Arithmetic

Arithmetic is present in many aspects of daily life, for example, to calculate change while shopping or to manage personal finances.

Ancient civilizations like the Egyptians and the Sumerians invented numeral systems to solve practical arithmetic problems in about 3000 BCE.

Starting in the 7th and 6th centuries BCE, the ancient Greeks initiated a more abstract study of numbers and introduced the method of rigorous mathematical proofs.

The ancient Indians developed the concept of zero and the decimal system, which Arab mathematicians further refined and spread to the Western world during the medieval period.

[8] However, in a more specific sense, number theory is restricted to the study of integers and focuses on their properties and relationships such as divisibility, factorization, and primality.

This helps to simplify mathematical computations by reducing the number of basic arithmetic operations needed to perform calculations.

One way to get an approximate result for the second operation is to employ Newton's method, which uses a series of steps to gradually refine an initial guess until it reaches the desired level of accuracy.

[106] In cases where absolute precision is not required, the problem of calculating arithmetic operations on real numbers is usually addressed by truncation or rounding.

Representing uncertainty using only significant digits is a relatively crude method, with some unintuitive subtleties; explicitly keeping track of an estimate or upper bound of the approximation error is a more sophisticated approach.

[116] The precision of numerical quantities can be expressed uniformly using normalized scientific notation, which is also convenient for concisely representing numbers which are much larger or smaller than 1.

[123] Forms of arithmetic can also be distinguished by the tools employed to perform calculations and include many approaches besides the regular use of pen and paper.

[124] One such technique is the compensation method, which consists in altering the numbers to make the calculation easier and then adjusting the result afterward.

This system relies on mark making, such as strokes drawn on a surface or notches carved into a wooden stick, to keep track of quantities.

[132][f] Sectors and slide rules are more refined calculating instruments that rely on geometric relationships between different scales to perform both basic and advanced arithmetic operations.

[145] Axiomatic foundations of arithmetic try to provide a small set of laws, called axioms, from which all fundamental properties of and operations on numbers can be derived.

[162] This changed with the ancient Greek mathematicians, who began to explore the abstract nature of numbers rather than studying how they are applied to specific problems.

[174] The Indian decimal system was further refined and expanded to non-integers during the Islamic Golden Age by Middle Eastern mathematicians such as Al-Khwarizmi.

[177] In the 16th century, the mathematician Gerolamo Cardano conceived the concept of complex numbers as a way to solve cubic equations.

Elementary arithmetic aims to give students a basic sense of numbers and to familiarize them with fundamental numerical operations like addition, subtraction, multiplication, and division.

[185] It is usually introduced in relation to concrete scenarios, like counting beads, dividing the class into groups of children of the same size, and calculating change when buying items.

Common tools in early arithmetic education are number lines, addition and multiplication tables, counting blocks, and abacuses.

[187] They also show how arithmetic operations are employed in other branches of mathematics, such as their application to describe geometrical shapes and the use of variables in algebra.

It examines how mathematical problems are understood and solved and how arithmetic abilities are related to perception, memory, judgment, and decision making.

This concerns pre-verbal and pre-symbolic cognitive processes implementing arithmetic-like operations required to successfully represent the world and perform tasks like spatial navigation.

A lack of numeracy can inhibit academic success and lead to bad economic decisions in everyday life, for example, by misunderstanding mortgage plans and insurance policies.

[201] Some proponents of this view state that arithmetic knowledge is innate while others claim that there is some form of rational intuition through which mathematical truths can be apprehended.

[202] A more recent alternative view was suggested by naturalist philosophers like Willard Van Orman Quine, who argue that mathematical principles are high-level generalizations that are ultimately grounded in the sensory world as described by the empirical sciences.

In daily life, it is required to calculate change when shopping, manage personal finances, and adjust a cooking recipe for a different number of servings.

These principles also play a key role in calculus in its attempt to determine rates of change and areas under curves.

These operations are used in calculations, problem-solving, data analysis, and algorithms, making them integral to scientific research, technological development, and economic modeling.