# Chapter 1. Foundations of Arithmetic

## Counting and Quantity

The natural numbers arise from the basic act of counting. When we count objects in a collection, we assign successive numbers:

$$
1,2,3,4,\ldots
$$

These numbers are called the natural numbers. The standard notation for the set of natural numbers is

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

Some authors include $0$ in $\mathbb{N}$, while others begin with $1$. In this text, we shall usually write

$$
\mathbb{N}_0=\{0,1,2,3,\ldots\}
$$

when zero is included explicitly.

Natural numbers measure the size of finite collections. If a set contains five objects, we say its cardinality is $5$. Thus the natural numbers provide the first mathematical model of discrete quantity.

## The Successor Principle

The natural numbers are ordered sequentially. Every natural number has a unique next number, called its successor. If $n$ is a natural number, its successor is

$$
n+1.
$$

The process never terminates. Starting from $1$, repeated application of the successor operation generates all natural numbers:

$$
1,2,3,4,\ldots
$$

This simple observation contains the essential infinite character of arithmetic.

The natural numbers satisfy the following fundamental properties:

1. $1$ is a natural number.
2. Every natural number has a unique successor.
3. Distinct natural numbers have distinct successors.
4. $1$ is not the successor of any natural number.
5. Any set containing $1$ and closed under successors contains all natural numbers.

The fifth property is the principle of mathematical induction, which will later become one of the central proof techniques in number theory.

## Arithmetic Operations

The basic arithmetic operations on natural numbers are addition and multiplication.

Addition combines quantities. For example,

$$
3+2=5.
$$

Multiplication represents repeated addition:

$$
3\cdot4=4+4+4=12.
$$

These operations satisfy several fundamental laws.

### Commutative Laws

$$
a+b=b+a
$$

and

$$
ab=ba.
$$

### Associative Laws

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

and

$$
(ab)c=a(bc).
$$

### Distributive Law

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

These algebraic identities form the structural foundation of arithmetic.

## Order Structure

Natural numbers possess a natural ordering. Given two natural numbers $a$ and $b$, exactly one of the following holds:

$$
a<b,\qquad a=b,\qquad a>b.
$$

The order relation is compatible with arithmetic. If

$$
a<b,
$$

then

$$
a+c<b+c
$$

for every natural number $c$. Similarly, if $c>0$, then

$$
ac<bc.
$$

The ordered structure of the natural numbers allows arithmetic to interact with geometry, measurement, and analysis.

## Infinity of the Natural Numbers

The set of natural numbers is infinite. No largest natural number exists. Indeed, if $n$ is any natural number, then

$$
n+1>n.
$$

Thus every natural number is followed by a larger one.

This property distinguishes arithmetic from finite systems. The infinite progression of the natural numbers forms the basis for much of mathematics, including algebra, analysis, topology, and modern number theory.

## Natural Numbers in Number Theory

Number theory studies the arithmetic properties of integers, especially divisibility and prime numbers. The natural numbers provide the starting point for all such investigations.

Questions such as

$$
\text{Is } n \text{ prime?}
$$

or

$$
a^2+b^2=c^2
$$

begin entirely within the arithmetic of natural numbers.

Although modern number theory eventually reaches deep areas such as modular forms, elliptic curves, and automorphic representations, its foundations remain rooted in the elementary structure of counting numbers.