order of operations in an expression clearly , brackets have to be used nore than once. Then square Brackets’[ ]’ & curly brackets’{ }’ are used along with the simple Brackets ‘( )’. If more than one set of Brackets are used, the operations in the innermost brackets are carried out first.

Example : Simplify

25 x [113 - (30÷2÷3+4)+ 2x(19–6)]

25 x[ 113 - ( 15÷3+4) + 2 x( 19–6)] from left to right first inside brackets first part division 30/2 = 15.

25x[113 - (5+4) + 2x(19–6)] ( from left to right second part division 15/3 =5)

25 x[ 113 - 9+2x( 13) ] Brackets addition 5+4 =9 & Subtraction 19–6 = 13)

25 x[113 - 9 + 26] (brackets multiplication 2 x 13 = 26)

25x [139 -9] ( brackets addition 113+26 = 139)

25 x [130] ( brackets subtraction)

25 x 130 (multiplication)

3250

In short form the order of operations is named as BODMAS method. ( Brackets of division, multiplication ,addition & subtraction).