Algebra

The real numbers

The set of all real numbers, "R," includes the sets of natural numbers, integers, rational, and irrational numbers.

Numbers we can actually count in nature (positive integers)
1, 2, 3, 4, ...
We can classify natural numbers as even (2, 4, 6, ...) and odd (1, 3, 5, ...). (What about zero? Zero is an even integer, but it is neither positive nor negative)
The symbol for the natural numbers is simply "N." When we mean to include the zero we write "N₀" while when we want to clearly exclude the zero we write "N₊".

The set of all natural numbers, zero, and the negative of all natural numbers (negative integers)
0, $\pm$1, $\pm$2, ...
The number line
The symbol for the set of all integers is "Z"

The set of all numbers that can be expressed "rationally," that is, that can be expressed as a fraction of two integers
$\frac{a}{b}$, where a, b are integers. Important: $\frac{a}{0}$ is not defined for any $a \in R$
The set of rational numbers is represented by the symbol "Q"

The set of irrational numbers is the set of all numbers that cannot be expressed as fractions
$2^{.5}$ and $\pi$ are examples of irrational numbers

We say that 1 is a member of N, which is a subset of Z etc. if $1\in N \subset Z \subset Q \subset R$

$a^n=a\cdot a\cdot ... \cdot a$ (multiply an integer number by itself for n times)
$a^0=1, \forall a \in \mathbb{R}$ and $a\neq 0$.
$0^0$is undefined.
$a^1=a, a^2=a\cdot a$ and so forth

$(\frac{p}{q})^n=\frac{p}{q}\cdot \frac{p}{q} \cdot ... \cdot\frac{p}{q}$ and also $(\frac{p}{q})^n=\frac{p^n}{q^n}$

Powers with the same base are multiplied by adding the exponents: $a^r\cdot a^s=a^{r+s}$
A power of a power is solved by multiplying the exponents: $(a^r)^s=a^{r\cdot s}$
These rules also applies to divisions. Thus $a^r\div a^s= a^{r-s}$
Also, the power of a product is equal to the product of the powers: $(a\cdot b)^{r}=a^r\cdot b^r$
Do example 1 p6

The rule of compound interest is very important in economic applications (from intertemporal choice theory development economics)
Say we have $100. If we put this money into our saving account we get an interest of 1.5% a month. What do we have after one year? After 5 years? after 30 years?
General formula: $x_t=x_0(1+r)^t$
So, after one year we get $x_1=100(1.015)^1=101.5$
After five years we get $x_1=100(1.015)^5\approx 107.73$
After forty years: $x_1=100(1.015)^{40}\approx 181.40$
Of course, the growth factor (the element within the parentheses) can also imply a negative rate of growth: $x_t=x_0(1-r)^t$

We can use negative exponents to answer the question: how much money should I have deposited, say, five years ago in order to get $110 today?
$x_0=x_t(1=r)^{-t}$
Note that this is just obtained by rearranging the formula above!

I trust that you guys all know the rules of algebra. If you need to refresh your memory the book does a pretty good job. If you have any doubts/difficulties, you can email me or come to office hours.

The inverse of the square of a real number, we can write a square root in many ways: $a^{.5}\equiv a^{1/2} \equiv \sqrt{a}$
I personally prefer the first as it makes multiplication of powers more intuitive
Of course, a square root is only valide for $a\geq 0$.
The same rules we saw for integer powers also apply for fractional powers (product rule, ratio rule)
Important: $\sqrt{a+b}\neq \sqrt{a}+\sqrt{b}$ !
Also important: $a^2=\pm \sqrt{a^2}$
Examples

We say that a is larger than b if $a>b$
We say that a is larger or equal to b if $a\geq b$
These implies the following rules
If $a, b>0 \Longrightarrow a+b>0$
Similarly, if $a, b>0 \Longrightarrow a\cdot b>0$
Also, if $a>b \Longrightarrow a-b>0$
We can also prove that if $a>b \Longrightarrow a+c>b+c$
Proof: a-b= a+c-(b+c), which implies that, for a>b, a-b>0. But then a+c-(b+c)>0 and rearranging, a+c>b+c. QED

Proof of 3: $a+d>b+d$ and $c+a>d+a$. Combining the two we get $a+c>a+d>b+d \Longrightarrow a+c>b+d$

The open interval $a<x<b$ is identified by the notation $(a, b)$. We write $x \in (a, b)$
The closed interval $a\geq x\geq b$ is identified by the notation $[a, b]$. We write $x \in [a, b]$
We can also have half-open intervals (to the left, to the right)
All of the above are bounded intervals. An interval that goes to positive or negative infinite is, by definition unbounded

The distance between two values is equal to the absolute value of their difference: $|x_a-x_b|$