A numeral system is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner.

The same sequence of symbols may represent different numbers in different numeral systems. For example, "11" represents the number eleven in the decimal or base-10 numeral system (today, the most common system globally), the number three in the binary or base-2 numeral system (used in modern computers), and the number two in the unary numeral system (used in tallying scores).

The number the numeral represents is called its value. Additionally, not all number systems can represent the same set of numbers; for example, Roman, Greek, and Egyptian numerals don't have an official representation of the number zero.

Ideally, a numeral system will:

For example, the usual decimal representation gives every nonzero natural number a unique representation as a finite sequence of digits, beginning with a non-zero digit.

Numeral systems are sometimes called number systems, but that name is ambiguous, as it could refer to different systems of numbers, such as the system of real numbers, the system of complex numbers, various hypercomplex number systems, the system of p-adic numbers, etc. Such systems are, however, not the topic of this article.

Early numeral systems varied across civilizations, with the Babylonians using a base-60 system, the Egyptians developing hieroglyphic numerals, and the Chinese employing rod numerals. The Mayans independently created a vigesimal (base-20) system that included a symbol for zero. Indian mathematicians, such as Brahmagupta in the 7th century, played a crucial role in formalizing arithmetic rules and the concept of zero, which was later refined by scholars like Al-Khwarizmi in the Islamic world. As these numeral systems evolved, the efficiency of positional notation and the inclusion of zero helped shape modern numerical representation, influencing global commerce, science, and technology. The first true written positional numeral system is considered to be the Hindu–Arabic numeral system. This system was established by the 7th century in India, but was not yet in its modern form because the use of the digit zero had not yet been widely accepted. Instead of a zero sometimes the digits were marked with dots to indicate their significance, or a space was used as a placeholder. The first widely acknowledged use of zero was in 876. The original numerals were very similar to the modern ones, even down to the glyphs used to represent digits.

By the 13th century, Western Arabic numerals were accepted in European mathematical circles (Fibonacci used them in his Liber Abaci). Initially met with resistance, Hindu–Arabic numerals gained wider acceptance in Europe due to their efficiency in arithmetic operations, particularly in banking and trade. The invention of the printing press in the 15th century helped standardize their use, as printed mathematical texts favored this system over Roman numerals. They began to enter common use in the 15th century. By the 17th century, the system had become dominant in scientific works, influencing mathematical advancements by figures like Isaac Newton and René Descartes. In the 19th and 20th centuries, the widespread adoption of Arabic numerals facilitated global finance, engineering, and technological developments, forming the foundation for modern computing and digital data representation. By the end of the 20th century virtually all non-computerized calculations in the world were done with Arabic numerals, which have replaced native numeral systems in most cultures.