数论在纯数学中扮演着重要的角色，具有简单性以及与许多其他数学领域之间的关联性，比如代数几何和算术几何。学习它的主要目的是发现自然数之间有趣和意想不到的关系。通过熟悉数论中的许多概念，如算术基本定理、二次剩余理论、丢番图逼近理论等，能够学习更多的前沿课题，如椭圆曲线理论、代数和解析数论。数论是纯粹数学的分支之一，主要研究整数的性质。整数可以是方程式的解(丢番图方程)。有些解析函数(像黎曼ζ函数)中包括了一些整数、质数的性质，透过这些函数也可以了解一些数论的问题。透过数论也可以建立实数和有理数之间的关系，并且用有理数来逼近实数(丢番图逼近)。按研究方法来看，数论大致可分为初等数论和高等数论。初等数论是用初等方法研究的数论，它的研究方法本质上说，就是利用整数环的整除性质，主要包括整除理论、同余理论、连分数理论。高等数论则包括了更为深刻的数学研究工具。它大致包括代数数论、解析数论、计算数论等等。数论早期称为算术。到20世纪初，才开始使用数论的名称，而算术一词则表示"基本运算"，不过在20世纪的后半，有部份数学家仍会用"算术"一词来表示数论。1952年时数学家Harold Davenport仍用"高等算术"一词来表示数论，戈弗雷·哈罗德·哈代和爱德华·梅特兰·赖特在1938年写《数论介绍》简介时曾提到"我们曾考虑过将书名改为《算术介绍》，某方面而言是更合适的书名，但也容易让读者误会其中的内容"。
Number theory plays an important role in pure mathematics, with its simplicity and connections to many other mathematical fields, such as algebraic geometry and arithmetic geometry. The main purpose of studying it is to discover interesting and unexpected relationships between natural numbers. By being familiar with many concepts in number theory, such as the fundamental theorem of arithmetic, quadratic residue theory, Diophantine approximation theory, etc., I can learn more frontier topics, such as elliptic curve theory, algebra and analytic number theory.Number theory is a branch of pure mathematics that deals with the properties of integers. Integers can be solutions to equations (Diophantine equations). Some analytic functions (such as the Riemann zeta function) include some properties of integers and prime numbers, which can also be used to understand some of the problems of number theory.
The relationship between real numbers and rational numbers can also be established through number theory, and the relation numbers can be used to approximate the real numbers (DIopHANtine APPROXIMATION). According to the research methods, number theory can be roughly divided into elementary number theory and advanced number theory. Elementary number theory is the theory of number studied by elementary methods. Its research methods essentially say that it uses the divisible properties of integer rings, including the theory of division, the theory of congruence and the theory of continued fractions. Advanced number theory includes more profound mathematical research tools. It roughly includes algebraic number theory, analytic number theory, computational number theory and so on.
In the early days of number theory it was called arithmetic. It was not until the beginning OF the 20TH CENTURY that THE name NUMBER THEORY began to be used, and the term ARITHMETIC meant "fundamental operations," although in the second half of the 20th century, some mathematicians still used the term ARITHMETIC to mean number theory. In 1952 the mathematician Harold Davenport was still using the term "higher arithmetic" to mean number theory, and Godfrey Harold Hardy and Edward Maitland Wright wrote in their 1938 Introduction to Number Theory that "we have considered changing the title to Introduction to Arithmetic, which in some ways would be a more appropriate title, But it's also easy for the reader to misunderstand the content."