We will start the exposition like a philosophy paper and define precisely what we are going to be talking about. At the beginning of all matters we have to define the players in the world of math: numbers.

Numbers are the basic objects which you can type into a calculator and which you use to calculate things. Mathematicians like to classify the different kinds of number-like objects into sets:

• The Naturals: $\mathbb{N} = \{0,1,2,3,4,5,6,7, \ldots \}$,
• The Integers: $\mathbb{Z} = \{\ldots, -3,-2,-1,0,1,2,3 , \ldots \}$,
• The Rationals: $\mathbb{Q} = \{-1,0,0.125,1,1.5, \frac{5}{3}, \frac{22}{7}, \ldots \} $,
• The Reals: $\mathbb{R} = \{-1,0,1,e,\pi, -1.539..,\ 4.94.., \ \ldots \}$,
• The Complex numbers: $\mathbb{C} = \{ -1, 0, 1, i, 1+i, 2+3i, \ldots \}$.

These categories of numbers should be somewhat familiar to you. Think of them as neat classification labels for everything that you would normally call a number. Each item in the above list is a set. A set is a collection of items of the same kind. Each collection has a name and a precise definition. We don't need to go into the details of sets and set notation for our purposes, but you have to be aware of the different categories. Note also that each of the sets in the above list contains all the sets above it.

Why do you need so many different sets of numbers? The answer is partly historical and partly mathematical. Each of the set of numbers is associated with more and more advanced mathematical problems.

The simplest kind of numbers are the natural numbers $\mathbb{N}$, which are sufficient for all your math needs if all you are going to do is count things. How many goats? Five goats here and six goats there so the total is 11. The sum of any two natural numbers is also a natural number.

However, as soon as you start to use subtraction (the inverse operation of addition), you start to run into negative numbers, which are numbers outside of the set of natural numbers. If the only mathematical operations you will ever use are addition and subtraction then the set of integers $\mathbb{Z} = \{ \ldots, -2, -1, 0, 1, 2, \ldots \}$ would be sufficient. Think about it. Any integer plus or minus any other integer is still an integer.

You can do a lot of interesting math with integers. There is an entire field in math called number theory which deals with integers. However, if you restrict yourself to integers you would be limiting yourself somewhat. You can't use the notion of 2.5 goats for example. You would get totally confused by the menu at Rotisserie Romados which offers $\frac{1}{4}$ of a chicken.

If you want to use division in your mathematical calculations then you will need the rationals $\mathbb{Q}$. The rationals are the set of quotients of two integers: \[ \mathbb{Q} = \{ \text{ all } z \text{ such that } z=\frac{x}{y}, x \text{ is in } \mathbb{Z}, y \text{ is in } \mathbb{N}, y \neq 0 \}. \] You can add, subtract, multiply and divide rational numbers and the result will always be a rational number. However even rationals are not enough for all of math!

In geometry, we can obtain quantities like $\sqrt{2}$ (the diagonal of a square with side 1) and $\pi$ (the ratio between a circle's circumference and its diameter) which are irrational. There are no integers $x$ and $y$ such that $\sqrt{2}=\frac{x}{y}$, therefore, $\sqrt{2}$ is not part of $\mathbb{Q}$. We say that $\sqrt{2}$ is irrational. An irrational number has an infinitely long decimal expansion. For example, $\pi = 3.1415926535897931..$ where the dots indicate that the decimal expansion of $\pi$ continues all the way to infinity.

If you add the irrational numbers to the rationals you get all the useful numbers, which we call the set of real numbers $\mathbb{R}$. The set $\mathbb{R}$ contains the integers, the fractions $\mathbb{Q}$, as well as irrational numbers like $\sqrt{2}=1.4142135..$. You will see that using the reals you can compute pretty much anything you want. From here on in the text, if I say number I will mean an element of the set of real numbers $\mathbb{R}$.

The only thing you can't do with the reals is take the square root of a negative number—you need the complex numbers for that. We defer the discussion on $\mathbb{C}$ until Chapter 3.

You can add and subtract numbers. I will assume you are familiar with this kind of stuff. \[ 2+5=7,\ 45+56=101,\ 65-66=-1,\ 9999 + 1 = 10000,\ \ldots \]

The visual way to think of addition is the number line. Adding numbers is like adding sticks together: the resulting stick has length equal to the sum of the two constituent sticks.

Addition is commutative, which means that $a+b=b+a$. It is also associative, which means that if you have a long summation like $a+b+c$ you can compute it in any order $(a+b)+c$ or $a+(b+c)$ and you will get the same answer.

Subtraction is the inverse operation of addition.

You can also multiply numbers together. \[ ab = \underbrace{a+a+\cdots+a}_{b \ times}=\underbrace{b+b+\cdots+b}_{a \ times}. \] Note that multiplication can be defined in terms of repeated addition.

The visual way to think about multiplication is through the concept of area. The area of a rectangle of base $a$ and height $b$ is equal to $ab$. A rectangle which has height equal to its base is a square, so this why we call $aa=a^2$ “$a$ squared.”

Multiplication of numbers is also commutative $ab=ba$, and associative $abc=(ab)c=a(bc)$. In modern notation, no special symbol is used to denote multiplication; we simply put the two factors next to each other and say that the multiplication is implicit. Some other ways to denote multiplication are $a\cdot b$, $a\times b$ and, on computer systems, $a*b$.

Division is the inverse of multiplication. \[ a/b = \frac{a}{b} = \text{ one } b^{th} \text{ of } a. \] Whatever $a$ is, you need to divide it into $b$ equal pieces and take one such piece. Some texts denote division by $a\div b$.

Note that you cannot divide by $0$. Try it on your calculator or computer. It will say error divide by zero, because it simply doesn't make sense. What would it mean to divide something into zero equal pieces?

Very often you have to multiply things together many times. We call that exponentiation and denote that with a superscript: \[ a^b = \underbrace{aaa\cdots a}_{b\ times}. \]

We can also have negative exponents. The negative in the exponent does not mean “subtract”, but rather “divide by”: \[ a^{-b}=\frac{1}{a^b}=\frac{1}{\underbrace{aaa\cdots a}_{b\ times}}. \]

An exponent which is a fraction means that it is some sort of square-root-like operation: \[ a^{\frac{1}{2}} \equiv \sqrt{a} \equiv \sqrt[2]{a}, \qquad a^{\frac{1}{3}} \equiv \sqrt[3]{a}, \qquad a^{\frac{1}{4}} \equiv \sqrt[4]{a} = a^{\frac{1}{2}\frac{1}{2}}=\left(a^{\frac{1}{2}}\right)^{\frac{1}{2}} = \sqrt{\sqrt{a}}. \] Square root $\sqrt{x}$ is the inverse operation of $x^2$. Similarly, for any $n$ we define the function $\sqrt[n]{x}$ (the $n$th root of $x$) to be the inverse function of $x^n$.

It is worth clarifying what “taking the $n$th root” means and what this operation can be used for. The $n$th root of $a$ is a number which, when multiplied together $n$ times, will give $a$. So for example a cube root satisfies \[ \sqrt[3]{a} \sqrt[3]{a} \sqrt[3]{a} = \left( \sqrt[3]{a} \right)^3 = a = \sqrt[3]{a^3}. \] Do you see now why $\sqrt[3]{x}$ and $x^3$ are inverse operations?

The fractional exponent notation makes the meaning of roots much more explicit: \[ \sqrt[n]{a} \equiv a^{\frac{1}{n}}, \] which means that $n$th root is equal to one $n$th of a number with respect to multiplication. Thus, if we want the whole number, we have to multiply the number $a^{\frac{1}{n}}$ times itself $n$ times: \[ \underbrace{a^{\frac{1}{n}}a^{\frac{1}{n}}a^{\frac{1}{n}}a^{\frac{1}{n}} \cdots a^{\frac{1}{n}}a^{\frac{1}{n}}}_{n\ times} = \left(a^{\frac{1}{n}}\right)^n = a^{\frac{n}{n}} = a^1 = a. \] The $n$-fold product of $\frac{1}{n}$ fractional exponents of any number products the number with exponent one, therefore the inverse operation of $\sqrt[n]{x}$ is $x^n$.

The commutative law of multiplication $ab=ba$ implies that we can see any fraction $\frac{a}{b}$ in two different ways $\frac{a}{b}=a\frac{1}{b}=\frac{1}{b}a$. First we multiply by $a$ and then divide the result by $b$, or first we divide by $b$ and then we multiply the result by $a$. This means that when we have a fraction in the exponent, we can write the answer in two equivalent ways: \[ a^{\frac{2}{3} }=\sqrt[3]{a^2} = (\sqrt[3]{a})^2, \qquad a^{-\frac{1}{2}}=\frac{1}{a^{\frac{1}{2}}} = \frac{1}{\sqrt{a}}, \qquad a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}. \]

Make sure the above notation makes sense to you. As an exercises try to compute $5^{\frac{4}{3}}$ on your calculator, and check that you get around 8.54987973.. as an answer.

There is a standard convention for the order in which mathematical operations have to be performed. The three basic operations have the following precedence:

1. Exponents and roots.
2. Products and divisions.
3. Additions and subtractions.

This means that the expression $5\times3^2+13$ is interpreted as “first take the square of $3$, then multiply by $5$ and then add $13$.” If you want the operations to be carried out in a different order, say you wanted to multiply $5$ times $3$ first and then take the square you should use parentheses: $(5\times 3)^2 + 13$, which now shows that the square acts on $(5 \times 3)$ as a whole and not on $3$ alone.

We can define all kinds of operations on numbers. The above three are special since they have a very simple intuitive feel to them, but we can define arbitrary transformations on numbers. We call those functions. Before we learn about functions, let us talk about variables first.