Addition Totally Explained

Everything about Addition totally explained

Addition is the mathematical process of putting things together. The plus sign "+" means that two numbers are added together. For example, in the picture on the right, there are 3 + 2 apples — meaning three apples and two other apples — which is the same as five apples, since 3 + 2 = 5. Besides counts of fruit, addition can also represent combining other physical and abstract quantities using different kinds of numbers: negative numbers, fractions, irrational numbers, vectors, and more.
As a mathematical operation, addition follows several important patterns. It is commutative, meaning that order doesn't matter, and it's associative, meaning that one can add more than two numbers (see Summation). Repeated addition of 1 is the same as counting; addition of 0 (number) doesn't change a number. Addition also obeys predictable rules concerning related operations such as subtraction and multiplication. All of these rules can be proven, starting with the addition of natural numbers and generalizing up through the real numbers and beyond. General binary operations that continue these patterns are studied in abstract algebra.
Performing addition is one of the simplest numerical tasks. Addition of very small numbers is accessible to toddlers; the most basic task, 1 + 1, can be performed by infants as young as five months and even some animals. In primary education, children learn to add numbers in the decimal system, starting with single digits and progressively tackling more difficult problems. Mechanical aids range from the ancient abacus to the modern computer, where research on the most efficient implementations of addition continues to this day.

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, ť

1 + 1 = 2

(verbally, "one plus one equals two")

2 + 2 = 4

(verbally, "two plus two equals four") ť

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:

A column of numbers, with the last number in the column underlined, usually indicates that the numbers in the column are to be added, with the sum written below the underlined number.
A whole number followed immediately by a fraction indicates the sum of the two, called a mixed number. For example,
3½ = 3 + ½ = 3.5.
This notation can cause confusion since in most other contexts denotes multiplication instead.

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 didn't 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 Latin. "" and "" 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 numbers, there are many possible interpretations and even more visual representations.

Combining sets

Possibly the most fundamental interpretation of addition lies in combining sets:

When two or more collections are combined into a single collection, the number of objects in the single collection is the sum of the number of objects in the original collections. This interpretation is easy to visualize, with little danger of ambiguity. It is also useful in higher mathematics; for the rigorous definition it inspires, see Natural numbers below. However, it isn't obvious how one should extend this version of addition to include fractional numbers or negative numbers.
One possible fix is to consider collections of objects that can be easily divided, such as pies or, still better, segmented rods. Rather than just combining collections of segments, rods can be joined end-to-end, which illustrates another conception of addition: adding not the rods but the lengths of the rods.

Extending a length

A second interpretation of addition comes from extending an initial length by a given length:

When an original length is extended by a given amount, the final length is the sum of the original length and the length of the extension. a + b can be interpreted as a binary operation that combines a and b, in an algebraic sense, or it can be interpreted as the addition of b more units to a. Under the latter interpretation, the parts of a sum a + b play asymmetric roles, and the operation a + b is viewed as applying the unary operation +b to a. Instead of calling both a and b addends, it's more appropriate to call a the augend in this case, since a plays a passive role. The unary view is also useful when discussing subtraction, because each unary addition operation has an inverse unary subtraction operation. and vice versa.

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 ť a + b = b + 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's a commutative law of multiplication. However, many binary operations are not commutative, such as subtraction and division, so it's 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 ť "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's true that ť (a + b) + c = a + (b + c).

For example, (1 + 2) + 3 = 3 + 3 = 6 = 1 + 5 = 1 + (2 + 3). Not all operations are associative, so in expressions with other operations like subtraction, it's important to specify the order of operations.

Zero and one

When adding zero to any number, the quantity doesn't change; zero is the identity element for addition, also known as the additive identity. In symbols, for any a, ť a + 0 = 0 + 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 used words rather than 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

b^).

In this sense, the maximum operation is a dequantized version of addition.

Other ways to add

Incrementation, also known as the successor operation, is the addition of 1 to a number. Summation describes the addition of arbitrarily many numbers, usually more than just two. It includes the idea of the sum of a single number, which is itself, and the empty sum, which is zero. An infinite summation is a delicate procedure known as a series. Counting a finite set is equivalent to summing 1 over the set. Integration is a kind of "summation" over a continuum, or more precisely and generally, over a differentiable manifold. Integration over a zero-dimensional manifold reduces to summation. Linear combinations combine multiplication and summation; they're sums in which each term has a multiplier, usually a real or complex number. Linear combinations are especially useful in contexts where straightforward addition would violate some normalization rule, such as mixing of strategies in game theory or superposition of states in quantum mechanics. Convolution is used to add two independent random variables defined by distribution functions. Its usual definition combines integration, subtraction, and multiplication. In general, convolution is useful as a kind of domain-side addition; by contrast, vector addition is a kind of range-side addition.

In literature

In chapter 9 of Lewis Carroll's Through the Looking-Glass, the White Queen asks Alice, "And you do Addition? ... What's one and one and one and one and one and one and one and one and one and one?" Alice admits that she lost count, and the Red Queen declares, "She can't do Addition".

In George Orwell's Nineteen Eighty-Four, the value of 2 + 2 is questioned; the State contends that if it declares 2 + 2 = 5, then it's so. See Two plus two make five for the history of this idea.Further Information

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