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Arithmetics (from the Greek word ἀριθμός, arithmos "number") is the most ancient branch of mathematics, the science of numbers and operations on sets of numbers, used very popularly, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of operations that combine numbers. In common usage, it refers to the simpler properties when using the traditional operations of addition, subtraction, multiplication and division with smaller values of numbers. Arithmetic is closely connected with algebra, which includes the study of operations performed on numbers. Professional mathematicians sometimes use the term higher arithmetic when referring to more advanced results related to number theory, but this should not be confused with elementary arithmetic.

The art of computation arose and developed long before the times of the oldest written records extant. The prehistory of arithmetic is limited to a small number of artifacts which may indicate the conception of addition and subtraction, the best-known being the Ishango bone from central Africa, dating from somewhere between 20,000 and 18,000 BC, the Cahoon papyri and the famous Rhind papyrus (about 2000 BC).

The earliest written records indicate the Egyptians and Babylonians used all the elementary arithmetic operations as early as 2000 BC. The hieroglyphic system for Egyptian numerals, like the later Roman numerals, descended from tally marks used for counting. In both cases, this origin resulted in values that used a decimal base but did not include positional notation. Complex calculations with Roman numerals required the assistance of a counting board or the Roman abacus to obtain the results.

Early number systems that included positional notation were not decimal, including the clumsy hexadecimal (sexagesimal) base 60) system for Babylonian numerals and the vigesimal (base 20) system that defined Maya numerals. Because of this place-value concept, the ability to reuse the same digits for different values contributed to simpler and more efficient methods of calculation.

The continuous historical development of modern arithmetic starts with the Hellenistic civilization of ancient Greece, although it originated much later than the Babylonian and Egyptian examples. Prior to the works of Euclid around 300 BC, Greek studies in mathematics overlapped with philosophical and mystical beliefs. For example, Nicomachus summarized the viewpoint of the earlier Pythagorean approach to numbers and their relationships to each other in his Introduction to Arithmetic.

Greek numerals were used by Archimedes, Diophantus and others in a positional notation not very different from ours. Because the ancient Greeks lacked a symbol for zero (until the Hellenistic period), they used three separate sets of symbols. One set for the unit’s place, one for the ten’s place and one for the hundred’s. Then for the thousand’s place they would reuse the symbols for the unit’s place, and so on. Their addition algorithm was identical to ours, and their multiplication algorithm was only very slightly different. Their long division algorithm was the same, and Archimedes, who may have invented it, knew the square root algorithm that was once taught in school. He preferred it to Hero’s method of successive approximation because, once computed, a digit doesn’t change, and the square roots of perfect squares, such as 7485696, terminate immediately as 2736. For numbers with a fractional part, such as 546.934, they used negative powers of 60 instead of negative powers of 10 for the fractional part 0.934. The ancient Chinese used a similar positional notation. Because they also lacked a symbol for zero, they had one set of symbols for the unit’s place and a second set for the ten’s place. For the hundred’s place they then reused the symbols for the unit’s place, and so on. Their symbols were based on the ancient counting rods. It is a complicated question to determine exactly when the Chinese started calculating with positional representation, but it was definitely before 400 BC.

Very little is known about the initial development of arithmetic in India. The simplest fractions were utilized in India long before the Christian era. Our own decimal computation system is of Indian origin. The earliest written Indian mathematical records extant were compiled in the 5^th century and indicate that the knowledge of arithmetic in India of that period was of a high standard. Indian mathematicians operated on integers and fractions using methods very similar to our own. They solved many problems on proportions and on the rule-of-three and could compute percentages. Studies on negative numbers began in India in the 7^th century.

The gradual development of Hindu-Arabic numerals independently devised the place-value concept and positional notation, which combined the simpler methods for computations with a decimal base and the use of a digit representing 0. This allowed the system to represent both large and small integers. This approach eventually replaced all other systems. In the early 6^th century AD, the Indian mathematician Aryabhata incorporated an existing version of this system in his work and experimented with different notations. In the 7^th century, Brahmagupta established the use of 0 as a separate number and determined the results for multiplication, division, addition and subtraction of zero and all other numbers, except for the result of division by 0. His contemporary, the Syriac bishop Severus Sebokht described the excellence of this system as "... valuable methods of calculation which surpass description". The Arabs also learned this new method and called it hesab.

Leibniz’s Stepped Reckoner was the first calculator that could perform all four arithmetic operations.

Although the Codex Vigilanus described an early form of Arabic numerals (omitting 0) by 976 AD, Fibonacci was primarily responsible for spreading their use throughout Europe after the publication of his book Liber Abaci in 1202. He considered the significance of this "new" representation of numbers, which he styled the "Method of the Indians" (Latin Modus Indorum), so fundamental that all related mathematical foundations including the results of Pythagoras and the algorism describing the methods for performing actual calculations, were "almost a mistake" in comparison.

In the Middle Ages, arithmetic was one of the seven liberal arts taught in universities.

The flourishing of algebra in the medieval Islamic world and in Renaissance Europe was an outgrowth of the enormous simplification of computation through decimals notation.

Various types of tools exist to assist in numeric calculations. Examples include slide rules (for multiplication, division and trigonometry) and monographs in addition to the electrical calculator.

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