= Division
{disambiguate=mathematics}
{title2=$a/b$}
{wiki}
= Division
{synonym}
For <real numbers> $a$ and $b\ne0$, <division> $a/b$ asks for the unique <number> $x$ satisfying $bx=a$. Equivalently, $a/b=a b^{-1}$ uses the <multiplicative inverse> of $b$. This description separates the role of the <denominator> from the result: the <quotient> undoes a specified <multiplication>.
For example, $3/2$ is the <number> whose double is three, so it is exactly $1.5$. A second interpretation asks how many portions of size $2$ fit into an amount $3$; the answer is one and a half portions. Equal sharing and measuring with a specified portion size lead to the same <quotient>, although the units assigned to the result can differ.
In the <natural numbers>, exact <division> need not be possible. <Euclidean division> instead keeps a <remainder>: $17=5\cdot3+2$, with $0\leq2<5$. Thus seventeen objects form three complete groups of five with two left over. As a <rational number>, the same <quotient> is $17/5=3+2/5=3.4$. The <remainder> and the <fractional part> describe the same unfinished group in different ways.
<Division> by zero cannot retain the unique-solution rule. For $a\ne0$, the equation $0x=a$ has no solution; for $a=0$, every <real number> $x$ solves $0x=0$, so there is no unique <quotient>. This explains both failures through <multiplication>, rather than assigning $a/0$ a value that would contradict ordinary <arithmetic>. https://openstax.org/books/prealgebra-2e/pages/3-4-multiply-and-divide-integers[OpenStax, Prealgebra 2e, multiplication and division conventions]{ref}
Back to article page
