Notes on Natural, Whole, Integer, and Rational Numbers

Natural Numbers

• Definition: The set of positive counting numbers starting from 1; used for counting discrete objects.

• Notation: $\mathbb{N} = {1, 2, 3, \dots}$

• Key points:

• Do not include 0 under this convention (some texts include 0, but per this transcript the natural numbers begin at 1).

• Foundation for counting and basic arithmetic on positive quantities.

• Examples: 1, 7, 42, 1000.

Whole Numbers

• Definition: The set that extends natural numbers by including zero.

• Notation: $\mathbb{W} = {0, 1, 2, 3, \dots}$

• Key points:

• Zero is included as a non-positive starting point for counting scales.

• Relationship: every natural number is a whole number; $\mathbb{N} \subset \mathbb{W}$.

• Examples: 0, 1, 2, 100.

Integers

• Definition: The set that includes all positive and negative whole numbers, plus zero.

• Notation: $\mathbb{Z} = {\dots, -2, -1, 0, 1, 2, \dots}$

• Key points:

• Extends the whole numbers to represent deficits, differences, temperatures below zero, etc.

• Supports operations that model "gains and losses" around zero.

• Examples: -5, 0, 13.

The Concept of Rational Numbers

• Context: After introducing natural, whole, and integer numbers, a broader class emerges to express ratios.

• Definition: A rational number is any number that can be expressed as a ratio of two integers with nonzero denominator.

• Notation: \mathbb{Q} = \left{ \frac{a}{b} \middle| \ a \in \mathbb{Z}, \ b \in \mathbb{Z} \setminus {0} \right}

• Key points:

• Includes all integers (as fractions with denominator 1): for any $a \in \mathbb{Z}$, $\frac{a}{1} = a \in \mathbb{Q}$.

• Represents ratios such as fractions and decimals that repeat or terminate.

• Decimal representations:

• A rational number has a decimal expansion that either terminates or becomes periodic (repeats).

• Examples:

  • $\frac{3}{4} = 0.75$ (terminating)

  • $-\frac{7}{2} = -3.5$ (terminating)

  • $\frac{1}{3} = 0.\overline{3}$ (repeating)

• Sign and magnitude:

• Can be positive or negative depending on the signs of the numerator and denominator.

• Relationships to other sets:

• Every integer is rational: $\mathbb{Z} \subset \mathbb{Q}$

Relationships among number sets (hierarchy)

• Chain of inclusions:

• $\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$

• Meaning:

• Each larger set contains all elements of the previous one, plus new elements (e.g., negatives, fractions).

• Real numbers include all rationals and irrationals (though irrational numbers are not discussed in this transcript).

Significance and practical implications

• Foundational role in arithmetic and algebra: understanding what kinds of numbers can be used to model quantities, ratios, and measurements.

• Zero as a foundational concept: introduced to build the system of whole numbers; enables representation of no quantity and balance around zero (gains/losses).

• Negative numbers enable representation of deficits, temperatures below zero, debt, etc., facilitating a full arithmetic system.

• Rational numbers underpin fractions, ratios, and many real-world measurements; crucial for topics like percentages, conversions, and precise representations of parts of a whole.

Connections to broader mathematical study

• Sets N, W, Z, Q provide stepping stones to more advanced number systems (irrationals, reals, complex numbers).

• Understanding these sets supports later topics such as number theory, algebra, and analysis.

Quick recap (definitions in one line)

• Natural numbers: $\mathbb{N} = {1,2,3,\dots}$

• Whole numbers: $\mathbb{W} = {0,1,2,3,\dots}$

• Integers: $\mathbb{Z} = {\dots,-2,-1,0,1,2,\dots}$

• Rational numbers: \mathbb{Q} = \left{ \frac{a}{b} \middle| \ a \in \mathbb{Z}, \ b \in \mathbb{Z} \setminus {0} \right}

Note about scope (from the transcript)

• The content introduces and defines the basic number sets and the concept of rational numbers, setting the stage for applications and deeper study in later sections.

• Definition: The set of positive counting numbers starting from 1.
• Notation: $\mathbb{N} = {1, 2, 3, \dots}$
• Key points: Does not include 0.

• Definition: The set that extends natural numbers by including zero.
• Notation: $\mathbb{W} = {0, 1, 2, 3, \dots}$
• Relationship: $\mathbb{N} \subset \mathbb{W}$. Every natural number is a whole number.

• Definition: The set that includes all positive and negative whole numbers, plus zero.
• Notation: $\mathbb{Z} = {\dots, -2, -1, 0, 1, 2, \dots}$
• Key points: Extends whole numbers to represent deficits and differences.

• Definition: Any number that can be expressed as a ratio of two integers with a nonzero denominator.
• Notation: \mathbb{Q} = \left{ \frac{a}{b} \middle\| \ a \in \mathbb{Z}, \ b \in \mathbb{Z} \setminus {0} \right}
• Key points:
• Includes all integers (as fractions with denominator 1).
• Decimal representations either terminate or repeat.
• Relationship: $\mathbb{Z} \subset \mathbb{Q}$ Every integer is rational.

• Chain of inclusions: $\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$

• These sets are foundational for arithmetic, algebra, and modeling quantities, ratios, and measurements.

• Natural numbers: $\mathbb{N} = {1,2,3,\dots}$
• Whole numbers: $$\