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They are the numbers that we use to count positive quantities: \(1, 2, 3, \ldots\)
The "\(\ldots\)" mean that the list continues forever! Some people include \(0\) in the natural numbers, others don't, each party has good reasons for doing so. When you mention the natural numbers, make sure your reader knows your position on this matter. Notationally, you can explicitly show that you include \(0\) by writing \(\mathbb{N}_{\geq 0}\) or \(\mathbb{Z}_{\geq 0}\). Similarly, you can show that you do not include \(0\) by writing \(\mathbb{N}_{\geq1}\) or \(\mathbb{Z}_{\geq 1}\), or even \(\mathbb{N}_{>0}\) or \(\mathbb{Z}_{>0}\).

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Integer numbers are \(\ldots, -3,-2,-1,0,1,2,3,\ldots\).
Note that the list is now infinite in both directions! So \(-463534\) and \(0\) are both integers.

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Rational numbers are all the numbers that can be written as a ratio of two integers. But beware, zero can never be in the denominator. The rationals contain the integers, which means that every integer is also a rational number. In set notation we write \(\mathbb{Z} \subset \mathbb{Q}\) or equivalently \(\mathbb{Q} \supset \mathbb{Z}\).

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The real numbers are all the numbers that we encounter in our everyday life. They include the integers, the rationals, and what we call the irrational numbers.
Any real number has a decimal representation.
The number \(\pi\) is an example of irrational number. Did you know that \(\pi\) is defined as the ratio of the circumference of any circle to its diameter?
Anyway, we have, \(\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}\).

Even if in many cases we don't require you to specify the set of numbers in which you are working, make it a habit to at least be clear about it yourself. And at the beginning, it will never hurt to write a little \(\forall x \in \mathbb{R}\) next to your equation to mean that you consider this equation to be true "for all real numbers \(x\)".

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https://www.encyclopediaofmath.org/index.php/Main_Page
Yes, you can also find a lot of detailed information about mathematics on Wikipedia, but not everything is right there so be careful.

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A prime number is a positive integer greater than 1 which admits only \(1\) and itself as positive divisors.

The first 10 prime numbers are:
\(2,3,5,7,11,13,17,19,23,29,\ldots\)
Can you find a few more down the list???

Prime numbers are hugely important among the natural numbers because every natural number greater than 1 may be expressed as a unique product of prime numbers.

Hence, if I give you three prime numbers, say \(2,3,5\), and you multiply them together, you get a unique natural number, in this case \(30\); and conversely, if I give you a natural number greater than \(1\), say \(32\), you can find the unique list of prime numbers that divide it, in this case \(2,2,2,2,2\). Did you get it?

Try to find the prime factor decomposition of the following natural numbers: \(4,6,7,14,15,72\).

You will need this fact a great deal every time you will want to simplify fractions and equations.

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I use the words function and map interchangeably. A function always comes with a domain of definition (or just domain for short) and a codomain. For the purpose of this definition I will give names to these 2 sets. Let's call \(D\subset \mathbb{R}\) the domain (note that \(\subset \mathbb{R}\) means that my domain is a subset of \(\mathbb{R}\)) and \(C \subset\mathbb{R}\) the codomain. It will also prove useful to give a name to the function I am defining. I will call it \(f\).
The fact that \(f\) is a function from \(D\) to \(C\) means that to each element \(x\in D\) of the domain corresponds an element \(f(x)\in C\) in the codomain.
We write \(f:D \to C\).

It will be VERY INSTRUCTIVE to read as much as you can of the Wikipedia page on functions (you may stop at section 7 of this link).

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As you probably have noticed by now, we often use letters in mathematics to represent something else than letters. "WTH!?" might some of you be thinking. NOT TO WORRY, IT IS ALL FINE. Really, don't be afraid of it.

Why do we use letters? They allow us to reach generality!

A few examples:

If a property $$P$$ is true for every natural number, we write \(P(n), \forall n \in \mathbb{N}\) instead of writing "\(P(1)\) and \(P(2)\) and \(P(3)\) and so on for ever..."

To denote the set of zeros of the cosine function we write: \(\left\{\frac{\pi}{2}+k\pi \ :\ k\in \mathbb{Z} \right\}\). Do you understand everything in this expression? Try to write out the zeros of the sine function and the real numbers that are excluded from the domain of definition of the tangent function

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Don't take this post too lightly. If you deeply understand the logical meaning of an equation, your math skills will jump very high!

An equation is a logical statement (see the post Logic and Math a few blocks to the left) which involves an equality. As any logical statement, it might be TRUE or FALSE. For instance, \(-3=-5\) is a FALSE equation whereas \(5+5=\frac{20}{2}\) is a TRUE equation.

All these examples might seem obvious to you. Now, what can I say about the truth value of the following equation?
\(3+x=5\)
Well, without some specifics about \(x\) I cannot say anything! If I am told that \(x\in \mathbb{R}\), then I can specifically list the cases in which the equation is TRUE and the cases in which it is FALSE. Namely, for \(x=2\) the equation is TRUE and for every \(x \neq 2\) the equation is FALSE. This is called solving the equation and \(2\) is called a solution to the equation.
If instead of \(x \in \mathbb{R}\) the assumption becomes \(x < 1\) then I can say that the equation is FALSE for all \(x < 1\). In this case, the equation would have no solution.

A classical TRUE statement in number theory is that the equation \(x^2=2\) has no solution in \(\mathbb{Q}\). This is because \(\sqrt{2}\) is irrational. Another classical result is that the imaginary number \(i\) is defined to be a solution to the equation \(x^2=-1\). This paragraph is just here as a teaser. You don't need to understand the proofs of these statements for this course. But do you understand their meaning? It is important that you do.

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To denote that \(x\) is an element of the set \(A\) we write, \begin{equation} x\in A\end{equation}

Given a set \(A\) and a subset \(B \subset A\) of it, we can form the set of all the elements of \(A\) that are not in \(B\) and we write, \begin{equation}A\setminus B \end{equation}

If \(P\) is a property that can either hold TRUE or FALSE for any real number \(x\). Then you can form the set of all reals having this property, \begin{equation} \{x \in \mathbb{R} \ : \ P(x)\}\end{equation} As a more concrete example, consider the set of zeros of the sine function \(\{k\pi \ : \ k\in \mathbb{Z}\}\).

To state that a property \(P\) holds true for all the elements of a particular set \(A\), write, \begin{equation} \forall x \in A,\ P(x) \end{equation} As a concrete example, we may define \(A\) to be the set of all ratios of the form \(\frac{1}{n}\) for \(n \in \mathbb{N}\) - with symbols we write \(A:=\left\{\frac{1}{n}\ :\ n\in \mathbb{N}_{>0}\right\}\). Then to say that all the elements of \(A\) are between \(0\) strictly and \(1\), we write, \begin{equation} \forall x \in A, \ 0<x\leq 1\end{equation}

To affirm that there exists an element \(x\) in a particular set \(A\) for which the property \(P\) holds TRUE we write, \begin{equation}\exists x \in A \text{ such that }P(x)\end{equation} As a concrete example, we might want to say that the function \(f\) defined on a subset \(D\) of \(\mathbb{R}\) (write \(f:D \subset \mathbb{R} \to \mathbb{R}\)) has limit \(l\) as \(x \to x^*\in \mathbb{R}\). In this case we write, \( \forall \epsilon>0\text{ }\exists \delta >0 \text{ such that, } \forall x\in D\setminus \{x^*\},\) \begin{equation}|x-x^*|<\delta \Rightarrow |f(x)-l|< \epsilon\end{equation}

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In mathematics, everything can be expressed in a rigorous, unambiguous way. This is why we call it a formal discipline. In this respect, mathematical language is very different than everyday language. Each word has a precise and - as much as possible - unique meaning. Now, in their everyday life, mathematicians don't use the formal language always, we say that they use informal arguments. But what is very important is that a mathematician always knows how to rewrite his informal work into a formal format, when needed. And why would we need this????

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In mathematics, every statement is either TRUE or FALSE. We call this the truth value of the statement. Here are 3 true statements,

\(3\) is prime

\(0<5\)

Every point \((x,y)\in \mathbb{R}^2\) lying on the unit circle satisfies \(x^2+y^2=1\)

and here are 3 false statements,

\(-3\) is prime

\(4\) divides \(5\)

The number \(1\) divided by any integer yields a rational number

Mathematicians like to know whether a statement is true or false, given some assumptions. This is what proofs are for! A mathematical proof is a logical argument that establishes the truth of a statement, provided some assumptions.

For simple statements, a proof amounts to checking a few definitions. Here is an informal proof of the statement "\(3\) is prime": The number \(3\) is a positive integer and its only (positive) divisors are \(1\) and \(3\). Therefore by definition, \(3\) is prime. Now comes an informal disproof of the statement "\(-3\) is prime": The number \(-3\) is negative, therefore by definition it is not prime. This is how I knew that the statement was FALSE.

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Sets are the building blocks of mathematical arguments. Any mathematician uses them EVERY DAY! A set is a collection of elements. For our purposes, we will mainly deal with sets of numbers. Hence, most of the time, a set is a collection of numbers!

Here is one way of writing a 4-element set using mathematical symbols:
\(\{1,4,35900,-249.22\}\). The elements of this set are \(1,4,35900\) and \(-249.22\). It is customary to give a letter name to a set that we use in an argument. Let's call the previous set \(A\). When giving a name to a specific set, mathematicians use the \(:=\) symbol as follows:

\(A:=\{1,4,35900,-249.22\}\)

Now we can make TRUE or FALSE statements about our newly defined set, for illustration purposes. The symbol \(\in\) is used to say that an element belongs to a set. Hence, \(4 \in A\) is TRUE while \(-33 \in A\) is FALSE. We can turn the last statement into a TRUE statement by modifying it as follows: \(-33 \notin A\).

Another crucial relation between sets is the relation of inclusion.

The empty set is the only set which has no element at all. We write it \(\emptyset\).

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In the following, I use the symbol \(\equiv\) to mean "equivalently written".

\(AND \equiv \wedge\)
example: \(a\wedge b\) reads "a and b"

\(OR \equiv \vee\)
example: \(a\vee b\) reads "a or b"

\(NOT \equiv \neg\)
example: \(\neg a\) reads "not a" or "the negation of a"

\(\Rightarrow\) (implication sign)
(logicians use \(\rightarrow\), please avoid this notation in our labs as we do mathematics)
example: \(a\Rightarrow b\) reads "a implies b" or "if a then b" or "a is sufficient for b" or "b is necessary for a"

\(\Leftarrow\)  (implication sign)
(logicians use \(\leftarrow\), please avoid it as well)
example: \(a\Leftarrow b\) is the same as \(b \Rightarrow a\). The statement \(a \Leftarrow b\) is called the converse of the statement \(a \Rightarrow b\). This is why you will see many times the word "conversely" in proofs of equivalence claims.

\(\Leftrightarrow\) (equivalence sign)
(logicians use \(\leftrightarrow\), please avoid it as well)
example: \(a\Leftrightarrow b\) read "a if and only if  b" or "a is equivalent to b" or "a is necessary and sufficient for b" or the same sentence with a and b swaped.