The smallest number that is 0 is, simply put, 0 itself. Zero is the numerical digit used to represent the absence of quantity or value. It is the starting point of the number line and is considered the smallest non-negative integer. In mathematics, zero is the additive identity, meaning that adding zero to any number does not change the value of that number. This concept is foundational in various mathematical operations and theories.

The significance of zero in mathematics is further underscored by its role in the development of the positional number system, which is the most commonly used numeral system in the world today. This system allows for the representation of any number using a combination of digits, with the value of each digit determined by its position within the number. Zero is crucial in this system as it enables the representation of numbers with fewer digits than would be required in other numeral systems, such as the Roman numeral system.

Data supporting the importance of zero in mathematics can be found in the historical context of its development. For instance, the concept of zero was first introduced in ancient India around the 3rd century AD by Brahmagupta, an Indian mathematician. Brahmagupta's work, "Brahmasphuta-Siddhanta," was the first to provide a systematic treatment of zero and negative numbers, as well as the rules for their arithmetic operations. This work had a profound impact on the development of mathematics in India and later in the Islamic world, and eventually in Europe.

In modern mathematics, zero is a critical component of various mathematical structures, including rings, fields, and vector spaces. For example, in ring theory, zero is the additive identity, and in field theory, it is the neutral element for multiplication. These properties make zero an essential element in the study of abstract algebra and other advanced mathematical topics.

Moreover, zero plays a vital role in the concept of infinity. In calculus, for instance, the limit of a function as it approaches zero can reveal important properties of the function, such as its continuity or differentiability. In set theory, the empty set, which contains no elements, is often denoted by the symbol ∅, and its cardinality is defined as zero.

In conclusion, zero is the smallest number that is 0, and its significance in mathematics is undeniable. Its introduction revolutionized the way we represent and manipulate numbers, and its properties are fundamental to various mathematical theories and applications.