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The numerator is the upper entry $a$ in a fraction $a/b$.

Fix a prime number $p$. Write a nonzero rational number as $x=p^{k}a/b$, where $k\in \mathbb{Z}$ and the nonzero integers $a,b$ are not divisible by $p$. The exponent $k$ is uniquely determined. Its P-adic valuation is this signed exponent $v_p(x)=k$: equivalently, for any nonzero integer numerator $A$ and denominator $B$,Factors of $p$ in the numerator decrease the p-adic absolute value; factors in the denominator increase it. For $p=2$ the computations are:

This is an absolute value on a field measuring divisibility, rather than the magnitude on a real number number line. It satisfiesThe proof comes directly from P-adic valuations: exponents add in a product of numbers, and extracting a common power of $p$ from a sum gives $v_p(x+y)\ge \min \{v_p(x),v_p(y)\}$. If the two P-adic valuations differ, the remaining term with smaller exponent cannot cancel modulo $p$, so equality holds. Thus the p-adic absolute value is a Non-Archimedean absolute value. ^[ref]

The metric $d_p(x,y)=|x-y{|}_p$ makes two rational numbers close when their difference contains a high power of $p$. This packages successive modular congruences into a numerical error estimate, allowing convergence in a metric space and convergent series to encode exact arithmetic information.

A p-adic unit is an invertible element of the ring ${\mathbb{Z}}_p$. For a p-adic integer $x$, this is equivalent to $v_p(x)=0$, $|x{|}_p=1$, or a nonzero initial p-adic digit $b_0$. A rational number $a/b$ is a p-adic unit when neither its reduced numerator nor its denominator is divisible by $p$.

Thus $3$ and $1/3$ are p-adic units at $p=2$, while $2$ is a p-adic integer with no inverse inside ${\mathbb{Z}}_2$. The modular inversion converges to a p-adic inverse construction computes the inverse of an integer unit to arbitrary finite precision.

Continue the transmission amplitude to complex wavenumber $q=i\kappa$ with $\kappa >0$, equivalently complex momentum $k=i\hslash \kappa$. Then $E=-{\hslash }^2{\kappa }^2/(2m)<0$. In the left exterior region the $A$ term grows at negative infinity and the $B$ term decays; in the right exterior region the $D$ term grows and the $C$ term decays. A bound state thus requires $A=D=0$, with nonzero $B,C$. The transfer matrix relation gives $M_{22}(i\kappa )=0$. Equivalently, a zero of the incoming coefficient is a bound-state pole of $t(q)$, whose numerator is nonzero at $q=i\kappa$.

The denominator found in part (c) givesFactorization produces two branches. To identify their parity, use an even interior wavefunction $A\cosh (\kappa x)$ or an odd interior wavefunction $A\sinh (\kappa x)$ for $|x|<a$, joined to a decaying exterior wavefunction at $a$. Continuity and the derivative jump give, respectively,Rearrangement gives the implicit energy eigenvalue equationsOnly strictly positive equation roots $\kappa$ are retained. With $\nu =-\mu >0$, the even equation root lies above $\nu$ and the odd equation root, when it exists, lies below $\nu$, so the even bound state has lower energy. The formal zero equation root in the odd equation does not itself establish a normalizable wavefunction.

With $t$ as in i, the foot of a perpendicular lies strictly between the endpoints precisely when $0<t<1$. Since the denominator is positive, this givesEquivalently, both dot productsare positive: the angles of the triangle at $A$ and $B$ are both less than $\pi /2$. If “between” includes the endpoints, replace both strict inequalities by non-strict ones; equality gives $P=A$ or $P=B$.

Write $z=x+iy$, with $y\ne 0$ because the triangle is nondegenerate. Let $w=u+iv$ be its circumcenter. Equality of the complex modulus values $|w|=|w-1|$ gives $u=1/2$. Equality of $|w|=|w-z|$, after expanding the squares, givesThus the real part and imaginary part determineUsing the complex conjugate $\bar{z}=x-iy$, we also haveThereforeThe nonzero denominator is exactly the nondegeneracy condition. If $z$ is real, the three distinct collinear points do not have a finite circumcenter.

Assume $h\ne 0$. The velocity in polar coordinates is $\dot{\mathbf{r}}=\dot{r}{\mathbf{e}}_r+r\dot{\theta }{\mathbf{e}}_{\theta }$. HenceTaking the dot product of the conserved Laplace-Runge-Lenz vector with $\mathbf{r}$ givesConsequently the polar equation of a conic isOnly angles for which the denominator is positive describe the physical trajectory. Its eccentricity is $A/(mk)$, giving an ellipse, parabola, or hyperbola when this ratio is respectively less than, equal to, or greater than $1$.

If $A=0$, the angle $\gamma$ is undefined, but the same dot product identity gives the circular orbit $\boxed{r=m{h}^2/k}$. If $h=0$, the trajectory is radial and the displayed division degenerates; for $k>0$, $\mathbf{A}=-mk{\mathbf{e}}_r$ and $\gamma =\pi$ on a fixed ray, so the identity does not determine $r$ as a function of a varying angle.

The ratio of the sum of the positive divisors of a positive integer to that integer. For coprime $a,b$, the ratio is multiplicative because the divisor function $\sigma$ is multiplicative. The contribution of a prime power $p^e$ is $(1+p+\cdots +p^e)/p^e$, so a prescribed quotient can be searched through its remaining numerator and denominator.

A conic section with a focus of a conic at the origin has the polar coordinates equation $r=\ell /(1+e\cos \theta )$, where $\ell >0$ is the semilatus rectum and $e\ge 0$ the eccentricity. The directrix of a conic definition $r=e(d-r\cos \theta )$ gives the formula with $\ell =ed$ when $e>0$; $e=0$ is the circle limit. The cases $0<e<1$, $e=1$, and $e>1$ give an ellipse, parabola, and hyperbola. The physical radius requires a positive denominator. For an attractive inverse-square central force, the Laplace-Runge-Lenz vector gives $\ell =m{h}^2/k$ and $e=A/(mk)$.

A nonnegative fraction with positive denominator and numerator smaller than its denominator, so its value lies in $[0,1)$. The zero value has the reduced fraction representation $0/1$. The term is sometimes extended to negative fractions by comparing absolute values; the nonnegative convention is the one used here.

Exact arithmetic on rational numbers represented by integer numerators and nonzero integer denominators. Common factors can be removed with the greatest common divisor. Exact rational arithmetic avoids rounding error when deciding equality or integrality.

A rational number is a quotient of two integers with nonzero denominator.

A reduced fraction $a/b$ has coprime integers $a,b$ and positive denominator $b$. Distinct reduced fractions with denominators at most $Q$ differ by at least $Q^{-2}$, since their difference has a nonzero integer numerator over a denominator at most $Q^2$.

Given consecutive continued fraction convergents $p_{k-1}/q_{k-1}$ and $p_k/q_k$, the intermediate fractions $(t{p}_k+p_{k-1})/(t{q}_k+q_{k-1})$ for $1\le t\le a_{k+1}$ are semiconvergents, where $a_{k+1}$ is the next partial quotient. The endpoint is the next convergent. These fractions matter in best rational approximation when a denominator bound cuts between consecutive convergents.

A binary tree enumerating every positive reduced fraction exactly once. Begin with boundary fractions $0/1$ and $1/0$; between neighboring fractions $a/b<c/d$, insert their mediant $(a+c)/(b+d)$ and recurse into the two new intervals. The determinant relation $bc-ad=1$ is preserved, and the new denominator is $b+d$.

A fraction of the form $1/n$ for a positive integer $n$. Its numerator is 1 and its denominator determines it uniquely.

Consider an original abstract data type whose values are rational numbers. Store an ordered pair $(n,d)$ of integers, interpreted as a numerator and a denominator. Use exact, unbounded integer arithmetic in this model. The storage space, representation invariant and abstraction function areThe representation invariant does not require a positive denominator or a reduced fraction. Consequently $(1,2)$, $(2,4)$ and $(-1,-2)$ are three different concrete states with the same abstract rational number $1/2$. Similarly, $(0,1)$ and $(0,-3)$ both represent zero. The abstraction function is a surjection: every rational number has an integer numerator and a nonzero integer denominator, so every abstract value has a representation.

Both $(1,0)$ and $(0,0)$ violate the representation invariant; neither has a value under this abstraction function. By contrast, $(2,4)$ is completely valid. Simplifying it to the reduced fraction $1/2$ may be a useful implementation choice, but is not a requirement of this representation. Strengthening the representation invariant to require a positive denominator and a greatest common divisor of one would choose a unique stored representative for each rational number; every constructor and operation would then need to establish that stronger condition.

Equality of abstract values is therefore different from equality of stored ordered pairs. For valid $(a,b)$ and $(c,d)$,This defines an equivalence relation on $R_{\mathrm{valid}}$: each equivalence class contains all representations of one rational number. An API equality operation can compare these cross-products using exact integer arithmetic. Comparing the two fields individually would incorrectly declare $(1,2)$ and $(2,4)$ unequal as rational numbers. The general reason is that an abstraction function may map several concrete values to one abstract value.^[ref]