Real Numbers

Number Sets

Each number set contains the number set before it: $/a$

• N: Natural Numbers - ie all whole numbers.
  • For example: 0,1,2... onwards
• Z: Integers - ie all N plus negative integers.
  • For example -2,-1,0,1,2...
• D: Decimals - ie all Z plus fractions that can be written with a finite number of decimals.
  • For example $11/4=2.75$
• Q: Rationals - ie all D, plus those that do recur (but may never terminate.
  • For example $1/3$ ( = 0.333333) or $143/999$ ( = 0.143143143)
• R: Reals - ie all Q plus the irrationals, which are decimals that neither terminate nor recur.
  • For example $\pi ,\sqrt{2},3\sqrt{3},-\sqrt{2},-\sqrt{5}/2$, etc

N is a subset of Z is a subset of D is a subset of Q is a subset of R

Intervals

Intervals are ways of specifying the range of possible values for x:

Interval Type	Interval	Values for the real number x
Closed	$x\in [a,b]$ $x\in [a,b[$ $x\in ]a,b]$ $x\in ]a,b[$	$a\le x\le b$ $a\le x<b$ $a<x\le b$ $a<x<b$
Open	$x\in [a,+\mathrm{\infty }]$ $x\in ]a,+\mathrm{\infty }[$ $x\in ]-\mathrm{\infty },a]$ $x\in ]-\mathrm{\infty },a[$	$x\ge a$ $x>a$ $x\le a$ $x<a$

The interval $[a,a]$is equal to $[a]$. The interval $[b,a]$ where $b>a$ is empty. The intersection of two intervals is always an interval or the empty set. If the intersection of two intervals is not empty, then the union of them is an interval.