Arithmetic is the shared toolbox under higher math
Arithmetic is the branch of mathematics built on adding, subtracting, multiplying and dividing, and more broadly on powers, roots and logarithms. Its name comes from Greek arithmos, number. It sits beneath algebra, calculus and statistics, and turns up whenever someone counts change at a shop or balances a household budget.
People have been doing arithmetic for thousands of years, possibly tens of thousands. Egyptians and Sumerians built numeral systems for practical sums around 3000 BCE. From the 7th and 6th centuries BCE Greek thinkers began treating numbers abstractly and demanded rigorous proof. Indian mathematicians developed zero and decimal notation, which Arab scholars refined and passed westward in the Middle Ages. Mechanical calculators appeared in the 17th century.
Scholars disagree about its boundaries. A narrow view limits it to natural numbers; the more common one takes in integers, fractions, real numbers and sometimes complex ones. Number theory, once called higher arithmetic, narrows the focus to integers and questions such as divisibility and primes. Cardinal numbers answer how many, ordinals which position. A rational number is a ratio of integers and always gives a decimal that ends or repeats, while irrationals such as pi or the diagonal of a unit square run on forever without a pattern.
How numbers are written matters. Early systems were all non-positional: tally marks repeat one stroke, Egyptian hieroglyphs had separate signs for ten, a hundred, a thousand and ten thousand, so 10,405 needed one, four and five of the relevant symbols, and Roman numerals used seven letters from I to M. In positional systems a digit's place sets its value, which is why 532, 325 and 253 differ despite sharing digits. The Babylonians invented the first such system, in base 60. Everyday life runs on base 10, computers on base 2, with each binary digit a single bit.
Each operation has an identity that changes nothing, such as zero for adding, and inverses that undo it: minus six cancels six, and subtraction reverses addition. Addition and multiplication can be reordered and regrouped freely. Applied to variables rather than specific numbers, the same moves become algebraic operations.
Source: Arithmetic
