Real Numbers: Difference between revisions

Revision as of 16:37, 24 September 2025

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 $π, \sqrt{2}, 3 \sqrt{3}, - \sqrt{2}, - \sqrt{5} / 2$ , etc

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

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


Interval Type	Interval	Values for the real number x
Closed	$x \in [a, b]$ $x \in [a, b [$ $x \in] a, b]$ $x \in] a, b [$	$a \leq x \leq b$ $a \leq x < b$ $a < x \leq b$ $a < x < b$
Open	$x \in [a, + \infty]$ $x \in] a, + \infty [$ $x \in] - \infty, a]$ $x \in] - \infty, a [$	$x \geq a$ $x > a$ $x \leq a$ $x < a$

The interval $[a, a]$ is equal to $[a]$

The interval $[b, a] w h e r e b > a i s e m p t y$

@@ Line 31: / Line 31: @@
 |<math> x \geq a</math><br><math>x > a</math><br><math>x \leq a</math><br><math>x < a</math>
 |}
+The interval <math>[a,a]</math>is equal to <math>[a]</math>
+The interval <math>[b,a] where b > a is empty</math>
 == Solving Inequalities ==