Numbers – The Basic Introduction

Numbers are the alphabet of mathematics. Logic is the grammar. And everything we build in math, from simple arithmetic to quantum field theory, comes from arranging these symbols according to rules.

But here’s what most people never learn. All the complicated numbers you’ll ever encounter (fractions, negatives, irrationals, imaginary numbers) can be built from the simplest ones. The counting numbers. One, two, three, four.

Leopold Kronecker, one of the great 19th-century mathematicians, put it this way:

“God created the natural numbers, and all the rest is the work of man.”

He wasn’t being poetic. He was making a precise mathematical claim. Every type of number you’ll ever use can be constructed from the positive integers. Rationals, reals, complex numbers. All of them. They’re human inventions built on divine foundations.

This article walks through that construction. We’ll start with the numbers God gave us and build everything else. Then we’ll explore the zoo of special numbers that mathematicians have catalogued over centuries: primes, perfect numbers, Mersenne numbers, Carmichael numbers, and more.

Some of these are genuinely useful. Some are mathematical curiosities. All of them reveal something about the deep structure of mathematics itself.

Natural Numbers: Where Everything Begins

Natural numbers are the counting numbers. One, two, three, four. The numbers you use when you count sheep, fingers, or dollar bills.

Eight pens. Eighteen trees. Three thousand people. These quantities are measured with natural numbers. The word “natural” isn’t arbitrary. These numbers arise naturally from the act of counting things in the physical world.

We write the set of natural numbers as \( \mathbb{N} \), using a special “blackboard bold” font that mathematicians love. The formal definition looks like this:

$$\mathbb{N} := \{1, 2, 3, 4, \ldots, n, \ldots\}$$

The set goes on forever. There’s no largest natural number. Add one to any natural number, and you get another natural number. This property (closure under addition) is one of the first things you learn in abstract algebra.

When we write \( n \in \mathbb{N} \), we’re saying “n is a natural number” or “n belongs to the set of natural numbers.”

One controversy worth mentioning. Some mathematicians (and Wolfram Research, the people behind Mathematica) include zero in the natural numbers. They define \( \mathbb{N} := \{0, 1, 2, 3, \ldots\} \). This isn’t wrong, just a different convention. I’ll stick with the more traditional definition that starts at 1, because it matches what “natural” actually means. You don’t count zero sheep.

The natural numbers have been studied for thousands of years. The ancient Greeks were obsessed with them. Euclid’s Elements contains theorems about natural numbers that are still taught today. And despite millennia of study, we still don’t understand them completely. The Riemann Hypothesis, one of the greatest unsolved problems in mathematics, is ultimately a question about the distribution of prime numbers within \( \mathbb{N} \).

Integers: Adding Negatives and Zero

Natural numbers can’t handle subtraction properly. What’s 3 minus 5? There’s no natural number answer. You need negative numbers.

The integers extend the natural numbers by adding zero and all the negative whole numbers. The set is written as \( \mathbb{Z} \) (from the German word “Zahlen,” meaning numbers).

$$\mathbb{Z} := \{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}$$

Or, in terms of natural numbers:

$$\mathbb{Z} := \{\pm n : n \in \mathbb{N}\} \cup \{0\}$$

Examples of integers include 1, -1, 8, 0, -37, and 5943. Any whole number, positive, negative, or zero, is an integer.

But here’s the interesting part. How do you actually construct negative numbers from natural numbers? Kronecker’s quote implies it should be possible, and it is.

The key insight is that every integer can be written as a difference of two natural numbers. The number 5 is 8 minus 3. The number -2 is 1 minus 3. Zero is 3 minus 3.

But there’s a problem. The number 5 is also 9 minus 4, and 10 minus 5, and infinitely many other differences. So we need to define when two differences represent the same integer.

Here’s the formal construction. Take ordered pairs of natural numbers \( (a, b) \), which we think of as representing \( a – b \). Define two pairs \( (a, b) \) and \( (c, d) \) to be equivalent if \( a + d = b + c \). This is an equivalence relation, and the equivalence classes are the integers.

So the integer “5” is really the equivalence class containing (6,1), (7,2), (8,3), (9,4), and infinitely many other pairs. We’ve built negative numbers and zero purely from positive counting numbers.

This construction might seem overly complicated for something as simple as negative numbers. But the technique generalizes. We’ll use similar constructions to build rationals from integers and reals from rationals.

Rational Numbers: Fractions and Ratios

Integers can’t handle division properly. What’s 1 divided by 2? There’s no integer answer. You need fractions.

The rational numbers are all numbers that can be expressed as a ratio of two integers, where the denominator isn’t zero. The set is written as \( \mathbb{Q} \) (for “quotient”).

A rational number is written as \( \frac{p}{q} \) where \( p \) and \( q \) are integers and \( q \neq 0 \).

Examples include \( \frac{6}{19} \), \( \frac{-1}{2} \), \( 3\frac{2}{3} \) (which equals \( \frac{11}{3} \)), and \( 5 \) (which equals \( \frac{5}{1} \)).

Every integer is a rational number. Just put it over 1. So \( \mathbb{Z} \subset \mathbb{Q} \), meaning the integers are a subset of the rationals.

The construction from integers works similarly to how we built integers from naturals. We take ordered pairs \( (p, q) \) of integers with \( q \neq 0 \), thinking of them as representing \( \frac{p}{q} \). Two pairs \( (p, q) \) and \( (r, s) \) are equivalent if \( ps = qr \) (cross-multiplication). The equivalence classes are the rational numbers.

So \( \frac{1}{2} \), \( \frac{2}{4} \), \( \frac{3}{6} \), and \( \frac{-7}{-14} \) are all the same rational number. They’re different representatives of the same equivalence class.

The rational numbers are dense in a precise sense. Between any two distinct rationals, there’s another rational. In fact, there are infinitely many rationals between any two rationals. You might think this means the rationals fill up the number line completely.

They don’t. And that brings us to irrational numbers.

Irrational Numbers: The Gaps in the Rationals

An irrational number is a real number that cannot be expressed as a ratio of two integers. It’s not \( \frac{p}{q} \) for any integers \( p \) and \( q \).

The most famous examples are \( \sqrt{2} \), \( \pi \), and \( e \).

The ancient Greeks discovered irrational numbers, and it freaked them out. The Pythagoreans believed that all numbers were ratios of whole numbers. When they discovered that \( \sqrt{2} \) (the diagonal of a unit square) couldn’t be expressed this way, legend has it they tried to suppress the discovery. One story says they drowned the mathematician who proved it.

Here’s a quick proof that \( \sqrt{2} \) is irrational. Suppose it were rational, so \( \sqrt{2} = \frac{p}{q} \) in lowest terms. Then \( 2 = \frac{p^2}{q^2} \), so \( p^2 = 2q^2 \). This means \( p^2 \) is even, so \( p \) is even. Write \( p = 2k \). Then \( 4k^2 = 2q^2 \), so \( q^2 = 2k^2 \), meaning \( q \) is also even. But we assumed \( \frac{p}{q} \) was in lowest terms. Contradiction. So \( \sqrt{2} \) is irrational.

Other irrational numbers include \( \sqrt{3} \), \( \sqrt{5} \), \( \sqrt{11} \), and in general \( \sqrt{n} \) for any non-perfect-square integer \( n \). The numbers \( \pi \) and \( e \) are also irrational, though proving this is much harder.

There’s also a distinction between algebraic irrationals (like \( \sqrt{2} \), which is a root of \( x^2 – 2 = 0 \)) and transcendental numbers (like \( \pi \) and \( e \), which aren’t roots of any polynomial with integer coefficients). Transcendental numbers are even more exotic than algebraic irrationals.

Despite being “gaps” in the rationals, irrational numbers are actually more numerous than rationals in a precise sense. The rationals are countably infinite (you can list them), while the irrationals are uncountably infinite (you can’t list them). Almost every real number is irrational.

Real Numbers: The Complete Number Line

The real numbers combine rationals and irrationals into a single system. They’re all the numbers you can represent as points on a number line.

The set of real numbers is written as \( \mathbb{R} \). It includes every rational and every irrational number.

Constructing the reals from the rationals is more subtle than the previous constructions. The standard approach uses “Dedekind cuts,” named after Richard Dedekind who invented the technique in 1872.

The idea is that every real number is uniquely determined by the set of rationals less than it. A Dedekind cut is a subset \( L \) of \( \mathbb{Q} \) that is nonempty, bounded above, has no maximum element, and is “downward closed” (if \( y \in L \) and \( x < y \), then \( x \in L \)).

For a rational number like \( \frac{1}{2} \), the cut is just the set of all rationals less than \( \frac{1}{2} \). For an irrational like \( \sqrt{2} \), the cut is the set of all rationals whose square is less than 2, plus all negative rationals.

This construction might seem abstract, but it’s incredibly important. It’s what makes calculus rigorous. The real numbers have a property called “completeness”: every bounded, increasing sequence converges to a real number. The rationals don’t have this property. Without completeness, you can’t prove the fundamental theorems of calculus.

The study of real numbers and their properties is called real analysis. It’s one of the foundational courses in a mathematics degree, and it’s where most students first encounter rigorous proofs about limits, continuity, and convergence.

Complex Numbers: Beyond the Number Line

What’s the square root of -1? There’s no real number answer. The square of any real number is non-negative.

Mathematicians solved this problem by inventing a new number, called \( i \), defined by the property that \( i^2 = -1 \). Numbers involving \( i \) are called imaginary numbers. Combine them with real numbers, and you get complex numbers.

A complex number has the form \( a + bi \) where \( a \) and \( b \) are real numbers. The number \( a \) is called the real part, and \( b \) is called the imaginary part.

The set of complex numbers is written as \( \mathbb{C} \). Formally:

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

Examples include \( 3 + 2i \), \( -1 + 0i = -1 \), \( 0 + i = i \), and \( 5 – 7i \). There are some truly strange ones too, like \( i^i \), which turns out to be a real number (approximately 0.2079).

Complex numbers can be visualized as points in a two-dimensional plane, called the complex plane. The horizontal axis represents the real part, and the vertical axis represents the imaginary part. This geometric interpretation makes complex numbers surprisingly useful for problems involving rotations and periodic phenomena.

The construction from reals is straightforward. A complex number is simply an ordered pair of real numbers \( (a, b) \), with addition defined componentwise and multiplication defined so that \( (0, 1) \times (0, 1) = (-1, 0) \). This makes \( (0, 1) \) play the role of \( i \).

Complex numbers complete the algebraic picture. Every polynomial equation with complex coefficients has a solution in the complex numbers. This is the Fundamental Theorem of Algebra. You don’t need to invent any more number systems beyond \( \mathbb{C} \) to solve polynomial equations.

And with that, we’ve verified Kronecker’s claim. We started with the natural numbers and built the integers, then the rationals, then the reals, then the complex numbers. Every number is, in some sense, a “sub-product” of the counting numbers 1, 2, 3, 4, …

The Number Hierarchy

Here’s the containment relationship between these number sets:

$$\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}$$

Every natural number is an integer. Every integer is a rational. Every rational is a real. Every real is a complex number (with imaginary part zero).

This hierarchy represents thousands of years of mathematical development. The natural numbers have been around as long as humans could count. Negative numbers took centuries to be accepted (even in the 18th century, some mathematicians called them “absurd”). Irrational numbers were discovered by the Greeks around 500 BC. Complex numbers weren’t fully accepted until the 19th century.

Each extension solved a problem. Integers let you subtract. Rationals let you divide. Reals let you take limits. Complex numbers let you take square roots of negatives and solve all polynomial equations.

There are further extensions (quaternions, octonions), but they come with tradeoffs. Quaternions sacrifice commutativity. Octonions sacrifice associativity. For most purposes, complex numbers are where the story ends.

Even and Odd Numbers

Now let’s explore special types of numbers within the integers. The simplest classification is even versus odd.

An integer is even if it’s divisible by 2. Formally, \( n \) is even if \( n = 2k \) for some integer \( k \).

Examples: 0, \( \pm 2 \), \( \pm 4 \), \( \pm 6 \), \( \pm 8 \), …, \( \pm 2n \), …

An integer is odd if it’s not divisible by 2. Formally, \( n \) is odd if \( n = 2k + 1 \) for some integer \( k \).

Examples: \( \pm 1 \), \( \pm 3 \), \( \pm 5 \), \( \pm 7 \), …, \( \pm (2n + 1) \), …

Every integer is either even or odd. Never both. Never neither.

The arithmetic of even and odd numbers follows simple rules. Even plus even is even. Odd plus odd is even. Even plus odd is odd. Even times anything is even. Odd times odd is odd.

These rules are often the key to solving competition math problems. If you need to prove something about parity (evenness or oddness), these rules are your tools.

Prime Numbers: The Atoms of Arithmetic

A prime number is an integer greater than 1 whose only positive divisors are 1 and itself.

The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, …

The number 2 is special. It’s the only even prime. Every other prime is odd.

A number greater than 1 that isn’t prime is called composite. It can be written as a product of smaller factors. For example, 12 = 2 × 2 × 3.

The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be written uniquely as a product of primes (up to reordering). This is why primes are called the “atoms” of arithmetic. They’re the indivisible building blocks from which all other integers are constructed.

There are infinitely many primes. Euclid proved this around 300 BC, and his proof is still one of the most elegant in mathematics. Suppose there are only finitely many primes. Multiply them all together and add 1. This new number isn’t divisible by any prime in your list (it leaves remainder 1 when divided by each). So either it’s a new prime, or it has a prime factor not in your list. Either way, contradiction.

Despite knowing there are infinitely many primes, we still don’t fully understand their distribution. The Prime Number Theorem says roughly how many primes there are below a given number, but the exact positions of primes remain mysterious. The Riemann Hypothesis, worth a million-dollar prize, would tell us much more about this distribution.

Twin Primes

Twin primes are pairs of primes that differ by exactly 2.

Examples: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73), …

The Twin Prime Conjecture asserts that there are infinitely many twin prime pairs. This remains unproven, despite being one of the most famous open problems in number theory.

In 2013, Yitang Zhang proved that there are infinitely many prime pairs differing by at most 70 million. This was a breakthrough. The bound has since been reduced to 246 through the work of James Maynard and the Polymath project. But getting it down to 2 (which would prove the Twin Prime Conjecture) remains out of reach.

Mersenne Primes

A Mersenne number has the form \( M_n = 2^n – 1 \) for positive integer \( n \).

The first few Mersenne numbers are: \( M_1 = 1 \), \( M_2 = 3 \), \( M_3 = 7 \), \( M_4 = 15 \), \( M_5 = 31 \), \( M_6 = 63 \), \( M_7 = 127 \), …

When a Mersenne number happens to be prime, it’s called a Mersenne prime. The Mersenne primes in the list above are 3, 7, 31, and 127.

For \( M_n \) to be prime, \( n \) itself must be prime (though this isn’t sufficient). When \( n \) is composite, say \( n = ab \), then \( 2^a – 1 \) divides \( 2^{ab} – 1 \).

Mersenne primes are important because they’re connected to perfect numbers (more on that later) and because they’re often the largest known primes. The Great Internet Mersenne Prime Search (GIMPS) has found many record-breaking primes. As of 2024, the largest known prime is \( 2^{82,589,933} – 1 \), a Mersenne prime with over 24 million digits.

Only 51 Mersenne primes are known. Whether infinitely many exist is an open question.

Pseudoprimes and Carmichael Numbers

Ancient Chinese mathematicians proposed a primality test. A number \( n \) is prime if and only if \( n \) divides \( 2^n – 2 \).

This works for small numbers. It’s true for all \( n \leq 340 \). But it fails at 341.

The number 341 = 11 × 31 is composite, but \( 341 \mid 2^{341} – 2 \). Numbers like this are called pseudoprimes (specifically, Fermat pseudoprimes to base 2). They pass the test but aren’t actually prime.

There are infinitely many pseudoprimes. Others include 561, 645, 1105, and 1387.

Even worse, some composite numbers are pseudoprimes to every base. These are called Carmichael numbers or absolute pseudoprimes.

A Carmichael number \( n \) satisfies \( n \mid a^n – a \) for all integers \( a \). The first Carmichael number is 561. Others include 1105, 1729, 2465, 2821, and 6601.

Carmichael numbers fool the Fermat primality test completely. You need more sophisticated tests (like Miller-Rabin) to handle them. There are infinitely many Carmichael numbers, as proven in 1994 by Alford, Granville, and Pomerance.

Sophie Germain Primes

A Sophie Germain prime is an odd prime \( p \) such that \( 2p + 1 \) is also prime.

The prime \( 2p + 1 \) is called a safe prime.

Examples: 2 (since 5 is prime), 3 (since 7 is prime), 5 (since 11 is prime), 11 (since 23 is prime), 23 (since 47 is prime), …

These primes are named after Sophie Germain, an 18th-19th century French mathematician who made significant contributions to number theory despite facing enormous obstacles as a woman in mathematics. She used Sophie Germain primes in her work toward proving Fermat’s Last Theorem.

Sophie Germain primes have applications in cryptography. Safe primes (which are \( 2p + 1 \) for Sophie Germain primes \( p \)) are used in Diffie-Hellman key exchange because they have certain security properties.

Whether there are infinitely many Sophie Germain primes is unknown.

Relatively Prime Numbers

Two integers are called relatively prime (or coprime) if their greatest common divisor is 1. They share no common prime factors.

Examples: 7 and 9 are relatively prime. Their divisors are {1, 7} and {1, 3, 9}, with only 1 in common.

Similarly, 15 and 49 are relatively prime. The divisors of 15 are {1, 3, 5, 15} and the divisors of 49 are {1, 7, 49}. Only 1 is shared.

Note that neither number needs to be prime. The numbers 8 and 15 are both composite, but they’re relatively prime because \( \gcd(8, 15) = 1 \).

The concept of relative primality is central to modular arithmetic and cryptography. The RSA encryption system, which secures most of internet commerce, relies fundamentally on properties of relatively prime numbers.

Perfect Numbers

A perfect number equals the sum of its proper divisors (all divisors except itself).

The smallest perfect number is 6. Its proper divisors are 1, 2, and 3. And indeed, \( 1 + 2 + 3 = 6 \).

The next perfect number is 28. Its proper divisors are 1, 2, 4, 7, and 14. Check: \( 1 + 2 + 4 + 7 + 14 = 28 \). Perfect.

The first few perfect numbers are 6, 28, 496, 8128, 33550336, and 8589869056. They get rare very quickly.

Euclid discovered a remarkable connection. If \( 2^p – 1 \) is prime (a Mersenne prime), then \( 2^{p-1}(2^p – 1) \) is perfect. This generates all known perfect numbers.

For example, \( 2^2 – 1 = 3 \) is prime, so \( 2^1 \times 3 = 6 \) is perfect. And \( 2^3 – 1 = 7 \) is prime, so \( 2^2 \times 7 = 28 \) is perfect.

Euler proved the converse: every even perfect number has this form. So finding even perfect numbers is equivalent to finding Mersenne primes.

What about odd perfect numbers? No one has ever found one. No one has proven they don’t exist. It’s one of the oldest open problems in mathematics, dating back over 2000 years. If an odd perfect number exists, it must be larger than \( 10^{1500} \) and have very specific properties. Most mathematicians suspect none exist.

Triangular Numbers

The triangular numbers are 1, 3, 6, 10, 15, 21, 28, 36, 45, …

The \( n \)-th triangular number is the sum of the first \( n \) natural numbers:

$$T_n = 1 + 2 + 3 + \ldots + n = \frac{n(n+1)}{2}$$

The name comes from geometry. You can arrange \( T_n \) dots in a triangular pattern with \( n \) dots on each side.

For example, \( T_4 = 10 \) looks like:

• (1 dot)
• • (2 dots)
• • • (3 dots)
• • • • (4 dots)

Total: 1 + 2 + 3 + 4 = 10 dots.

Triangular numbers have many interesting properties. Every perfect number is triangular (since \( 2^{p-1}(2^p – 1) = T_{2^p – 1} \)). The sum of two consecutive triangular numbers is always a square number. And the sum of the first \( n \) cubes equals the square of the \( n \)-th triangular number: \( 1^3 + 2^3 + \ldots + n^3 = T_n^2 \).

Square Numbers

A square number is \( n^2 \) for some natural number \( n \).

The first few are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, …

Geometrically, \( n^2 \) is the number of dots in an \( n \times n \) square grid.

Square numbers have distinctive properties. The difference between consecutive squares is always odd: \( n^2 – (n-1)^2 = 2n – 1 \). Every square leaves remainder 0 or 1 when divided by 4. And no square can end in 2, 3, 7, or 8.

Fermat proved that every prime of the form \( 4k + 1 \) can be written as a sum of two squares, while primes of the form \( 4k + 3 \) cannot. This is a beautiful result connecting number theory and geometry.

Catalan Numbers

The Catalan numbers form one of the most ubiquitous sequences in combinatorics:

1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, …

The \( n \)-th Catalan number is given by:

$$C_n = \frac{1}{n+1}\binom{2n}{n} = \frac{(2n)!}{n!(n+1)!}$$

These numbers count an astonishing variety of things. The number of ways to correctly match \( n \) pairs of parentheses. The number of different binary trees with \( n \) nodes. The number of ways to triangulate a convex polygon with \( n + 2 \) sides. The number of paths from (0,0) to (n,n) that never go above the diagonal.

The Catalan numbers appear in probability, computer science, biology, and physics. Their ubiquity suggests they represent something fundamental about the structure of discrete objects.

Palindromic Numbers

A palindromic number reads the same forwards and backwards.

Examples: 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, 131, …

Larger examples include 12321, 45654, 1234321, and 999999999.

Palindromic primes are prime numbers that are also palindromes. Examples: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, …

An interesting conjecture (still unproven): take any number, reverse its digits, and add. Repeat until you get a palindrome. It’s believed this process always terminates, but the number 196 has been tested through billions of iterations without producing a palindrome. Whether it ever does remains unknown.

E-Primes (Even Primes)

Here’s an unusual definition. An e-prime is an even positive integer that cannot be expressed as the product of two other even integers.

The e-primes are 2, 6, 10, 14, 18, 22, 26, … (the numbers of form \( 4k + 2 \)).

Why? Consider 12. It can be written as 2 × 6 or 4 × 3. But 4 × 3 uses an odd number, so that doesn’t count. And 2 × 6 uses 2 and 6, both even. So 12 is e-composite.

But 6 cannot be written as a product of two even numbers (you’d need 2 × 3, but 3 is odd). So 6 is e-prime.

This is mostly a curiosity, but it illustrates how the concept of “prime” depends on the multiplication system you’re working in.

The Endless Zoo

I’ve only scratched the surface. Mathematicians have named and studied hundreds of special number types. Fibonacci numbers, Lucas numbers, happy numbers, amicable numbers, sociable numbers, untouchable numbers, weird numbers, vampire numbers, Keith numbers, narcissistic numbers, and on and on.

Some of these are deeply important (primes, perfect numbers, Catalan numbers). Some are recreational curiosities (palindromes, happy numbers). But each one reveals something about the structure lurking within the integers.

And remember Kronecker’s insight. All of this complexity emerges from the simplest possible starting point: the counting numbers 1, 2, 3, 4, …

God created the natural numbers. Everything else is mathematics.

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