Information AboutAddition |
| CATEGORIES ABOUT ADDITION | |
| elementary arithmetic | |
| binary operations | |
| SHOPPER'S DELIGHT | |
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Addition is the basic Operation of Arithmetic . In its simplest form, addition combines two Number s, the ''addends'' or '' Terms '', into a single number, the ''sum''. Adding more than two numbers can be viewed as repeated addition; this procedure is known as Summation and includes ways to add infinitely many numbers in an Infinite Series . Repeated addition of the number One is the most basic form of Counting . Addition is also defined for mathematical objects other than numbers — for example, Matrices or Polynomial s, often by analogy. NOTATION AND TERMINOLOGY Addition is written using the Plus Sign "+" between the terms; that is, in Infix Notation . The result is expressed with an Equals Sign . For example, :1 + 1 = 2 :2 + 2 = 4 :5 + 4 + 2 = 11 (see "associativity" Below ) :3 + 3 + 3 + 3 = 12 (see "multiplication" Below ) There are also situations where addition is "understood" even though no symbol appears:
The numbers or the objects to be added are generally called the "terms", the "addends", or the "summands"; this terminology carries over to the summation of multiple terms. This is to be distinguished from ''factors'', which are Multiplied . Some authors call the first addend the ''augend''. In fact, during the Renaissance , many authors did not consider the first addend an "addend" at all. Today, due to the symmetry of addition, "augend" is rarely used, and both terms are generally called addends. All of this terminology derives from " are English words derived from the Latin Verb ''addere'', which is in turn a Compound of ''ad'' "to" and ''dare'' "to give", from the Indo-European Root ''do-'' "to give"; thus to ''add'' is to ''give to''. Using the Gerundive Suffix ''-nd'' results in "addend", "thing to be added". Likewise from ''augere'' "to increase", one gets "augend", "thing to be increased". "Sum" and "summand" derive from the Latin Noun ''summa'' "the highest, the top" and associated verb ''summare''. This is appropriate not only because the sum of two positive numbers is greater than either, but because it was once common to add upward, contrary to the modern practice of adding downward, so that a sum was literally higher than the addends. ''Addere'' and ''summare'' date back at least to Boethius , if not to earlier Roman writers such as Vitruvius and Frontinus ; Boethius also used several other terms for the addition operation. The later Middle English terms "adden" and "adding" were popularized by Chaucer . INTERPRETATIONS Addition is used to model countless physical processes. Even for the simple case of adding Natural Number s, there are many possible interpretations and even more visual representations. Combining sets Possibly the most fundamental interpretation of addition lies in combining sets:
This interpretation is easy to visualize, with little danger of ambiguity. However, it is not obvious how one should extend this version of addition to include fractional numbers or negative numbers. See this article for an example of the sophistication involved in adding with sets of "fractional cardinality". One possible fix is to consider collections of objects that can be easily divided, such as Pie s or, still better, segmented rods. Rather than just combining collections of segments, rods can be joined end-to-end. Extending a measure
Under this interpretation, the parts of a sum ''a'' + ''b'' play asymmetric roles; instead of calling both ''a'' and ''b'' addends, it is more appropriate to call ''a'' the augend, since ''a'' plays a passive role. In Geometry , ''a'' might be a Point and ''b'' a Vector ; their sum is then another point, the translation of ''a'' by ''b''. In Analytic Geometry , ''a'' and ''b'' might both be represented by Ordered Pair s of numbers, but they remain conceptually different. Here, the addition operation is not so much a s : This formula is a special case of a law of Exponentiation that may be more familiar for numbers. The unary view is useful, for example, when discussing Subtraction . Addition and subtraction are not inverses as binary operations, but they ''are'' inverses as families of unary operations. This section is under construction. Combining translations
This section is under construction. PROPERTIES Commutativity Addition is Commutative , meaning that one can reverse the terms in a sum left-to-right, and the result will be the same. Symbolically, if ''a'' and ''b'' are any two numbers, then a The fact that addition is commutative is known as the "commutative law of addition". This phrase suggests that there are other commutative laws: for example, there is a commutative law of multiplication. However, many Binary Operation s are not commutative, such as subtraction and division, so it is misleading to speak of an unqualified "commutative law". Associativity A somewhat subtler property of addition is Associativity , which comes up when one tries to define repeated addition. Should the expression :"''a'' + ''b'' + ''c''" be defined to mean (''a'' + ''b'') + ''c'' or ''a'' + (''b'' + ''c'')? That addition is associative tells us that the choice of definition is irrelevant. For any three numbers ''a'', ''b'', and ''c'', it is true that : (''a'' + ''b'') + ''c'' = ''a'' + (''b'' + ''c''). Not all operations are associative, so in expressions with operations other than addition, it is important to specify the Order Of Operations . Zero and one If one adds Zero to any number, the quantity won't change; zero is the Identity Element for addition, also known as the Additive Identity . In symbols, for any ''a'', a This law was first identified in Brahmagupta 's '' Brahmasphutasiddhanta '' in 628 , although he wrote it as three separate laws, depending on whether ''a'' is negative, positive, or zero itself, and he preferred words to algebraic symbols. Later Indian Mathematicians refined the concept; around the year 830 , Mahavira wrote, "zero becomes the same as what is added to it", corresponding to the unary statement 0 + ''a'' = ''a''. In the 12th century, Bhaskara wrote, "In the addition of cipher, or subtraction of it, the quantity, positive or negative, remains the same", corresponding to the unary statement ''a'' + 0 = ''a''. In the context of integers, addition of One also plays a special role: for any integer ''a'', the integer (''a'' + 1) is the least integer greater than ''a'', also known as the successor of ''a''. Because of this succession, the value of some ''a'' + ''b'' can also be seen as the successor of ''a'', making addition iterated succession. Units In order to numerically add certain types of numbers, such as Vulgar Fraction s and physical quantities with Units , they must first be expressed with a Common Denominator . For example, if a measure of 5 feet is extended by 2 inches, the sum is 62 inches, since 60 inches is synonomous with 5 feet. On the other hand, it is usually meaningless to try to add 3 meters and 4 square meters, since those units are incomparable; this sort of consideration is fundamental in Dimensional Analysis . PERFORMING ADDITION Innate ability Studies on mathematical development starting around the 1980s have exploited the phenomenon of s look longer at situations that are unexpected. A seminal experiment by Karen Wynn in 1992 involving Mickey Mouse dolls manipulated behind a screen demonstrated that five-month-old infants ''expect'' 1 + 1 to be 2, and they are comparatively surprised when a physical situation seems to imply that 1 + 1 is either 1 or 3. This finding has since been affirmed by a variety of laboratories using different methodologies. Another 1992 experiment with older Toddler s, between 18 to 35 months, exploited their development of motor control by allowing them to retrieve Ping-pong balls from a box; the youngest responded well for small numbers, while older subjects were able to compute sums up to 5. Even some nonhuman Animal s show a limited ability to add, particularly Primate s. In a 1995 experiment imitating Wynn's 1992 result (but using Eggplant s instead of dolls), Rhesus Macaque s and Cottontop Tamarin s performed similarly to human infants. More dramatically, after being taught the meanings of the Arabic Numerals 0 through 4, one Chimpanzee was able to compute the sum of two numerals without further training. Elementary methods Typically children master the art of Counting first, and this skill extends into a form of addition called "counting-on"; asked to find three plus two, children count two past three, arriving at four ''five''. This strategy seems almost universal; children can easily pick it up from peers or teachers, and some even invent it independently. Those who count to add also quickly learn to exploit the commutativity of addition by counting up from the larger number. Decimal system The prerequisitive to addition in the Decimal System is the internalization of the 100 single-digit "addition facts". Conceivably one could Memorize all the facts, but many strategies besides Rote Learning are more enlightening and, for most people, more efficient:
To add multidigit numbers, one typically aligns the addends vertically and adds the columns, starting from the ones column on the right. If a column exceeds ten, the extra digit is "carried" into the next column. For a more detailed description of this algorithm, see ''''. An alternate strategy starts adding from the most significant digit on the left; this route makes carrying a little clumsier, but it is faster at getting a rough estimate of the sum. Computers for details.]] Analog Computer s work directly with physical quantities, so their addition mechanisms depend on the form of the addends. A mechanical adder might represent two addends as the positions of sliding blocks, in which case they can be added with an Averaging Lever . If the addends are the rotation speeds of two Shafts , they can be added with a Differential . A hydraulic adder might need to add the Pressure s in two chambers, to be done by balancing forces on an assembly of Piston s via Newton's Second Law . The most common situation for a general-purpose analog computer is to add two Voltage s (referenced to Ground ); this can be accomplished roughly with a Resistor Network , but a better design exploits an Operational Amplifier . Addition is not tremendously important to analog computers, whose essential function is Integration . By contrast, addition is fundamental to the operation of Digital Computers . For digital computers, the efficiency of addition, in particular the carry mechanism, is an important limitation to overall performance. In fact, addition was not only a tool but also a basic goal for the earliest automatic, digital computers, and as late as the 20th century, mechanical calculators have been called ". By 1674 Gottfried Leibniz made the first mechanical multiplier; it was still powered, but not motivated, by addition. including the addition and carry mechanisms]] Adders execute integer addition in electronic digital computers, usually using Binary Arithmetic . The simplest architecture is the ripple carry adder, which follows the standard multi-digit algorithm taught to children. One slight improvement is the ''carry skip'' design, again following human intuition; one does not perform all the carries in computing 999 + 1, but one bypasses the group of 9s and skips to the answer. This old method predates electronic computing; it was known to Charles Babbage as "carriage anticipating". Since they compute digits one at a time, the above methods are too slow for most modern purposes. In modern digital computers, integer addition is typically the fastest arithmetic instruction, yet it has the largest impact on performance, since it underlies all the Floating-point operations as well as such basic tasks as Address generation during Memory access and fetching Instructions during Branching . To increase speed, modern designs calculate digits in Parallel ; these schemes go by such names as carry select, Carry Lookahead , and the Ling pseudocarry. Almost all modern implementations are, in fact, hybrids of these last three designs. Unlike addition on paper, addition on a computer often changes the addends. On the ancient Abacus and Adding Board , both addends are destroyed, leaving only the sum. The influence of the abacus on mathematical thinking was strong enough that early Latin texts often claimed that in the process of adding "a number to a number", both numbers vanish. In modern times, the ADD instruction of a Microprocessor replaces the augend with the sum but preserves the addend. In a High-level Programming Language , evaluating ''a'' + ''b'' does not change either ''a'' or ''b''; to change the value of ''a'' one uses the addition assignment operator ''a'' += ''b''. DEFINITIONS AND PROOFS FOR THE REAL NUMBERS In order to prove the usual properties of addition, one must first ''define'' addition for the context in question. Addition is first defined on the s, the Rational Number s, and the Real Number s. (In Mathematics Education , positive fractions are added before negative numbers are even considered; this is also the historical route.) Naturals There are two popular ways to define the sum of two natural numbers ''a'' and ''b''. If one defines natural numbers to be the cardinalities of finite sets, then it is appropriate to define their sum as follows:
Here, ''A'' U ''B'' is the Union of ''A'' and ''B''. An alternate version of this definition allows ''A'' and ''B'' to possibly overlap and then takes their Disjoint Union , a mechanism which allows any common elements to be separated out and therefore counted twice. The other popular definition is recursive:
Again, there are minor variations upon this definition in the literature. Taken literally, the above definition is an application of the Recursion Theorem on the Poset N2. On the other hand, some sources prefer to state a restricted Recursion Theorem that applies only to the natural numbers. One then considers ''a'' to be temporarily "fixed", applies recursion on ''b'' to define a function "''a'' + ", and pastes these unary operations for all ''a'' together to form the full binary operation. This recursive formulation of addition was developed by Dedekind as early as 1854, and he would expand upon it in the following decades. He proved the associative and commutative properties, among others, through Mathematical Induction ; for examples of such inductive proofs, see '' Addition Of Natural Numbers ''. Integers The simplest conception of an integer is that it consists of an Absolute Value (which is a natural number) and a Sign (generally either positive or negative). The integer zero is a special third case, being neither positive nor negative. The corresponding definition of addition must proceed by cases: | ||
| Define <math>a+b | \{q+r q\in a, r\in b\}</math> |