Every real number is a living, breathing creature with a home... on the number line.
Not all real numbers have the same status though. Like humans, they divide themselves into classes.
Some real numbers are quite sane, and rational.
On the other hand, the vast majority are quite unruly and insane. For lack of a better description, they are called irrational.
Of course, rational numbers, who know what's best for all real numbers must further subdivide themselves in categories. Some numbers live in nice houses and drive nice cars. These are the most cheerful and positive; they are an elite ruling class called natural numbers, or the positive integers.
Every positive integer has a grumpy, pessimistic twin called an opposite. All the grumpy twins are called the negative integers.
There is a number, which lives at the center of the universe. It is the most powerful number, capable of unravelling the fabric of the universe. With a single multiplication, it can devour any other number, rational, or irrational. When added, is can hide in the inky shadows, undetected. Indeed, it is the most powerful number, yet, it is the most rational, sane, and cool-headed number there is.
Can you guess what it is?
The positive integers and negative integers together with the all-powerful, yet benevolent 0, are simply called the integers.
Although zero may be the good-natured and sublimely content ruler of the universe, do not mess with zero.
Like all positive integers, every positive real number (this includes irrationals too) also has a a foul-mooded twin, and when they "get married," they make 0. $$ \begin{array}{c|c|c|c} \mbox{Pos. Real Number} & \mbox{Grumpy Opposite} & \mbox{They ``marry''} & \mbox{Result}\\\hline 2 & -2 & 2+(-2) & 0 \\\hline \frac{2}{3} & -\frac{2}{3} & \frac{2}{3}+(-\frac{2}{3}) & 0 \\\hline \sqrt{5} & -\sqrt{5} & \sqrt{5}+(-\sqrt{5}) & 0 \\\hline \pi & -\pi & \pi+(-\pi) & 0 \\\hline \end{array} $$
Opposites, are called additive inverses of one another because when they marry, they "undo" one another to give us zero.
I also like to think that opposites "balance," "complement," or even "complete" one another.
Who here has seen Superman 3 or The Dark Crystal?
If you ever get married to zero, you'd never know it because its touch is so slight, so subtle, that you'd never feel it.
In fact, when zero marries any real number, it doesn't even notice: $$ \begin{array}{c|c|c|c} \mbox{Real number $x$} & Zero & \mbox{$x$ ``marries" $0$} & \mbox{Result}\\\hline 2 & 0 & 2+0 & 2 \\\hline -\frac{2}{3} & 0 & -\frac{2}{3}+0 & -\frac{2}{3} \\\hline \sqrt{5} & 0 & \sqrt{5}+0 & \sqrt{5} \\\hline -\pi & 0 & 0+(-\pi) & -\pi \\\hline \end{array} $$
Now, the grumpier or pessimistic a negative number is, the more space its cheerful, positive counterpart must make to get some space from from it. For example, a grumpy $-2$ lives $2$ doors to the left of supreme $0$. Therefore, to get get some breathing room from its surly and unpleasant counterpart, $2$ has decided to live $2$ doors to the right of $0$.
Another example: $-2178$ is in an even worse mood than $-2$. It lives $2178$ doors left of $0$. Accordingly, the very positive and cheerful $2178$ must live $2178$ doors to the right of $0$ in order not to hear its twin's constant bellyaching.
The thing to notice every number and its twin live the same distance from 0. That distance is called the absolute value.
How do we say precisely that one number is more cheerful (positive) than another? Well, the further the number is to the right, the more cheerful and positive it is. So, for example $5$ is clearly more cheerful than $-2.1$, so we write, $-2.1<5.3$, or $5.3>-2.1$.
Of course, as we already know, $-2718$ is in a really bad mood compared to $-2$. We can say that $-2.1$ is "less grumpy" than $-2178$. But what's another way to say "less grumpy?"
Yes! The number $-2.1$ is "more cheerful" than $-2178$. Therefore, $-2.1>-2178$, and $-2178<-2.1$
Note: We can also say that $-2.1 \geq -2178$ since $-2.1$ is just as, if not more, cheerful than $-2178$.
Just like their human counterparts, numbers organize themselves into neighborhoods.
These neighborhoods are just pieces of the real line (the real number universe), just like our neighborhoods are pieces of the city.
These pieces of the real line, or neigborhoods, are called intervals.
Puzzle Time: What number times itself is 4?
Puzzle Time: What number times itself is 36?
Puzzle Time: What number times itself is 10?
Answer Time: The principal square root of 4 is 2. We denote this as $\sqrt{4}=2$.
$\sqrt{4}$: Rational.
Answer Time: The principal square root of 36 is 6. We denote this as $\sqrt{36}=6$.
$\sqrt{36}$: Rational.
Answer Time: The principal square root of 10 is close to 3. We denote this as $\sqrt{10}\approx 3$.
$\sqrt{10}$: Irrational.
Quiz Time: Only one brave soul has ever dared to venture into this admittedly crazy number society.
Any guesses?
