Real number: Difference between revisions

Latest revision as of 01:16, 29 April 2022

Calculus is based in the system of real numbers and their properties.

Classification

Real numbers are classified as rational numbers (denoted by $Q$ ), integers ( $Z$ ), whole numbers ( $W$ ), natural numbers, and irrational numbers. In order of inclusion, non-irrational real numbers can be ordered as follows:

$N \subseteq W \subseteq Z \subseteq Q$

Properties

Distributive Property

$a . (b + c) = a . b + a . c$

Commutative property of addition

$a + b = b + a$

Commutative property of multiplication

$a . (b . c) = (a . b) . c$

Aditive identity property

$a + 0 = a$

Multiplicative identiy property

$a . 1 = a$

Multiplicative identity property

$a . 1 = a$

Additive inverse property

$a + (- a) = 0$

Multiplicative inverse property

$a . \frac{1}{a} = 1$ where $a \neq 0$

Zero property of multiplication

$a . 0 = 0$

Closure property of addition

$a + b$ is a real number

Closure property of multiplication

$a . b$ is a real number

Addition property of equality

If $a = b$ , then $a + c = b + c$

Substitution property

If $a = b$ , then $a$ may be substituted for $b$ or conversely

Reflexive (or identity) property of equality

$a = a$

Symmetric property of equality

If $a = b$ , then $b = a$

Transitive property of equality

If $a = b$ and $b = c$ , then $a = c$

Law of trichotomy

Exactly one of the following holds: $a < b, a = b, a > b$

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+Calculus is based in the system of real numbers and their properties.
 == Classification ==
-Real numbers are classified as [[rational number]]s (denoted by {{math|'''Q'''}}), [[integer]]s, [[whole number]]s, [[natural number]]s, and [[irrational number]]s.
+Real numbers are classified as [[rational number]]s (denoted by <math>Q</math>), [[integer]]s (<math>Z</math>), [[whole number]]s (<math>W</math>), [[natural number]]s, and [[irrational number]]s. In order of inclusion, non-irrational real numbers can be ordered as follows:
+<math>N \subseteq W \subseteq Z \subseteq Q</math>
+== Properties ==
+=== Distributive Property ===
+<math>a.(b+c)=a.b+a.c</math>
+=== Commutative property of addition ===
+<math>a+b=b+a</math>
+=== Commutative property of multiplication ===
+<math>a.(b.c)=(a.b).c</math>
+=== Aditive identity property ===
+<math>a+0=a</math>
+=== Multiplicative identiy property ===
+<math>a.1=a</math>
+=== Multiplicative identity property ===
+<math>a.1=a</math>
+=== Additive inverse property ===
+<math>a+(-a)=0</math>
+=== Multiplicative inverse property ===
+<math>a.\frac{1}{a}=1</math> where <math>a\neq0</math>
+=== Zero property of multiplication ===
+<math>a.0=0</math>
+=== Closure property of addition ===
+<math>a+b</math> is a real number
+=== Closure property of multiplication ===
+<math>a.b</math> is a real number
+=== Addition property of equality ===
+If <math>a=b</math>, then <math>a+c=b+c</math>
+=== Substitution property ===
+If <math>a=b</math>, then <math>a</math> may be substituted for <math>b</math> or conversely
+=== Reflexive (or identity) property of equality ===
+<math>a=a</math>
+=== Symmetric property of equality ===
+If <math>a=b</math>, then <math>b=a</math>
+=== Transitive property of equality ===
+If <math>a=b</math> and <math>b=c</math>, then <math>a=c</math>
-<math>Q \subseteq [[whole number]]s \subseteq [[integer]]s \subseteq [[rational number]]s</math>
+=== Law of trichotomy ===
-== Representation ==
+Exactly one of the following holds: <math>a<b, a=b, a>b</math>