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Addition — Grokipedia Fact-checked by Grok 3 months ago Addition Ara Eve Leo Sal 1x Addition is one of the four basic operations of arithmetic , alongside subtraction , multiplication , and division; it consists of combining two or more quantities, known as addends, to produce their total, called the sum, and is typically denoted by the plus sign (+). [1] This operation exhibits key properties that underpin its role in mathematics. Addition is commutative , meaning the order of addends does not change the sum: a + b = b + a a + b = b + a a + b = b + a . It is associative , allowing the grouping of addends to vary without affecting the result: ( a + b ) + c = a + ( b + c ) (a + b) + c = a + (b + c) ( a + b ) + c = a + ( b + c ) . Furthermore, zero acts as the additive identity element , such that a + 0 = a a + 0 = a a + 0 = a for any addend a a a . These properties hold for real numbers and extend to other algebraic structures. [2] [3] The origins of addition trace back to ancient civilizations, where it served practical purposes like counting and measurement. In Egyptian mathematics, documented in papyri such as the Moscow Papyrus (c. 1850 B.C.), addition was performed using a base-10 grouping system to combine symbols representing powers of 10, facilitating tasks in accounting and land surveying. The modern plus symbol (+) first appeared in a 1456 German manuscript , evolving from earlier notations like the word "et" for combining terms. [4] [5] Beyond basic arithmetic , addition generalizes to abstract domains, including vector addition in geometry —where vectors are combined head-to-tail—and matrix addition in linear algebra , aligning corresponding elements. It forms the foundation for advanced concepts, such as limits and integrals in calculus , and is implemented in computing through algorithms like binary addition for digital circuits. [6] Notation and Terminology Notation The plus sign (+) serves as the standard binary operator for addition in mathematics, denoting the operation of combining two quantities. This symbol, derived from the Latin word "et" meaning "and," was first introduced in print by the German mathematician Johannes Widmann in his 1489 arithmetic treatise Behende und hupsche Rechnung auf allen kauffmanschafft to represent surplus or addition in accounting contexts. [7] In inline notation, addition is typically expressed as a + b a + b a + b , where a a a and b b b are the operands, such as in the arithmetic example 2 + 3 = 5 2 + 3 = 5 2 + 3 = 5 . For the summation of multiple terms, the uppercase Greek letter sigma ( Σ \Sigma Σ ) is used in display form, as introduced by Leonhard Euler in 1755 to compactly represent repeated additions, for instance ∑ i = 1 n i = n ( n + 1 ) 2 \sum_{i=1}^{n} i = \frac{n(n+1)}{2} ∑ i = 1 n ​ i = 2 n ( n + 1 ) ​ . This distinguishes finite summation from binary addition, though Σ \Sigma Σ generalizes the concept of + + + over a sequence. [5] Variations appear in specialized mathematical structures. For vector addition, the operator remains + + + , written as a ⃗ + b ⃗ \vec{a} + \vec{b} a + b , combining corresponding components. In matrix addition, the + operator is used to add corresponding elements element-wise. In Boolean algebra and logic, the vee symbol ∨ \vee ∨ denotes disjunction, serving as an analogy to addition under modulo-2 arithmetic. The notation supports commutativity, where a + b = b + a a + b = b + a a + b = b + a . [5] Terminology In mathematics , the numbers or quantities being added together in an operation are known as addends, with each individual operand referred to as an addend. [8] [9] The result of this addition is called the sum, which represents the total obtained by combining the addends. [10] When addition involves a sequence of multiple terms, such as in summation , each term in the sequence is termed a summand, a usage that emphasizes the additive process over multiple elements. [11] In some contexts, particularly historical or specific instructional materials , the term addendum is used interchangeably with addend to denote each number being added, though it is less common today. [12] An older distinction identifies the first addend as the augend, to which subsequent addends are applied, as seen in expressions like augend + addend = sum; however, due to the commutative nature of addition, this terminology is rarely emphasized in modern usage. [10] [13] Addition is fundamentally a binary operation , involving exactly two operands, whereas extending it to more than two terms results in n-ary summation , where multiple summands are combined iteratively. [14] [15] For example, in the equation 3 + 4 = 7, the addends are 3 and 4, and the sum is 7. [8] Definitions and Interpretations Combining Sets In set theory , addition of natural numbers can be understood as the operation of combining two disjoint sets to form their union, with the resulting size given by the sum of the individual sizes, or cardinalities . For disjoint sets A A A and B B B , the cardinality of the union satisfies ∣ A ∪ B ∣ = ∣ A ∣ + ∣ B ∣ |A \cup B| = |A| + |B| ∣ A ∪ B ∣ = ∣ A ∣ + ∣ B ∣ , providing a foundational interpretation of addition where the natural numbers represent sizes of finite sets./01%3A__Sets/1.04%3A_Set_Operations_with_Two_Sets) This perspective traces back to the Peano axioms , formulated by Giuseppe Peano in 1889, which axiomatize the structure of natural numbers and admit models in set theory where numbers are constructed as sets (for instance, via the von Neumann ordinals) and addition aligns with disjoint union of such sets. Example Consider the disjoint sets A = { 1 , 2 } A = \{1, 2\} A = { 1 , 2 } and B = { 3 , 4 } B = \{3, 4\} B = { 3 , 4 } . Their union is { 1 , 2 , 3 , 4 } \{1, 2, 3, 4\} { 1 , 2 , 3 , 4 } , which has cardinality 4, matching ∣ A ∣ + ∣ B ∣ = 2 + 2 |A| + |B| = 2 + 2 ∣ A ∣ + ∣ B ∣ = 2 + 2 ./01%3A__Sets/1.04%3A_Set_Operations_with_Two_Sets) To accommodate repetitions, the interpretation extends to multisets, where addition combines two multisets by summing the multiplicities of shared elements, yielding a cardinality that is the sum of the input cardinalities (each defined as the total of multiplicities). [16] Extending Lengths In the geometric interpretation of addition, lengths are added by concatenating line segments on a number line, where the sum represents the total distance from the origin to the endpoint of the combined segments. For instance, starting at 0 and moving 2 units to the right places one at point 2; adding another 3 units extends the path further right to point 5, illustrating that 2 + 3 = 5. This model emphasizes addition as a process of successive displacements or extensions along a continuous line, providing an intuitive basis for understanding positive integers before extending to other numbers. [17] A physical analogy for this interpretation involves combining tangible objects like rods or measuring tapes end-to-end to form a longer segment, where the total length equals the sum of the individual lengths. This approach mirrors real-world measurements, such as aligning two rods —one of 2 centimeters and another of 3 centimeters—to obtain a combined rod of 5 centimeters, directly observable and verifiable by rulers or calipers . Such manipulations highlight addition's role in quantifying cumulative extents in physical space, distinct from discrete counting but analogous in building totals incrementally. [1] [18] The segment addition postulate formalizes this in Euclidean geometry : if points A, B, and C are collinear with B between A and C, then the length of AC equals the sum of AB and BC. For example, if AB measures 2 cm and BC measures 3 cm, then AC measures 5 cm, as the segments AB and BC concatenate without overlap to span AC. This postulate underpins geometric proofs involving collinear points and extends the intuitive rod-joining idea to rigorous deduction. [19] This length-extension view connects to the real numbers through the construction of reals as limits of rational approximations, where addition of irrationals or transcendentals inherits the rational addition laws via convergence. Every real number serves as the limit of a sequence of rationals, allowing sums like √2 + π to be defined as the limit of sums of rational approximations, preserving the geometric continuity of the number line while filling gaps left by rationals alone. This ties the intuitive concatenation of finite lengths to the complete, dense structure of the reals. [20] Other Interpretations In logic, particularly within Boolean algebra , the disjunction operation (p ∨ q) can be interpreted as a form of addition of truth values, where the result is true if at least one of the propositions is true, analogous to Boolean addition that yields 1 (true) unless both inputs are 0 (false). [21] This view treats truth values as elements in a structure where disjunction acts like summation without carry-over, preserving the "or" semantics in computational and logical systems. [21] Addition also manifests in temporal contexts as the concatenation of durations, combining intervals of time to yield a total span, such as adding 2 hours to 3 hours to obtain 5 hours. [22] This process relies on additive principles similar to numerical summation but applied to measurable time units, often involving fractional components like minutes or seconds to ensure precise alignment. [22] In financial applications, addition serves to combine quantities or amounts, such as aggregating debts or assets to determine total obligations, exemplified by summing $75 owed to one party and $25 to another to reach a $100 total. [23] This interpretation underscores addition's role in accounting and economics for balancing ledgers or calculating net worth through the merger of monetary values. [23] A notable example appears in programming, where the plus operator (+) facilitates string concatenation, effectively "adding" textual elements end-to-end, as in combi