I have trouble remembering the number sets and what’s included in each of them especially when trying to memorize. After reading into the history and how they came about, I realized I didn’t need to memorize and my curiosity in the background will do that for me automatically.

Number Base

Base 10 became dominant primarily because humans have 10 fingers, making it the most intuitive counting system across cultures. The ancient Egyptians were among the earliest to systematically use decimal notation around 3000 BCE with their hieroglyphic numerals, while the ancient Chinese developed decimal systems around 1500 BCE, and the ancient Greeks adopted base 10 around 600 BCE.

Base 20 systems developed in cultures that counted both fingers and toes, with the Maya civilization using this vigesimal system from around the 4th century BCE through 900 CE. The Aztecs inherited base 20 from earlier Mesoamerican cultures, and some Celtic cultures also employed base 20 between 500 BCE and 500 CE. You can still see traces of base 20 in modern languages like French, where 80 is expressed as “quatre-vingts” meaning four twenties.

Base 60 was developed by the Sumerians in Mesopotamia around 3000 BCE and later refined by the Babylonians between 2000-1600 BCE. This sexagesimal system wasn’t based on anatomy but rather on mathematical convenience, since 60 has twelve divisors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. This abundance of factors made it excellent for trade calculations and astronomical measurements. The Babylonians used this system for mathematics and astronomy throughout the ancient Near East, and remarkably, we still use base 60 today for measuring time with 60 seconds and 60 minutes, as well as for angles with 360 degrees. https://www.storyofmathematics.com/sumerian.html/ 

Base 12 appeared in cultures where people counted the segments of their four fingers using their thumb as a pointer. This duodecimal influence persists in our 12-hour clock system, 12 months in a year, and the concept of dozens.

Here’s a markdown table showing examples of different base systems with their positional representations:

The global dominance of base 10 occurred through several key historical developments. The crucial turning point came with the development of Hindu-Arabic numerals in India during the 6th and 7th centuries CE, which included the revolutionary concept of zero. Fibonacci introduced these numerals to Europe around 1202 CE through his influential book “Liber Abaci,” though it took several centuries for them to replace Roman numerals completely. By the 15th and 16th centuries, base 10 had become dominant in European mathematics and commerce.

The Scientific Revolution of the 1600s and 1700s further reinforced base 10 as the standard for mathematical and scientific work, while the Industrial Revolution of the 1800s made standardized numerical systems essential for global trade and manufacturing. European colonization from the 1500s through the 1800s spread base 10 systems worldwide, ultimately establishing it as the global standard. Despite this dominance, remnants of other base systems persist in our daily lives through time measurement, angular degrees, and various cultural practices, demonstrating the lasting influence of these ancient mathematical innovations.

Before zero was recognised as a number, various ancient civilisations developed systems that hinted at its functional need, primarily as a placeholder in positional notation.

2,000 BC – The Babylonian sexagesimal system: The Babylonians, inheriting a sexagesimal (base-60) system from the Sumerians and Akkadians, were pioneers in using place value.¹ However, their early system lacked a dedicated symbol for an empty position, leading to ambiguity.²

1. The Number System Hierarchy

Each number system proceeded the next due to some limitation.

Operation Ladder

$\mathbb{N}$ — closed under $+$ and $\times$
  $\downarrow$ need $-$
$\mathbb{Z}$ — closed under $+,\, -,\, \times$
  $\downarrow$ need $\div$
$\mathbb{Q}$ — closed under $+,\, -,\, \times,\, \div$ (except $0$)
  $\downarrow$ need limits of Cauchy sequences
$\mathbb{R}$ — complete continuum
  $\downarrow$ need root of $x^{2}+1=0$
$\mathbb{C}$ — algebraically closed

1.1 Natural Numbers ($\mathbb{N}$)

Definition: $\mathbb{N} = \{1, 2, 3, 4, ...\}$

The natural numbers are humanity’s first mathematical abstraction. Every civilization independently developed counting numbers - from Babylonian cuneiform tablets to Incan quipus. The Sumerian base-60 positional numerals (3rd millennium BCE) set the template for written arithmetic; Egyptian hieroglyphs tallied powers of ten in a non-positional decimal script; Chinese counting-rod boards (at least 3rd century BCE) embodied decimal place value that could be physically shuffled for calculation—all three handled only the natural numbers

These numbers answered the fundamental question: “How many?”

Note: Some fields include 0, making $\mathbb{N} = \{0, 1, 2, 3, ...\}$ but we introduce them next so ignore. Limitation: You can’t subtract 5 from 3 and stay within $\mathbb{N}$. There is no concept of debt. Luckily that is introduced next.

1.2 Adding Zero and Negatives: Integers ($\mathbb{Z}$)

Definition: $\mathbb{Z} = \{..., -3, -2, -1, 0, 1, 2, 3, ...\}$
Etymology: Z from German “Zahlen” (numbers)

The origins of Zero https://www.diplomacy.edu/blog/journey-of-zero-evolution/  https://www.diplomacy.edu/blog/origins-of-zero-a-fascinating-story-of-science-and-spirituality-across-civilisations/ 

The concept of zero emerged independently in several civilizations, but it was the Indian mathematician Brahmagupta who, in his 628 CE work Brahmasphutasiddhanta, first gave rules for mathematical operations with zero and negative numbers. He described debts as negative numbers and fortunes as positive, establishing rules like:

• A debt minus zero is a debt
• A fortune minus zero is a fortune
• Zero minus zero is zero

What it solves: Closure under subtraction - any integer minus any integer yields an integer.

The concept of zero, termed ‘Shunya’ in Hindi, has deep roots in ancient Indian philosophical and religious traditions.  ‘Shunyata’, often translated as ’emptiness’ or ‘void’, holds significant importance in Buddhism.  The renowned philosopher Nagarjuna, active around the 2nd century CE, anchored Mahayana Buddhism on the principle of ’emptiness’, emphasizing the interdependent existence of phenomena.

This philosophical understanding of ’emptiness’ or ‘void’ laid the groundwork for the mathematical adoption of the number zero. By the 6th century AD, prominent Indian mathematicians like Aryabhata and Brahmagupta had begun employing zero as a placeholder in their calculations.

To date, archaeological efforts have unveiled two significant artefacts in India that demonstrate the early use of the numeral zero:

The more ancient of the two is the stone known as K-127, dated to 683 CE. Discovered in the Hindu temple complex of Sambor near the Mekong River, this stone features the numeral zero depicted as a dot amidst other numbers. Presently, K-127 is housed in the National Museum in Phnom Penh, Cambodia.

Subsequent to this is the ‘Gwalior zero’, found inscribed in the Chaturbhuj Temple in Gwalior, India. This artifact, dating to 876 CE, showcases the use of the number zero in a manner akin to modern usage, specifically to document a land grant. https://www.diplomacy.edu/blog/origins-of-zero-a-fascinating-story-of-science-and-spirituality-across-civilisations/ 

How integers ($\mathbb{Z}$) emerged

Placeholder → number
Babylonian cuneiform adopted a sexagesimal placeholder for “nothing” by the late 3rd millennium BCE, but scribes treated it only as an empty slot, not as an actual number. https://brill.com/display/book/9789004691568/BP000018.xml 

Zero treated arithmetically
The Bakhshali manuscript (3rd–4th cent.) shows a dot-zero symbol; Brahmagupta’s Brāhmasphuṭasiddhānta (628 CE) gave explicit rules such as $a+0=a$ and $a-0=a$, promoting zero to full number status.

Negative numbers
Chinese Nine Chapters (1st cent. CE) used red rods for positives and black for negatives, listing subtraction rules that yield negative results.

Cardano’s Ars Magna (1545) systematised negative numbers in Europe, though he still called them numeri ficti.

Symbol $\mathbb{Z}$
From the 1930s the Bourbaki group standardised the blackboard-bold $\mathbb{Z}$, taking Z from German Zahlen (“numbers”), a typographic choice rather than the work of a single German mathematician

Q for the set of rational numbers and Z for the set of integers are apparently due to N. Bourbaki. (N. Bourbaki was a group of mostly French mathematicians which began meeting in the 1930s, aiming to write a thorough unified account of all mathematics.) The letters stand for the German Quotient and Zahlen. These notations occur in Bourbaki’s Algébre, Chapter 1.

1.3 Rational Numbers ($\mathbb{Q}$)

Definition: $\mathbb{Q} = \{p/q \mid p, q \in \mathbb{Z}, q \neq 0\}$
Etymology: Q from “Quotient”

Ancient Egyptians worked with unit fractions (1/n) as early as 2000 BCE, using them for practical division of food and land. The Rhind Papyrus shows calculations like 2/3 and complex fraction arithmetic.

Key insight: Every rational number has a decimal expansion that either:

• Terminates: $3/4 = 0.75$
• Repeats: $1/3 = 0.333...$

What it solves: Closure under division (except by zero).

1.4 The Surprising Discovery: Irrational Numbers

Definition: Numbers that cannot be expressed as $p/q$ for any integers $p, q$

The discovery of irrational numbers caused the first major crisis in mathematics. Legend says the Pythagorean Hippasus was thrown overboard for proving $\sqrt{2}$ was irrational, shattering the Pythagorean belief that all numbers were ratios. The proof is elegant:

If $\sqrt{2} = p/q$ (in lowest terms), then $2q^2 = p^2$, making $p$ even. If $p = 2k$, then $2q^2 = 4k^2$, so $q^2 = 2k^2$, making $q$ even too. But this contradicts $p/q$ being in lowest terms.

Examples:

• Algebraic irrationals: $\sqrt{2}$, $\sqrt[3]{7}$, $(1+\sqrt{5})/2$ (golden ratio)
• Transcendental numbers: $\pi$, $e$

1.5 The Complete Line: Real Numbers ($\mathbb{R}$)

Definition: $\mathbb{R} = \mathbb{Q} \cup \{\text{Irrational Numbers}\}$

The real numbers weren’t rigorously defined until the 19th century. Richard Dedekind’s “cuts” (1872) and Georg Cantor’s Cauchy sequences finally gave precise meaning to “all points on the number line.”

Subcategories:

• Algebraic numbers: Solutions to polynomial equations with rational coefficients
  • Example: $\sqrt{2}$ solves $x^2 - 2 = 0$
• Transcendental numbers: Not algebraic
  • $\pi$ (proved transcendental by Lindemann, 1882)
  • $e$ (proved transcendental by Hermite, 1873)

1.6 Beyond the Line: Imaginary Numbers ($\mathbb{I}$)

Definition: $\mathbb{I} = \{bi \mid b \in \mathbb{R}, b \neq 0\}$ where $i^2 = -1$

Italian mathematician Gerolamo Cardano first used imaginary numbers around 1545 while solving cubic equations, though he called them “sophistic” and “useless.” It wasn’t until Euler popularized the notation $i = \sqrt{-1}$ that they gained acceptance.

The key identity: $i^2 = -1$, which means:

• $i^3 = -i$
• $i^4 = 1$ (the pattern repeats)

1.7 The Complete System: Complex Numbers ($\mathbb{C}$)

Definition: $\mathbb{C} = \{a + bi \mid a, b \in \mathbb{R}\}$

Complex numbers unify the entire number system. Gauss proved the Fundamental Theorem of Algebra (1799): every polynomial of degree $n$ has exactly $n$ complex roots (counting multiplicity).

Geometric interpretation: Complex numbers form a 2D plane where:

• Real part: horizontal axis
• Imaginary part: vertical axis
• Example: $3 + 4i$ is the point $(3, 4)$

N⊂Z⊂Q⊂R⊂C ↘ Irrational ⊂R\subset \mathbb{R} ⊂R ↓ Algebraic ↓ Transcendental

Key relationships: - Every natural number is an integer: $5 = 5/1 \in \mathbb{Z}$ - Every integer is rational: $-3 = -3/1 \in \mathbb{Q}$ - Every rational is real: $2/3 \in \mathbb{R}$ - Every real is complex: $\pi = \pi + 0i \in \mathbb{C}$ - But: $\sqrt{2} \in \mathbb{R}$ but $\sqrt{2} \notin \mathbb{Q}$

2. Set-Builder Notation as a Language

Set-builder notation is mathematics’ way of precisely describing infinite sets or sets with specific properties. Instead of listing elements, we describe the rules for membership.

2.1 Why We Need Set-Builder Notation

Consider these challenges:

1. Infinite sets: How do you write all even numbers? $\{2, 4, 6, ...\}$ is imprecise
2. Complex conditions: “All real numbers except 0 and 1”
3. Derived sets: “All $x$ where $x^2 < 10$”

Georg Cantor, the father of set theory, developed much of our modern notation in the 1870s while grappling with different sizes of infinity. His work sparked fierce debate - Henri Poincaré called set theory a “disease” from which mathematics would recover, while David Hilbert declared “No one shall expel us from the paradise that Cantor has created.”

2.2 Anatomy of Set-Builder Notation

The basic structure: $\{variable \in set \mid condition\}$

Read as: “The set of all [variable] in [set] such that [condition]”

Components:

• Variable: A placeholder (commonly $x$, but can be anything)
• Domain ($\in$ set): Which number system we’re drawing from
• Condition: The rule elements must satisfy
• Separator: | (read “such that”) or : (colon)

The vertical bar notation comes from the Zermelo-Fraenkel axioms of set theory (1908-1922), though Ernst Zermelo used different notation initially. The modern form was popularized by Nicolas Bourbaki’s group in the 1930s.

2.3 Basic Examples

Example 1: All positive real numbers $\{x \in \mathbb{R} \mid x > 0\}$

Example 2: Even integers
$\{n \in \mathbb{Z} \mid n = 2k \text{ for some } k \in \mathbb{Z}\}$ Or more concisely: $\{2k \mid k \in \mathbb{Z}\}$

Example 3: Rational numbers between 0 and 1 $\{q \in \mathbb{Q} \mid 0 < q < 1\}$

2.4 Translation Exercises

English → Set-Builder:

• “All integers except 5” → $\{n \in \mathbb{Z} \mid n \neq 5\}$
• “Square roots of positive integers” → $\{\sqrt{n} \mid n \in \mathbb{N}\}$
• “All real solutions to $x^2 = 4$” → $\{x \in \mathbb{R} \mid x^2 = 4\} = \{-2, 2\}$

Set-Builder → English:

• $\{x \in \mathbb{R} \mid |x| < 3\}$ → “All real numbers with absolute value less than 3”
• $\{p/q \mid p, q \in \mathbb{Z}, q \neq 0\}$ → “All ratios of integers (the rational numbers)“

2.5 Common Patterns and Conventions

Pattern 1: Domain restrictions $\text{Dom}(f) = \{x \in \mathbb{R} \mid x \neq 0\}$ for $f(x) = 1/x$

Pattern 2: Solution sets $\{x \in \mathbb{R} \mid x^2 - 5x + 6 = 0\} = \{2, 3\}$

Pattern 3: Ranges and intervals

• $\{x \in \mathbb{R} \mid a \leq x \leq b\}$ - Closed interval $[a, b]$
• $\{x \in \mathbb{R} \mid a < x < b\}$ - Open interval $(a, b)$

2.6 Advanced Set-Builder Techniques

Multiple conditions (use “and” or commas): $\{x \in \mathbb{R} \mid x > 0 \text{ and } x^2 < 16\} = \{x \in \mathbb{R} \mid 0 < x < 4\}$

Existence quantifiers: $\{x \in \mathbb{R} \mid \exists n \in \mathbb{Z}, x = \sqrt{n}\}$ This describes all real numbers that are square roots of some integer.

Universal quantifiers: $\{p \in \mathbb{N} \mid \forall d \in \mathbb{N}, (d|p \Rightarrow d = 1 \text{ or } d = p)\}$ This defines the set of prime numbers!

2.7 Historical Note: Russell’s Paradox

Bertrand Russell discovered in 1901 that naive set theory leads to contradictions. Consider: $R = \{x \mid x \notin x\}$ “The set of all sets that don’t contain themselves”

Is $R \in R$? If yes, then by definition $R \notin R$. If no, then by definition $R \in R$. This paradox led to more careful axiomatization of set theory and restrictions on what constitutes a valid set definition.

3. Connecting Sets and Notation

Now let’s express each number system using set-builder notation and explore common restrictions.

3.1 Number Sets in Set-Builder Form

Natural Numbers: $\mathbb{N} = \{n \in \mathbb{Z} \mid n > 0\}$ Or with 0: $\mathbb{N}_0 = \{n \in \mathbb{Z} \mid n \geq 0\}$

Integers (constructed from naturals): $\mathbb{Z} = \{a - b \mid a, b \in \mathbb{N}_0\}$

Rational Numbers: $\mathbb{Q} = \left\{\frac{p}{q} \mid p \in \mathbb{Z}, q \in \mathbb{Z}, q \neq 0\right\}$

Irrational Numbers: $\mathbb{R} \setminus \mathbb{Q} = \{x \in \mathbb{R} \mid x \notin \mathbb{Q}\}$

Algebraic Numbers: $\mathbb{A} = \{x \in \mathbb{C} \mid \exists \text{ polynomial } p \text{ with rational coefficients}, p(x) = 0\}$

Transcendental Numbers: $\mathbb{T} = \{x \in \mathbb{R} \mid x \notin \mathbb{A}\}$

Complex Numbers: $\mathbb{C} = \{a + bi \mid a, b \in \mathbb{R}\}$

3.2 Common Restrictions and Their Notation

Non-zero elements (often denoted with * or ×):

• $\mathbb{R}^* = \{x \in \mathbb{R} \mid x \neq 0\}$
• $\mathbb{Z}^* = \{n \in \mathbb{Z} \mid n \neq 0\}$

Positive elements (often denoted with +):

• $\mathbb{R}^+ = \{x \in \mathbb{R} \mid x > 0\}$
• $\mathbb{Z}^+ = \{n \in \mathbb{Z} \mid n > 0\} = \mathbb{N}$

Non-negative elements:

• $\mathbb{R}_{\geq 0} = \{x \in \mathbb{R} \mid x \geq 0\}$

Bounded sets:

• $\{x \in \mathbb{R} \mid |x| \leq M\}$ - All reals with absolute value at most $M$
• $\{z \in \mathbb{C} \mid |z| = 1\}$ - The unit circle in the complex plane

3.3 Edge Cases and Gotchas

Empty set: Can be written as $\{x \in \mathbb{R} \mid x \neq x\} = \emptyset$

Singleton sets: $\{x \in \mathbb{R} \mid x^2 = 0\} = \{0\}$

Implicit domain: When the domain is clear from context, it’s sometimes omitted:

• $\{x \mid x > 0\}$ (implicitly real numbers)

Careful with division: $\left\{\frac{1}{x} \mid x \in \mathbb{R}^*\right\} = \mathbb{R}^*$ Note we must exclude 0 from the domain!

3.4 Set Operations in Builder Notation

Union ($\cup$): $A \cup B = \{x \mid x \in A \text{ or } x \in B\}$

Intersection ($\cap$): $A \cap B = \{x \mid x \in A \text{ and } x \in B\}$

Set difference ($\setminus$): $A \setminus B = \{x \mid x \in A \text{ and } x \notin B\}$

Example: The irrational numbers are $\mathbb{R} \setminus \mathbb{Q}$

4. Real-World Applications

4.1 Function Domains

Square root function: $\text{Dom}(\sqrt{x}) = \{x \in \mathbb{R} \mid x \geq 0\}$

Rational function: $\text{Dom}\left(\frac{1}{x^2-4}\right) = \{x \in \mathbb{R} \mid x^2 \neq 4\} = \mathbb{R} \setminus \{-2, 2\}$

Logarithm: $\text{Dom}(\ln(x)) = \{x \in \mathbb{R} \mid x > 0\} = \mathbb{R}^+$

4.2 Solution Sets

Quadratic inequality: $\{x \in \mathbb{R} \mid x^2 - 5x + 6 < 0\} = \{x \in \mathbb{R} \mid 2 < x < 3\}$

Trigonometric equation: $\{x \in \mathbb{R} \mid \sin(x) = 0\} = \{n\pi \mid n \in \mathbb{Z}\}$

4.3 Describing Data Constraints

In computer science and databases:

• Valid ages: $\{a \in \mathbb{N} \mid 0 \leq a \leq 150\}$
• Valid percentages: $\{p \in \mathbb{R} \mid 0 \leq p \leq 100\}$
• Prime indices: $\{p \in \mathbb{N} \mid p \text{ is prime}\}$

Quick Reference Card

Common Modifiers:

• $\mathbb{R}^+$: Positive reals
• $\mathbb{Z}^*$: Non-zero integers
• $\mathbb{Q}_{\geq 0}$: Non-negative rationals
• $[a, b]$: $\{x \in \mathbb{R} \mid a \leq x \leq b\}$
• $(a, b)$: $\{x \in \mathbb{R} \mid a < x < b\}$

Equivalent Special Sets

Let’s recall the table of special sets.

As noted in the table, the following sets are equivalent:

• $\mathbb{Z}^+$ and $\mathbb{N}$

• $\mathbb{Z}_0^+$ and $\mathbb{N}_0$

• $x \in (-\infty, \infty)$ is the interval containing all real numbers, the same as $\mathbb{R}$.

• $x \in (0, \infty)$ is the interval containing all positive real numbers, the same as $\mathbb{R}^+$.

• $x \in [0, \infty)$ is the interval containing all positive real numbers and zero, the same as $\mathbb{R}_0^+$.

• $x \in (-\infty, 0)$ is the interval containing all negative real numbers, the same as $\mathbb{R}^-$.

• $x \in (-\infty, 0]$ is the interval containing all negative real numbers and zero, the same as $\mathbb{R}_0^-$.