Division is not commutative, and order changes the answer
Division splits a dividend by a divisor to yield a quotient. Share 20 apples among 4 people and each gets 5; divide 21 among 4 and 1 remains. Unlike multiplication, swapping numbers or regrouping operations can change results. Dividing by zero is never defined.
Division is one of arithmetic's four basic operations alongside addition, subtraction, and multiplication. The dividend is what gets divided; the divisor is what you divide by; the quotient is the result. Among natural numbers, division may leave a remainder: 21 apples shared by 4 people give 5 each with 1 left over.
Two interpretations help beginners. Quotition asks how many times the divisor fits into the dividend; partition asks how large each equal share is when a set is split. Extending to rational or real numbers lets division always yield one value when the divisor is nonzero, since division becomes the inverse of multiplication.
Division breaks familiar arithmetic rules. It is not commutative: a/b need not equal b/a. It is not associative: (24/6)/2 equals 2, but 24/(6/2) equals 8. It is right-distributive over addition—(a ± b)/c equals a/c ± b/c—but not left-distributive: 12/(2 + 4) is 2 while 12/2 + 12/4 is 9.
Euclidean division states that for integers a and b with b ≠ 0, unique quotient q and remainder r exist with a = bq + r and 0 ≤ r < |b|. Notation varies: fractions a/b, obelus a ÷ b introduced by Johann Rahn in 1659, or colon a : b from Leibniz in 1684. Long division, chunking, abacuses, slide rules, and modern algorithms all compute quotients. Integers are not closed under division unless the dividend is an exact multiple of the divisor.
Source: Division (mathematics)
