Round-off error arises when numbers or arithmetic results are rounded to a finite set of representable values. Its scale is associated with machine epsilon, and its effects depend on subsequent operations. Even a formula with zero truncation error can lose information. In decimal arithmetic rounded to three significant digits, $1.234$ becomes $1.23$ and $1.233$ becomes $1.23$: subtracting the stored numbers gives zero although the exact difference is $0.001$. The subtraction exposes errors already present in its nearly equal operands.

This example also has a large condition number: independently perturbing the two inputs of $a-b$ can produce a relative amplification $(|a|+|b|)/|a-b|=2467$. High sensitivity belongs to the input-output problem; numerical stability concerns the arithmetic used to solve it. Conversely, on a common binary system with 53 significant bits and rounding to nearest, $(10^{16}+1)-10^{16}$ evaluates to zero while $(10^{16}-10^{16})+1$ evaluates to one. Exact arithmetic makes both one: changing summation order changes the accumulated roundoff error. MIT's summation lecture explains why numerical stability and the condition number must be assessed separately.^[ref]

Computer aided geometric design develops representations and algorithms for creating, evaluating and modifying curves, parametric surfaces and solid geometry. A representation must support useful questions: where is a point on the shape, what is its tangent vector, do two pieces join smoothly, and can two objects intersect? Bézier curves, splines and subdivision surfaces trade local control, smoothness and computational cost in different ways.

For a simple Bézier curve, choose control points $P_0=(0,0)$, $P_1=(1,1)$ and $P_2=(2,0)$. Its Bernstein basis formula isThe endpoints are interpolated, but the middle control point generally is not: $C(1/2)=(1,1/2)$. Its position influences the bulge, while Bézier derivative control polygons specify endpoint tangents. This makes shape changes predictable from a small set of geometric inputs.^[ref]

A boundary representation records how faces, edges and vertices join to enclose a solid; constructive solid geometry records Boolean combinations of primitive regions. These topological and geometric data play different roles. For computational queries, a bounding volume supplies a conservative approximation, while an exact limit representation or a controlled error bound is needed for reliable collision detection and intersection of subdivision limit surfaces.

// Target: geometry-and-topology.bigb

A continuation method follows a solution branch of a parameter-dependent equation from a known solution, using nearby solutions to guide the next solve. This is useful when one wants to understand how solutions change, rather than obtain just one isolated answer. A continuation predictor-corrector step follows a tangent vector and then uses a Newton method correction on a transverse section. The same geometry supports both pseudo-arclength continuation through a continuation fold and transversal surface intersection tracing.

Consider the original scalar polynomial exampleIts solution branch bends back twice in $\lambda$. Increasing $\lambda$ alone eventually stops following the same nearby solutions; following the curve in the $(x,\lambda )$ plane can continue around both bends. The continuation folds areFor $|\lambda |<2/(3\sqrt{3})$ there are three distinct real zeros of a function at that fixed $\lambda$, all lying on this one connected curve. A continuation method keeps their geometric relationship visible.

Every continuation method needs a seed, supplied by an independent solve, an explicitly known solution, or another search. Following one solution branch does not guarantee finding every connected component of the solution set. A large step can miss detail or make the correction converge to a distant solution. At branch intersections or a loss of rank of the full Jacobian matrix, the regular-curve argument needs separate analysis. The seed and component limitations also matter in transversal surface intersection tracing.

Interval arithmetic calculates with closed intervals that contain the possible values of a quantity. If all we know is $x\in X=[a,b]$, an interval enclosure for a function $f$ must contain $f(x)$ for every allowed $x$. The result says what is guaranteed, including uncertainty in the input and, with outward rounding, roundoff error in the calculation. A wide answer can still be correct; a narrow answer is useful only if it contains every possible value.

The attraction is that a finite calculation can make a statement about infinitely many real numbers at once. An interval enclosure can certify that a function stays positive throughout a whole closed interval, or bound its maximum for mathematical optimization. For example, on $X=[0,1]$, the actual image of a function for $f(x)=x(1-x)$ is $[0,1/4]$, but direct interval arithmetic gives the wider $[0,1]$. Interval subdivision and the interval square range improve this answer in different ways.

Historically, Ramon E. Moore's 1962 Stanford dissertation studied interval arithmetic and automatic error analysis in digital computing; his book Interval Analysis followed in 1966. Eldon Hansen's account documents earlier reports as well: these dates describe important steps in developing and disseminating the subject, rather than a claim of sole invention in a particular year.^[ref] Moore, R. Baker Kearfott and Michael J. Cloud later wrote Introduction to Interval Analysis (2009).^[ref]

If a twice differentiable parametric curve $R:[a,b]\to {\mathbb{R}}^m$ satisfies $\Vert R^{\prime \prime }\Vert \le M$, its deviation from the linearly parameterized chord is at most $M(b-a{)}^2/8$. This follows from the Green function for the second derivative with zero endpoint values. It bounds the Hausdorff distance between the arc and its chord.

// Target: analysis.bigb

A bracketing method finds a zero of a function by maintaining a root bracket and shrinking it as new values of the function become available. The bisection method always tries the midpoint; its special attraction is a predictable bisection error bound without any derivative evaluations.

The Newton root-finding iteration seeks a zero of a function $F$ by repeatedly solving its tangent line approximation. At a current guess $t_r$ with derivative $F^{\prime }(t_r)\ne 0$, setGeometrically, follow the tangent line to the horizontal axis, then return to the function graph for the next tangent line. Close to a suitable zero of a function, the error becomes dramatically smaller.^[ref]

For a concrete square root calculation, take the polynomial $F(t)=t^2-2$ and start at $t_0=1$. The Newton root-finding iteration becomes $t_{r+1}=(t_r+2/t_r)/2$, givingThe errors use the unrounded iterates. After the first step, each iterate overestimates $\sqrt{2}$, and the exact identityexhibits quadratic convergence: once close, the number of correct decimal digits roughly doubles per step. This speed makes the Newton root-finding iteration useful when accurate derivatives are available.

Global convergence requires additional care. For the polynomial $F(x)=x^3-2x+2$, starting at $x_0=0$ gives $F(0)=2$, $F^{\prime }(0)=-2$, hence $x_1=1$. Then $F(1)=1$, $F^{\prime }(1)=1$ gives $x_2=0$. The sequence repeats $0,1,0,1,\dots$ forever, although a real polynomial root lies in $(-2,-1)$ by the intermediate value theorem. Neither point in the cycle is a zero of a function, and neither has zero derivative. A maintained sign-changing interval and fallback bisection method can protect a root solver from unsuitable Newton steps.

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 nucleotide sequence obtained by reversing the order and complementing each nucleobase according to the appropriate base pairing rules.

Base stacking is the close arrangement and interaction of neighbouring nucleobases with their approximately planar faces above one another. Along a nucleic acid double helix, neighbouring base pairs stack successively; base pairing instead associates partners across the two strands. Both affect duplex stability, and the interactions depend on the identities and arrangement of neighbouring nucleobases. See Brown, Genomes: base pairing and base stacking.

Two nucleic acid strands with opposite directional alignment: a $5^{\prime }$-to-$3^{\prime }$ direction on one is aligned with the $3^{\prime }$-to-$5^{\prime }$ direction on the other.

A nuclear pore complex is an assembly of proteins spanning the nuclear envelope and regulating passage between the cell nucleus and cytosol. It is a selective gateway, not simply an unregulated hole: messenger RNA export and import of nuclear proteins require appropriate transport machinery. See Alberts et al.: nuclear pore complexes.

The nuclear pore complex occupies a junction of both nuclear envelope membranes. Its protein scaffold surrounds an aqueous passage connecting cytosol to the cell nucleus, rather than the intermembrane space. Diffusion permits small soluble cargo to pass; selective receptors transport other cargo. Molecular mass alone is not a universal cutoff.^[ref]

Form Champernowne constant by using concatenation to join the decimal representations of the positive integers $1,2,3,\dots$ after the decimal point. Champernowne proved in 1933 that this particular real number is a normal number in base ten. The construction is striking because its rule is completely explicit, while all finite words have the prescribed limiting digit block frequencies. ^[ref] Copeland and Erdős explicitly recall this result at the start of their own paper. ^[ref]

The first $100000$ digits of Champernowne constant have the following exact counts, in digit order $0,1,\dots ,9$:For the $99999$ adjacent pairs fully inside that prefix, the words $00,01,12,99$ occur $642,1753,2088,753$ times respectively. Thus, for example, $F_{100000}(12)=2088/99999$, which is about $0.02088$, rather than $0.01$. The prefix ends partway through the decimal representation of $22222$; the many leading ones in earlier positive integers help explain the visible imbalance.

These are finite counts, not a proof of the normal number theorem: a finite prefix can be followed by an all-zero tail. The theorem for Champernowne constant concerns the infinite limit, and establishes base-ten normality, rather than absolute normality. Nor do computations prove normality for familiar constants: the normality of Pi and Euler's number remains unproved, as discussed by Bailey and Borwein. ^[ref]

A real number is simply normal in base $K$ if each individual digit in its positional notation has limiting digit block frequency $1/K$. This is the length-one part of the definition of a normal number. It places no equal-frequency requirement on longer words.

For example, $x=0.010101{\dots }_2=1/3$ is a simply normal number in base two: each binary digit has frequency $1/2$. But $00$ and $11$ never occur, and $01$ and $10$ each have frequency $1/2$. Thus $x$ is not a normal number in base two. Even a sequence that looks more complicated, such as the deterministic Thue–Morse sequence, must be tested against the actual digit block frequency definition; visual complexity is no substitute for it.

The decimal representation $y=0.01234567890123456789\dots$ makes the same point in base ten. Every digit has frequency $1/10$, yet the only adjacent pairs areEach has limiting digit block frequency $1/10$; the other ninety pairs have frequency zero. A normal number in base ten requires $1/100$ for every one of the hundred pairs. This rational number is a simply normal number in base ten, but it is not a normal number in that base.

For a finite prefix of $N\ge r$ digits in positional notation, the digit block frequency $F_N(w)$ of a length-$r$ word $w$ is its number of overlapping occurrences divided by $N-r+1$, the number of available starting positions. All such digit block frequencies sum to one. The infinite limit used for a normal number can instead divide counts by $N$: the factor $(N-r+1)/N$ tends to one for fixed $r$.

For the six binary digits $010101$, the single-digit counts are three zeros and three ones. Its five adjacent pairs are $01,10,01,10,01$, givingThis example explains both the overlapping convention and the end correction in a finite sample. The corresponding infinite binary expansion has pair frequencies $0,1/2,1/2,0$, rather than the four values $1/4$ required of a normal number in base two.

Fermi coordinates transport a local frame along a timelike worldline. Along a timelike geodesic, a parallel-transported orthonormal frame defines Fermi normal coordinates in which the metric tensor is Minkowskian and the Christoffel symbols vanish on the worldline. Curvature first appears in terms quadratic in the spatial distance from it, so finite laboratories retain tidal forces.

For a weak, nearly static conformal factor in Nordstrom theory of gravitation, the trace of the dust stress-energy tensor is negative in signature $(-+++)$, so its scalar field equation reduces to the Poisson equation for the Newtonian gravitational potential. The geodesic equation for a conformally flat metric gives $\ddot{\mathbf{x}}=-\nabla \Phi$ at small speeds. The constant one in $\varphi$ has no effect on the force.

An affected-sib-pair analysis examines marker identity by descent between affected full siblings. Without genetic linkage to the disease locus, the probabilities of sharing zero, one and two copies are $1/4,1/2,1/4$. Excess sharing supplies evidence for linkage under disease models that preferentially transmit a common risk allele to both siblings.

// Target: biology.bigb

For independent pairs of affected full siblings with unrelated, noninbred parents and a fully informative genetic marker, the no-linkage null gives shared identity by descent count $J\sim \mathrm{Bin}(2,1/2)$. Therefore $EJ=1$ and $\mathrm{Var}J=1/2$. An excess-sharing test uses the displayed approximately standard normal statistic and an upper-tail critical value. Pairs sharing family members are dependent, so their variance needs separate treatment. This targeted test differs from an omnibus test of all three sharing probabilities.

Young's inequality for products states that for conjugate exponents $q,q^{\prime }>1$ and nonnegative $a,b$,Its weighted form absorbs a lower power into a coercive higher power at the cost of a constant.