$\oint 1.1$ Properties of Real Numbers

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Sets of numbers

• Complex $a+bi$
  • Real $\mathbb R$
    • Rational $\mathbb Q -4, \frac{3}{4}, \frac{1}{3}$
      • Integers $\mathbb Z \{ 5, -2, -1, 0, 1\}$
        • Whole $0, 1, 2, 3\dots$
          • Counting $\mathbb N \{1, 2, 3\dots\}$
    • Irrational $\pi, \sqrt{2}$
  • Imaginary $i$

Digits

$\{0,1,2,3,4,5,6,7,8,9\}$

Natural/Counting $(\mathbb N)$

$\{1,2,3\dots\}$

Whole

$\{0,1,2,3\dots\}$

Integers $(\mathbb Z)$

$\{ \dots -2, -1, 0, 1, 2\dots\}$

Rational $(\mathbb Q)$

$\bigg\{ {\large\frac{p}{q}}\mid p, q \in \mathbb Z, q \neq 0 \bigg\}$

Irrational

Radical or Transcendental

Transcendental Number:

• A real or complex number that is not algebraic - not a root of a non-zero polynomial equation with real coefficients such as $\pi$ and $e$

Real

$\bigg\{ x \mid x$ is any number on the number line$\bigg\}$

Complex

$\bigg\{ a\pm bi \mid a,b\in\mathbb R, i=\sqrt{-1} \bigg\}$

Properties and Definitions

For all a and b belonging to the reals

$$\forall a,b \in \mathbb R$$

1. Commutative
   • Addition: $$a+b = b+a$$
   • Multiplication: $$ab = ba$$
2. Associatve
   • Addition: $$(a + b) + c = a + (b + c)$$
   • Multiplication: $$(ab)c = a(bc)$$
3. Identity
   • Additive Identity: $$a+0 = a$$
   • Multiplicative Identity: $$a \cdot 1 = a$$
4. Inverse
   • Additive Inverse: $$a + (-a) = 0$$
   • Multiplicative Inverse: $$a \cdot \frac{1}{a} = 1$$
5. Distributive
   • $$a(b \pm c) = ab \pm ac$$
6. Definition of Subtraction
   • $$a-b = a + (-b)$$
7. Definition of Division
   • $$\frac{a}{b} = \frac{1}{b} \cdot a, b \neq 0$$
8. Closure
   • We select a specific set of numbers an an operation
   • The set is closed wrt the operation if picking any numbers result in an answer that is also in the set

   • Is Digits closed wrt Addition?
     $8 + 6 = 14 \notin Digits$, thus no

Subset Notation

"$\subset$" is a symbol used to signify a subset

if $A \subset B$ then every element in A is also in B

Example:

Digits $\subset$ Whole

$\mathbb Z \subset \mathbb R$