Division (mathematics, $a/b$)

For real numbers $a$ and $b\ne 0$, 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: $17=5\cdot 3+2$, with $0\le 2<5$. 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 $a\ne 0$, 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. ^[ref]

Division of integers that returns an integer quotient rather than an exact fraction. A rounding convention is required for negative operands. For $n\ge 0$ and $d>0$, the quotient is $q=\lfloor n/d\rfloor$ and the remainder is $r=n-qd$ with $0\le r<d$.

For an $n$-bit unsigned dividend $a$ and positive divisor $b$, read the dividend bits from most to least significant. If the processed prefix is $p$, maintain $p=bQ+R$ with $0\le R<b$. Appending bit $d$ gives trial remainder $T=2R+d$; append quotient bit $[T\ge b]$ and replace the remainder by $T-b[T\ge b]$. This preserves the quotient–remainder identity.

The old dividend and new quotient can share one $n$-bit processor register: after $k$ iterations its value is $(a{2}^k\mathrm{mod}{2}^n)+Q_k$, where $Q_k<2^k$. The shifted dividend occupies the upper positions and the accumulated quotient occupies the lower positions, so no information is overwritten. The high processor register holds $R_k$. For an $n$-bit dividend the next trial remainder is no larger than the next prefix, hence fits in $n$ bits through the final iteration. This last bound is specific to the dividend width; it must not be assumed for unrestricted multiword dividends.

For signed two's complement integers, a conditional positive bias converts the rounding behaviour of an arithmetic right shift into C integer division. It can be formed without a conditional branch by turning the sign bit into either zero or a low-$k$-bit bit mask.

Proof. Nonnegative $a$ needs no adjustment. If $a<0$, write $-a=2^{k}q+r$, $0\le r<2^k$. Then $a+(2^k-1)=-2^{k}q+(2^k-1-r)$, whose quotient rounded down is $-q$, exactly the quotient truncated towards zero. The bias fits safely for $1\le k<n$ in an $n$-bit signed word: it is added only to negative inputs, and the largest possible adjusted input is $2^k-2$. The corresponding remainder is $((a+b)\mathrm{mod}{2}^k)-b$, where $b=(2^k-1)1_{a<0}$.

For representable integer quotients with nonzero divisor, C integer division discards the fractional part towards zero and defines the remainder by $a=bq+r$. Thus a nonzero remainder has the dividend's sign. For example, $-13=5(-2)+(-3)$, so division by five returns quotient $-2$ and remainder $-3$. This convention differs from division rounded down, which would return quotient $-3$ and nonnegative remainder $2$. Division by zero and an unrepresentable quotient are excluded.^[ref]

A ratio compares two quantities $a,b$, written $a:b$ and corresponding to division $a/b$ when $b\ne 0$. Equal ratios of corresponding lengths characterize the scaling of similar triangles.

For example, $6:4$ and $3:2$ specify the same ratio because $6/4=3/2$. Multiplying both entries by the same nonzero scale preserves the quotient: $(ca)/(cb)=a/b$. This is useful when comparing mixtures or scaling a recipe; doubling both ingredient amounts preserves their ratio, while adding the same amount to both generally does not. For quantities with the same physical units, their ratio is dimensionless, connecting this elementary calculation with dimensional analysis.

A percentage expresses a ratio in hundredths: $p$ as a fraction is $100p\%$. For example, $18/60=0.3=30\%$. Comparing two percentages requires specifying whether the change is an absolute difference in percentage points or a relative change divided by the initial value.

A percentage point is one hundredth on a probability or proportion scale. A change from $20\%$ to $25\%$ is $5$ percentage points, while the relative increase is $(25-20)/20=25\%$. These answer different questions and have different denominators.