Addition

In algebra, another area of mathematics, addition can also be performed on abstract objects such as vectors, matrices, subspaces and subgroups.

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 members of other animal species.

In primary education, students are taught 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.

For example, There are also situations where addition is "understood", even though no symbol appears: The sum of a series of related numbers can be expressed through capital sigma notation, which compactly denotes iteration.

Today, due to the commutative property of addition, "augend" is rarely used, and both terms are generally called addends.

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.

[25] Studies on mathematical development starting around the 1980s have exploited the phenomenon of habituation: infants look longer at situations that are unexpected.

[26] 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.

In a 1995 experiment imitating Wynn's 1992 result (but using eggplants instead of dolls), rhesus macaque and cottontop tamarin monkeys performed similarly to human infants.

As they gain experience, they learn or discover the strategy of "counting-on": asked to find two plus three, children count three past two, saying "three, four, five" (usually ticking off fingers), and arriving at five.

Eventually children begin to recall certain addition facts ("number bonds"), either through experience or rote memorization.

[b] 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.

Since the end of the 20th century, some US programs, including TERC, decided to remove the traditional transfer method from their curriculum.

Addition requires two numbers in scientific notation to be represented using the same exponential part, so that the two significands can simply be added.

This is similar to what happens in decimal when certain single-digit numbers are added together; if the result equals or exceeds the value of the radix (10), the digit to the left is incremented: This is known as carrying.

Analog computers work directly with physical quantities, so their addition mechanisms depend on the form of the addends.

A hydraulic adder can add the pressures in two chambers by exploiting Newton's second law to balance forces on an assembly of pistons.

The most common situation for a general-purpose analog computer is to add two voltages (referenced to ground); this can be accomplished roughly with a resistor network, but a better design exploits an operational amplifier.

The abacus, also called a counting frame, is a calculating tool that was in use centuries before the adoption of the written modern numeral system and is still widely used by merchants, traders and clerks in Asia, Africa, and elsewhere; it dates back to at least 2700–2300 BC, when it was used in Sumer.

[50] In a high-level programming language, evaluating a + b does not change either a or b; if the goal is to replace a with the sum this must be explicitly requested, typically with the statement a = a + b.

Such overflow bugs may be hard to discover and diagnose because they may manifest themselves only for very large input data sets, which are less likely to be used in validation tests.

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).

This way of defining integers as equivalence classes of pairs of natural numbers, can be used to embed into a group any commutative semigroup with cancellation property.

Originally, the Grothendieck group was, more specifically, the result of this construction applied to the equivalences classes under isomorphisms of the objects of an abelian category, with the direct sum as semigroup operation.

[67] Unfortunately, dealing with multiplication of Dedekind cuts is a time-consuming case-by-case process similar to the addition of signed integers.

The sum of two m × n (pronounced "m by n") matrices A and B, denoted by A + B, is again an m × n matrix computed by adding corresponding elements:[75][76] For example: In modular arithmetic, the set of available numbers is restricted to a finite subset of the integers, and addition "wraps around" when reaching a certain value, called the modulus.

The set of integers modulo 2 has just two elements; the addition operation it inherits is known in Boolean logic as the "exclusive or" function.

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.