Addition is commutative, associative, and toddler-early
Addition, marked by plus, is one of arithmetic's four basic operations beside subtraction, multiplication, and division. It combines counts or abstract numbers. Order does not matter, grouping does not matter, and tiny sums are among the first numerical skills children show.
The simplest sum, one plus one, is within reach of babies around five months old and even some animals. Schoolchildren then work up from single decimal digits to harder problems, helped by tools running from the abacus to the computer, and engineers still study the fastest ways for chips to add. Beyond counted objects, the same operation applies to integers, real and complex numbers, and in algebra to vectors, matrices and members of additive groups.
The vocabulary is Latin. Add comes from addere, joining ad, "to," with dare, "to give," so adding literally means giving to. The things being added are addends, or summands; Renaissance writers often called the first one the augend, from augere, to increase, though commutativity has made that word rare. Sum comes from summa, "the top," because Greek and Roman reckoners wrote a column's total above it. Boethius used addere and summare, and Chaucer spread the Middle English adden. A mixed number such as three and a half hides an unwritten plus, a quirk since placing symbols side by side usually means multiplying.
Picture it as merging separate collections: the combined pile holds as many items as the piles did together. That picture struggles with fractions and negatives, so another treats addition as joining rods end to end, or extending one length by another. A third view sees adding as repeated succession, so six plus two is the successor of the successor of six. Many terms added in a row form a summation, written compactly with a capital sigma; an infinite one is a series.
The order and grouping of terms never change the result, laws not shared by subtraction or division, and in the standard order of operations addition ranks below powers, multiplication and division but level with subtraction. Adding zero leaves a number alone, a rule Brahmagupta first stated in 628 AD in words and in three cases; Mahavira around 830 and Bhaskara in the 12th century refined it.
Source: Addition
