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The 2-adic absolute value is the p-adic absolute value for the prime number $p=2$. It is defined on the rational numbers by $|x{|}_2=2^{-v_2(x)}$ for $x\ne 0$ and $|0{|}_2=0$, and extends to their P-adic number completion ${\mathbb{Q}}_2$. For a nonzero integer, count factors of two until the remaining factor is odd; for a rational number, subtract the denominator's count from the numerator's count.

The associated metric reverses a familiar comparison:Although $13$ is eight units from $5$ on the ordinary number line, it agrees with $5$ modulo $8$, whereas $6$ already disagrees modulo $2$. The 2-adic absolute value measures that agreement. The value $|3/8{|}_2=8$ also shows why this Non-Archimedean absolute value must be distinguished from the ordinary absolute value $|3/8|=3/8$.

In the left panel, differences farther from the diagonal grow in ordinary absolute value. In the right panel, nonzero even differences are closer than odd differences, and each further factor of two halves the 2-adic absolute value. The highlighted pairs have opposite distance rankings. The repeated pattern is the nested congruence class structure of an ultrametric space, visible without adding any new points.

Catastrophic cancellation occurs when subtraction of nearly equal approximations removes their matching leading digits and exposes earlier roundoff error. The subtraction can itself be exact on the stored operands: the information may already have been lost while obtaining them. Goldberg distinguishes this from cancellation of accurately known quantities, which can be harmless.^[ref]

For $x>0$, consider the single real-valued functionThe rationalization of a square-root difference proves equality in exact arithmetic. The two evaluations use the same input $x$, but the direct version first forms a value near one and then subtracts one. The rearranged version keeps the small input in its numerator and divides by a denominator near two.

For binary64 x = 1e-16, the rounded sum 1.0 + x is one, so sqrt(1.0 + x) - 1.0 is zero. The rationalized evaluation returns approximately $5\times 10^{-17}$. A 100-digit Decimal calculation using the exact stored input gives relative errors of one and about $2.50\times 10^{-17}$ respectively. At x = 1e-12, the direct result is about $5.0004445029\times 10^{-13}$, with relative error $8.89\times 10^{-5}$; the rearranged result is about $4.99999999999875\times 10^{-13}$, with relative error $1.21\times 10^{-17}$.

The plot uses Decimal.from_float(x) to preserve the binary input exactly, computes $x/(\sqrt{1+x}+1)$ at 100 decimal digits, and verifies the reference at 160 digits. Exact conversion of a stored binary value is documented by Python.^[ref] Each error is $|\hat{f}-f|/|f|$, with the computed result converted exactly to Decimal before comparison. Hence the curves measure evaluation error for one common input, rather than a mixture of input conversion error and evaluation error. The plotted interval $10^{-16}\le x\le 10^{-1}$ is far above the subnormal number range.

The original problem is well conditioned with respect to relative changes in $x$. Using the derivative $f^{\prime }(x)=1/(2\sqrt{1+x})$ and rationalization of a square-root difference, its relative condition number isIt tends to one as $x$ tends to zero. The direct evaluation instead introduces the intermediate difference $s-1$ with $s=\sqrt{1+x}$; an error of order unit roundoff in $s$ is large relative to $f(x)\approx x/2$. This is a failure of the chosen algorithm's accuracy despite a small condition number for the same real-valued function. The rationalized evaluation avoids that intermediate catastrophic cancellation, retaining relative accuracy near the unit roundoff scale throughout the plotted interval. Recognizing this structure can improve a calculation far more than merely printing extra digits.

Two integers $a,b$ are coprime when their greatest common divisor is one:For positive integers, this says that the only positive common divisor is one. For signed integers, use $\gcd (a,b)=\gcd (|a|,|b|)$, with absolute values; changing either sign leaves coprimality unchanged. With the convention $\gcd (0,0)=0$, zero is coprime precisely to $1,-1$, because $\gcd (a,0)=|a|$. Both $1,-1$ are coprime to every integer. ^[ref]

The word concerns shared prime factors, not whether either input is a prime number. For instance $8=2^3$ and $9=3^2$ are coprime composite numbers, even though neither is a square-free integer. Conversely, $6=2\cdot 3$ and $10=2\cdot 5$ are square-free integers but are not coprime. Consecutive integers are always coprime: any common divisor of $n,n+1$ divides their difference one. Thus $14,15$ provide another pair of coprime composite numbers.

For positive integers, let $P(n)$ be the prime support of an integer. The equivalent test isThe set intersection of the two prime supports of an integer must be the empty set. Indeed, any positive common divisor greater than one has a prime factor, and any shared prime factor is a common divisor greater than one. Hence coprimality depends on which prime factors occur, rather than on their prime valuations; replacing either input by its radical of an integer preserves coprimality.

For a list of at least two integers, distinguish pairwise coprimality, requiring every pair to be coprime, from joint coprimality, requiring only the greatest common divisor of the entire list to be one. For example,No prime factor belongs to all three prime supports of an integer, yet each pair shares one. By contrast, $4,9,25$ have pairwise coprimality. The distinction is presented by Poritz under the names pairwise and mutually relatively prime. ^[ref]

One use is simplifying fractions. For a fraction $a/b$ with integer entries and $b>0$, divide its numerator and denominator by $d=\gcd (a,b)$ to obtain a reduced fraction with coprime entries. For example, $42/30=7/5$ since $d=6$, and $-42/30=-7/5$ uses the same $d$. A further common divisor would make $d$ too small, so the reduction is complete. In particular, the reduced fraction for zero is $0/1$.

Another use is reversible modular arithmetic: for an integer modulus $m>1$, an integer $a$ has a modular inverse precisely when $a,m$ are coprime. The proof outline uses Bézout's identity: $ua+vm=1$ makes $u$ a modular inverse, and any inverse relation $ua-km=1$ forces every common divisor to divide one. For example, $3\cdot 5-2\cdot 7=1$ gives $5^{-1}\equiv 3(\mathrm{mod}7)$, whereas $6$ has no modular inverse modulo $10$. Thus coprimality is the exact condition allowing division by a residue in modular arithmetic. ^[ref]

For a deterministic binary operation on an abstract data type, let $f$ be its specified operation on abstract values and $F$ the implementation on concrete states. For every permitted valid input pair $(r,s)$, require bothThe first condition preserves the representation invariant. The second makes a commutative diagram: either execute $F$ and then apply the abstraction function, or apply the abstraction function to both inputs and then execute the abstract operation $f$; both paths reach the same value. These conditions state functional correctness of a completed operation; termination is an additional obligation for algorithm correctness.^[ref]

For addition in the unreduced rational representation, chooseFor example, $F((1,2),(1,3))=(5,6)$ represents $5/6$, precisely the sum of $1/2$ and $1/3$. The short correctness argument uses nonzero denominators: $b\ne 0$ and $d\ne 0$ imply $bd\ne 0$, preserving the representation invariant, and

Now consider an incorrect implementation that uses addition on the numerators and multiplication on the denominators:It preserves the representation invariant for every valid input pair because its denominator is still $bd\ne 0$. Yet $B((1,2),(1,3))=(2,6)$ represents $1/3$, not the specified sum $5/6$. Even perfect checks of the representation invariant before and after this call would miss the error: the result is a legal representation of the wrong rational number.

The incorrect operation also exposes an accidental dependence on representation. Replace $(1,2)$ by the equivalent state $(2,4)$ while keeping $(1,3)$. The correct operation gives $(10,12)$, still representing $5/6$; the incorrect operation gives $(3,12)$, representing $1/4$ instead of its previous $1/3$. A correct addition must yield the same abstract result for equivalent inputs, although its concrete outputs may differ. This is why checks against the formal specification should compare abstract values, including unreduced representations and representations with a negative denominator, rather than demand identical stored pairs.

The calculation assumes exact integers. An implementation with bounded storage must additionally specify how it handles products and sums outside its supported range; the mathematical argument alone does not establish its functional correctness. The bounded queue with encapsulation illustrates the corresponding need to account for returned status values and failed operations, alongside the abstract state transition.

For a reduced fraction $p/q$ with $0<p<q$, $q>1$, and $q,10$ coprime integers, its decimal representation repeats immediately after the decimal point. The shortest repeating block has lengthHere $\mathrm{ord}$ denotes multiplicative order, and $\phi$ denotes Euler's totient function. Changing the numerator preserves the decimal period as long as the fraction remains reduced.^[ref]

For example, long division of $1/7$ gives the remainders $1,3,2,6,4,5,1$, with emitted digits $1,4,2,8,5,7$. ConsequentlyEach remainder determines both the next digit and the next remainder; returning to one restarts the same six steps.

More generally, initialize long division with $r_0=p$. At each step, division with remainder givesso $r_j\equiv p10^j(\mathrm{mod}q)$. Since $p,q$ are coprime integers, the modular inverse of $p$ shows that $r_L=r_0$ exactly when $10^L\equiv 1(\mathrm{mod}q)$. To see why the digit block cannot repeat sooner than the remainders, a purely repeating decimal with a block of $L$ digits represents $D/(10^L-1)$ for an integer $D$. A reduced fraction $p/q$ with that value forces $q\mid 10^L-1$. Hence the shortest digit repeat and the multiplicative order coincide.

Knowing the decimal period predicts how many digits must be computed and stored: one block determines the whole repeating decimal. It also constructs the block directly. With $L={\mathrm{ord}}_q(10)$, the integeris the block, padded on the left with zeros to $L$ digits. For $1/11$, $L=2$ and $D=9$, so the block is $09$ and $1/11=0.\overline{09}$. Reduction matters: $7/21=1/3=0.\overline{3}$ has decimal period one, whereas ${\mathrm{ord}}_{21}(10)=6$.

If the reduced denominator contains prime factors two or five, ten has no multiplicative order modulo that whole denominator. WriteThe decimal preperiod has exactly $k_0$ digits. If $m>1$, the remaining repeating decimal has decimal period ${\mathrm{ord}}_m(10)$. For instance, $28=2^2\cdot 7$ gives $1/28=0.03\overline{571428}$: two prefix digits, then six repeating digits. If $m=1$, the decimal representation is a terminating decimal with exactly $k_0$ fractional places. We use the representation ending in zeros, rather than its alternative ending in nines; for example $1/40=0.025$ terminates after three places.

A short derivation explains both cases. After $k$ digits, multiplying the reduced fraction by $10^k$ and discarding its integer part leaves a reduced fraction whose denominator is $q/\gcd (q,10^k)$. This denominator first becomes coprime to ten at $k=k_0$. It then equals $m$, giving an immediate repeat when $m>1$, or an integer when $m=1$. The same long division argument works for positional notation in any base $b\ge 2$: for a reduced denominator $q>1$ coprime to $b$, the shortest repeating block has length ${\mathrm{ord}}_q(b)$.

The shortest prefix before the repeating tail of a decimal representation. For a reduced fraction with denominator $2^u{5}^{v}m$, where $m>1$ is coprime to ten, its length is $\max (u,v)$; the tail length is given by decimal period from multiplicative order.

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]

Floating-point arithmetic represents a real number by a sign, a fixed number of significant digits, and a scale. It trades exactness for a compact representation spanning many orders of magnitude. The picture shows the essential compromise: with the same precision, large numbers have wider gaps between representable neighbors. Thus accuracy is naturally relative to magnitude, rather than a fixed distance along the number line.

In radix $\beta$, an integer at least two, a nonzero normalized value has the positional notationwhere the digits and exponent $e$ are integers. The parenthesized factor is the significand; $p$ counts all its significant digits, including the leading digit. Normalization means $d_0\ne 0$, so the same value has one such representation. The allowed exponents satisfy $e_{\min }\le e\le e_{\max }$. Finite precision restricts the available digits, while the exponent bounds restrict the range. Zero needs a separate representation. This scales a digit string in positional notation by a power of its base.^[ref]

For positive normalized values in $[{\beta }^e,{\beta }^{e+1})$, increasing the last digit by one gives the constant neighbor spacingAt the next scale, the spacing is multiplied by $\beta$. In the binary numeral system with $p=4$, the values in $[1,2)$ are $1,1.125,1.25,\dots ,1.875$; those in $[2,4)$ are $2,2.25,\dots ,3.75$; those in $[4,8)$ are $4,4.5,\dots ,7.5$. The respective gaps are $1/8$, $1/4$, and $1/2$. Although absolute resolution deteriorates with magnitude, ${\Delta }_e/y$ stays between ${\beta }^{-p}$ and ${\beta }^{1-p}$. This scale invariance is why floating-point arithmetic suits both microscopic and astronomical calculations.

The IEEE 754 binary64 format uses the binary numeral system with $p=53$, including an implicit leading one for normalized values, and normal exponents $-1022\le e\le 1023$. Its 64 stored binary digits comprise one sign digit, 11 exponent digits and 52 trailing significand digits. Subnormal numbers extend the range toward zero with a different leading-digit rule. In rounding to nearest, an operation returns the representable value closest to its exact result; a tie goes to the value whose last significant binary digit is even. Machine epsilon describes the gap just above one, whereas unit roundoff bounds a single relative rounding error in the normal range. These format and rounding rules are presented in MIT's floating-point lecture.^[ref]

Even an ordinary terminating decimal can require rounding in the binary numeral system. The rational number $0.1=1/10$ has the repeating positional notation $0.0001100110011\dots {}_2$. Every finite binary fraction has a reduced denominator that is a power of two, so the factor five in $10$ prevents an exact finite binary representation. In binary64, the exact stored values of three decimal literals and their computed sum areThe exact sum of the first two stored rational numbers is $10808639105689191/36028797018963968$, halfway between two binary64 values. The even-last-digit tie rule rounds it upward. The printed result is 0.30000000000000004, one step $2^{-54}$ above stored 0.3, so 0.1 + 0.2 == 0.3 is false. Both conversion of the inputs and rounding of the addition matter here.

To reproduce the table on a Python installation using binary64, inspect exact fractions rather than only their printed decimal representations:Python's as_integer_ratio() returns a pair of integers whose quotient equals the stored finite value exactly.^[ref] Exact rational numbers are also useful for checking floating-point arithmetic: a short decimal printout can hide the approximation being made.

Historically, David Goldberg's March 1991 ACM Computing Surveys paper explained representation, rounding and catastrophic cancellation together with the IEEE standard's design choices. Oracle provides an authorized edited reprint. Its enduring lesson is practical: use the spacing to understand roundoff error, and inspect intermediate values when mathematically equivalent algorithms behave differently.^[ref]

A fraction $a/b$ represents the quotient of its numerator $a$ by its nonzero denominator $b$. For integers $a,b$ this is a rational number. Different fractions can represent the same number: $252/198=14/11$. Dividing both entries by their greatest common divisor gives a reduced fraction, with the sign convention that the denominator is positive.

For integers $a,b$ not both zero, their greatest common divisor is the largest positive integer dividing both. It is unchanged by replacing either input by its negative. The convention $\gcd (0,0)=0$ is useful, but this exceptional value is not a largest positive common divisor. For positive inputs, the Euclidean algorithm computes the answer, and Bézout's identity expresses it as an integer linear combination of the inputs.

For $12=2^2\cdot 3$ and $18=2\cdot 3^2$, a common divisor can contain at most one copy of the prime number two and one of the prime number three. Taking the smaller prime valuation at each prime number therefore givesThe check $6=18-12$ is also a Bézout's identity: every common divisor divides this difference, and six itself divides both inputs. More generally, every common divisor divides the greatest common divisor, a stronger statement than merely being no larger than it. The companion least common multiple takes the larger prime valuations, as the diagram there illustrates. For an application, divide the numerator and denominator of the fraction $12/18$ by six to obtain $2/3$; these remaining integers are coprime.

For closed intervals $X=[a,b]$ and $Y=[c,d]$, exact addition, subtraction and multiplication giveThese interval arithmetic operations describe all results $u\circ v$ for $(u,v)$ in the Cartesian product $X\times Y$: the two inputs are allowed to vary independently. Independence here means freedom to choose either input anywhere in its closed interval; it does not assert anything about probability. For example, $[1,2]+[3,5]=[4,7]$ and $[1,2]-[3,5]=[-4,-1]$.

All four endpoint products matter when signs change. For $X=[-2,3]$ and $Y=[-4,1]$, the endpoint products are $8,-2,-12,3$, so $XY=[-12,8]$. Taking just $ac$ and $bd$, even after sorting them, would give $[3,8]$ and lose valid negative products.

For division, if $0\notin Y$, the multiplicative inverse gives $1/Y=[1/d,1/c]$ and $X/Y=X(1/Y)$. Thus $[1,2]/[3,4]=[1/4,2/3]$. This ordinary finite-closed interval rule requires a denominator that excludes zero. When the denominator contains zero, division is undefined for some pairs and can become unbounded near zero; these formulas cannot supply a finite interval enclosure of all quotients. The operations above use exact endpoints; a computer implementation must also apply outward rounding.

For positive integers $a,b$, their least common multiple is the smallest positive integer that is an integer multiple of both. Write it as $\mathrm{lcm}(a,b)$. A positive common integer multiple always exists, since $ab$ is one, but it can be larger than necessary.

For example, the positive integer multiples of $12$ begin $12,24,36,48,60,72$, while those of $18$ begin $18,36,54,72$. Their first positive coincidence is $36$, so $\mathrm{lcm}(12,18)=36$. Both also divide zero, but the word positive excludes zero from this minimum.

Suppose two events happen every twelve and eighteen days, and both happen at day $t=0$. Their next meeting is at day $36$, after three twelve-day intervals and two eighteen-day intervals. Through day $72$, their common event times are $0,36,72$.

The prime factorization explains why this coincidence occurs. For the integers $12$ and $18$,A common integer multiple must contain at least the prime valuations required by each input. Taking the larger prime valuation at each prime number meets both requirements with no extra factors. The greatest common divisor uses the smaller prime valuations instead. This is the same coordinate comparison developed under prime factorization, applied in opposite directions.

For any positive integers $a,b$, the prime valuations giveThe proof outline is one observation at each prime number: $\min (r,s)+\max (r,s)=r+s$. Hence the two sides have identical prime factorizations. Numerically, $6\cdot 36=12\cdot 18=216$. This also gives a calculation using the Euclidean algorithm for the greatest common divisor, without first finding the prime factors of both inputs.

Every positive common integer multiple is an integer multiple of the least common multiple, because its prime valuations must be at least the larger input prime valuations. Including negatives and zero, the common integer multiples of $12$ and $18$ are exactly $36k$ for integers $k$. For more than two positive integers, the same definition and rule of taking the largest prime valuations apply; one may also combine inputs successively. For example, $\mathrm{lcm}(12,18,20)=\mathrm{lcm}(36,20)=180$.

For nonzero signed integers, take absolute values: $\mathrm{lcm}(-12,18)=36$. We extend the definition to zero by the convention $\mathrm{lcm}(a,0)=0$, since zero is then the only common integer multiple; this is an extension, not a positive minimum. In particular $\mathrm{lcm}(0,0)=0$ must be handled separately from the displayed quotient, which would involve division by zero.

The least common multiple is useful for combining fractions with the smallest positive shared denominator. For example,Its scheduling interpretation requires care about starting times. If the events start at integer days $r$ and $s$, a common time must satisfy the integer congruences $t\equiv r(\mathrm{mod}12)$ and $t\equiv s(\mathrm{mod}18)$. The Chinese remainder theorem for noncoprime moduli says a solution exists exactly when $6\mid (r-s)$. If it exists, solutions form one congruence class modulo $36$, but the first meeting need not be day $36$. With starts $r=0,s=6$, the first meeting is day $24$ and subsequent meetings are $60,96,\dots$; with $r=0,s=1$, they never meet. The beauty of the least common multiple is the exact shared spacing, while the integer congruences determine whether and where the spacing begins.

Book VII, Proposition 34 of Euclid's Elements constructs the least common multiple of two numbers, and Proposition 36 extends the construction to three by first combining two of them. In modern notation this is $\mathrm{lcm}(a,b,c)=\mathrm{lcm}(\mathrm{lcm}(a,b),c)$. These are specific ancient treatments of the construction, rather than a claim about its first discovery. ^{[ref] [ref]}

Repeated division with remainder that determines successive digits of a quotient. To generate the fractional digits in base $b$, multiply the current remainder by $b$, divide by the denominator, and keep the resulting remainder for the next step.

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.