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 $\frac{11}{4}=2.75$
• Q: Rationals - ie all D, plus those that do recur (but may never terminate.
  • For example $\frac{1}{3}$ ( = 0.333333) or $\frac{143}{999}$ ( = 0.143143143)
• R: Reals - ie all Q plus the irrationals, which are decimals that neither terminate nor recur.
  • For example $\pi ,\sqrt{2},3\sqrt{3},-\sqrt{2},-\frac{\sqrt{5}}{2}$, 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.

Solving Inequalities

If the real number $a$ is :

• Positive, then all solutions to $ax+b>0$ is $]-b/a,+\inf [$.
• Negative, then all solutions to $ax+b>0$ is $]-\inf ,-b/a[$.