Number theory is the study of integers. The integers, denoted by $ \mathbb{Z} $ are the counting numbers $ 1, 2, 3, \dots $, their negatives $ -1, -2, -3, \dots $, and zero: $$ \mathbb{Z} = \{ \dots -3, -2, -1, 0, 1, 2, 3, \dots \}. $$
We will assume the familiar arithmetic of addition, subtraction, and multiplication of integers; integer division will be the first topic in this treatment.
We will also assume the principle of mathematical induction and the well-ordering principle.
The integers and their arithmetic are based on the Peano axioms , which uniquely define the natural numbers $$ \mathbb{N} = \{ 0, 1, 2, 3, \dots \}. $$ The integers $ \mathbb{Z} $ are then defined by extending the natural numbers so that each natural number $ n $ has an additive inverse $ -n $ satisfying $$ n + (-n) = 0. $$
Some treatments of number theory will define the natural numbers as beginning at $ 1 $. The original Peano axioms did just that. There is no consensus in the mathematical community, but the Peano axioms section of the appendix defines them as starting at $ 0 $. To avoid confusion throughout the text, we will denote by $ \mathbb{N}_k $ the natural numbers starting at the integer $ k $, so $$ \begin{align} \mathbb{N}_0 &= \{ 0, 1, 2, 3, \dots \} \\ \mathbb{N}_1 &= \{ 1, 2, 3, 4, \dots \} \\ \mathbb{N}_5 &= \{ 5, 6, 7, 8, \dots \} \\ \vdots \end{align} $$
The sections in the appendix linked above are introductory and not comprehensive. The interested reader might augment those topics with additional sources.