Mathematics, at hand.

Move it once

Drag the empty part of the top header, beside the Symbraix name. Resize from any edge or corner, including the upper edge and upper corners. The expression and result remain selectable. We remember the window position and restore it when you reopen the app. If a display is disconnected, the calculator is moved back into an available screen area.

The filled pin indicates Keep on Top. Click it, or use Window > Keep on Top, to allow other windows in front. Your choice is remembered. The window requests availability across desktop Spaces; full-screen applications, Stage Manager, and system overlays remain subject to macOS window management.

Text size in Settings

Choose Symbraix > Settings… or press Command-comma, including while typing an expression. The small native Settings window opens above the pinned calculator. Opening Settings again brings the same window forward. Closing it leaves Keep on Top unchanged.

Input font size changes the active multiline editor. Calculation display font size changes existing and future worksheet expressions and text answers, including vectors, matrices, units, uncertainty, symbolic and indexed tensor results. The controls are independent, accept 10–36 points, and take effect immediately. Use the numeric field (Return to commit) or the stepper. Tab moves between controls.

The defaults preserve the earlier appearance: input 17 pt and calculation display 22 pt. The display value is the main answer size; expressions and matrix answers retain their smaller relative sizing, with a 10 pt minimum. Font families and alignment stay the same. Toolbar buttons and plot labels do not grow.

Changing fonts preserves your unfinished text, selection, text undo history and stored values. It does not calculate again. Long input wraps and scrolls; wide answers scroll horizontally and multiline answers grow vertically. Both preferences survive quitting and reopening. New, Open and Save affect calculations, not these app preferences. Restore Defaults resets only the two text sizes, without clearing calculations or changing pinning or window geometry.

Resize at the visible boundary

Drag the visible top, bottom, left or right border to change one dimension, or a rounded corner to change both. The opposite edge or corner stays anchored, subject to the minimum usable size and available screen area. A directional resize cursor distinguishes the boundary from the header interior. Drag the empty interior header beside the name to move the whole window. These gestures work with Keep on Top on or off; the resized geometry is remembered.

Close, hide, or quit deliberately

Changing apps does not close the calculator. Standard Mac controls still let you close, hide, or minimize it intentionally. Window > Show Symbraix, Command-0, the Dock icon, or the menu-bar entry restores it. Command-Q quits the app. There is no automatic login launch.

These are app shortcuts, not system-wide hotkeys. Use the menu-bar entry when another app is active. The interface follows system light or dark appearance; Reduce Transparency uses a solid background.

Names store values, not live formulas

Names are case-sensitive. Use names such as x, velocity, or sample_2, beginning with a letter or underscore. Constants, function names, i, j, and ans are reserved. There can be up to 64 named variables. Reassigning an existing name replaces its value. Legacy saved variables that share a newly added function name remain usable; the variable name and a function call such as det(A) are distinct.

For example, after x = 7 and y = x + 1, y is 8. Changing x to 20 does not change y until you evaluate its assignment again. Recalling history reevaluates the expression using current variables, not the old values.

Operators and constants

Write 2pi or 3i, not 2pi or 3i. Use a decimal point and commas between arguments, independent of region settings. Scientific notation such as 1e-6 is supported. If any statement fails, no statement from that submission changes variables, ans, or the worksheet. Correct the highlighted input and evaluate again.

Define, calculate, continue

Type this on one line and press Return:

x = 4; y = 7; x+y**2

The worksheet records three results: 4, 7, and 53. The input clears after success and keeps keyboard focus, ready for the next calculation. Alternatively, use Shift-Return between statements and Return once to evaluate the whole input.

Reuse an earlier answer

Each new result is labeled Out[n]. Click that label to insert out(n) at the input cursor, or type the reference yourself. It uses the stored value, not the rounded screen text. On a new session, the example above gives Out[3] = 53; out(3)*2 then returns 106. Use the actual number shown in your worksheet.

The reuse-arrow recalls an expression for evaluation with current variables. Out[n] instead inserts the saved value, including its uncertainty correlations. Copy Result produces reusable text for ordinary values. Copying a measurement creates an independent measurement when pasted; use Out[n] or Save when identity matters.

Separators and errors

Semicolons inside brackets remain matrix row separators: A=[1,2;3,4]; A@A is two statements. New lines inside brackets or parentheses are whitespace; keep a semicolon between matrix rows. Wrap a long expression in parentheses to continue it over several lines.

A failed submission keeps your input and reports the offending statement and location. None of its assignments or results is saved. Earlier successful work remains unchanged. Blank lines and trailing semicolons are accepted; a trailing semicolon does not suppress output. This is a sequential worksheet, not a reactive spreadsheet.

A submission supports up to 128 statements and shares one work budget. The worksheet retains up to 1,000 entries and 200,000 result elements, removing the oldest entries first. Result numbers are never reassigned while a session continues; an evicted out(n) reports an unavailable-result error. Assign important results to named variables.

Components, magnitude, and phase

real(z) and imag(z) extract components. conj(z) (or conjugate(z)) takes the conjugate. abs(z) is the magnitude, and arg(z) (or phase(z)) is the phase in radians. phase(0) is defined as 0 in Symbraix. complex(a,b) constructs a + i*b from real inputs and broadcasts over vectors and matrices.

Branches are a deliberate convention

Complex square roots, logarithms, non-integer powers, and inverse functions return a principal value, not every possible value. For example, ln(-1) returns approximately pii and sqrt(-4) returns 2i. A negative real input can therefore produce a complex answer instead of a domain error.

Use an explicit small positive or negative imaginary component to approach either side of a branch cut. Exactly zero imaginary components are normalized to real values; signed-zero branch-side distinctions are not preserved in stored results. Tiny nonzero components are not silently discarded.

Integer powers use repeated multiplication, so i^2 is exactly -1. Expressions such as exp(i*pi) can retain a tiny imaginary rounding residual. cbrt(-8) returns the real cube root, -2; a genuinely complex input uses a principal cube-root value.

Complex numbers are not ordered. min, max, minimum, maximum, floor, ceil, round, sign, and % require real inputs. Select abs, real, or imag first when that is the operation you intend.

Elementwise is the default

Arithmetic and ordinary scientific functions act on each element. A scalar or a length-one vector expands across the other vector. Otherwise lengths must match. Empty vectors are supported where the operation is defined. A shape error explains incompatible lengths.

Two different inner products

dot(a,b), or a @ b, sums a[k]b[k] without conjugation. vdot(a,b) sums conj(a[k])b[k]. Both require equal-length vectors and do not broadcast. This follows the NumPy distinction, not MATLAB's default complex dot convention.

a = [1+i, 2]
dot(a,a)     -> 4 + 2*i
vdot(a,a)    -> 6

Access and combine elements

Indexing starts at zero: [10,20,30][0] is 10 and [10,20,30][-1] is 30. concat joins vectors. diff returns successive differences; cumsum returns running sums. cross accepts two three-element vectors. Matrix rows and entries are described on the next page. Slices are not supported.

Keep elementwise and matrix operations distinct

The operators +, -, *, /, ^ and ** act entry by entry. This keeps the vector rules consistent. Use @ or matmul(A,B) for a matrix product, and mpow(A,n) for an integer matrix power. A^-1 is entrywise reciprocal, not the inverse; use inv(A). sin(A) and exp(A) are entrywise functions, not matrix sine or matrix exponential.

Broadcast by shape, not by guesswork

Shapes align at their trailing dimension. A matrix with two columns can add [10,20] to each row. A column matrix [[10],[20]] adds 10 to the first row and 20 to the second. A dimension may expand only when its length is 1. For row-specific factors, write reshape([10,20],2,1). Mismatched dimensions produce an error.

Select, reshape, and copy

A[0,1] selects row 0, column 1. A[1] selects the second row; A[-1,-1] selects the last entry. row(A,r) and col(A,c) return vectors. shape(A), rows(A), and cols(A) inspect dimensions. reshape and flatten use row-major order. The screen displays matrix rows separately; Copy Result returns a complete nested-array expression.

Transpose is not the conjugate transpose

transpose(A) swaps rows and columns without conjugating. adjoint(A), also named ctranspose(A), conjugates as it transposes. conj(A) conjugates entries without transposing. vdot(A,B) flattens both arrays and conjugates the first; it requires equal total element counts. A rank-one vector remains rank one under transpose. Reshape it to a row or column matrix when orientation matters.

Solve first; invert only when you need the inverse

solve(A,b) solves A @ x = b for a square, nonsingular A. The right-hand side may be a vector or a matrix of multiple columns. Use it instead of inv(A) @ b when the solution is your goal. A singular system reports an error; this version does not solve rectangular least-squares problems.

det, inv, solve, rank, and mpow limit dense dimensions to 128. Near-singular pivot warnings are heuristic, not a condition-number estimate. rank uses pivoted elimination and an optional absolute tolerance, not an SVD. The default tolerance is max(rows,cols)epsmax(abs(entries)). Check residuals and problem conditioning before relying on sensitive results.

Other tools include eye/identity, zeros(r,c), ones(r,c), diag, outer, kron, hstack, and vstack. hstack/vstack require matrix arguments; reshape vectors explicitly. mpow accepts integer exponents, including negative values for invertible matrices. Work limits can reject a long compound calculation even when each matrix fits.

Sequence boundaries

arange excludes its stop value; its step cannot be zero. linspace includes both endpoints when count is at least 2; count 1 returns only the start and count 0 returns an empty vector. Floating-point steps may produce surprising boundaries; use linspace when endpoint inclusion matters.

sum([]) is 0, prod([]) is 1, and norm([]) is 0. mean, min, and max require at least one element. Zero to the zero power is defined as 1. Division by zero and logarithms of zero report an error rather than an infinity.

Small contributions can disappear

2^53 + 1

At this scale, adjacent representable values are no longer one unit apart. The result can round back to 2^53. Warnings flag some large-value, lost-contribution, and underflow cases, but they are not complete error bounds. Extra printed digits do not recover lost accuracy.

Likewise, sqrt(8)^42/100 is roughly 9.22e16. Adding several units to a value this large cannot be represented reliably by binary64. Reformulate the calculation or use a validated higher-precision tool when those units matter.

Finite results and understandable errors

Overflow, division by zero, undefined names, incompatible array shapes, invalid syntax, and exceeded resource limits show an error near the input. Edit the expression and try again. Your previous successful values are preserved. Expressions cannot execute Python, shell commands, file operations, or arbitrary code.

Parser-depth and work limits prevent unbounded calculation. A storage warning means the latest state was not saved. Reduce stored data before relying on relaunch recovery.

What this version does not do

This version does not provide a full computer algebra system, symbolic integration or tensor geometry, arbitrary-precision arithmetic, RPN, user-defined function declarations, general nonlinear solving, or Python/MATLAB execution. Units and uncertainty are real-scalar features; plots are two-dimensional sampled snapshots.

Local by design

We save variables, the last result, the worksheet, and the unfinished input locally. Out[n] references survive normal relaunch. Older preview-only history can be reevaluated for reusable values. Save and Open preserve separate snapshots in files you choose; opening does not execute their expressions.

Symbraix has no account system, advertising, analytics SDK, cloud sync, or calculation network service. Apple manages purchases and any operating-system diagnostics you choose to share. Copy Result uses the system clipboard only when requested; other apps and system clipboard features may access copied content.

Clear or remove your calculations

Choose New in the header or File menu, or press Command-N. Confirm when the chain is not empty. This clears variables, ans, results, numbering, errors, and the input draft. It keeps window and appearance preferences. The reset cannot be undone and does not erase exported files, clipboard data, or your backups.

To remove the app, first clear the session if you also want the saved calculations removed, then quit and move the application to Trash using Finder. Removing the app alone may leave its local data and preferences for a later reinstall. No separate self-uninstaller is included.

Troubleshooting

Text editing uses standard Mac selection, Copy/Paste, Undo, and Redo. Undo applies to editing, not completed calculations or New. Save a useful chain before resetting it. Remove sensitive calculations before sharing a chain or support report.

Construct units explicitly

Write products, quotients, and powers inside the quotes: "kg*m/s^2", "m^2", or "m/s". SI base units, SI prefixes, and common derived units such as N, Pa, J, W, Hz, V, ohm, and T are recognized. Selected practical units include min, h, day, L, in, ft, yd, mi, lb, oz, bar, atm, and eV. Use u for the micro prefix.

force = quantity(2,"kg") * quantity(3,"m/s^2")
convert(force,"N")

The result is 6 N. Multiplication and division calculate in canonical SI units; convert chooses a compatible display unit. Addition retains the left-hand display unit. Adding a length to a time is an error, not an implicit conversion.

Extract values deliberately

valueof(q) returns the nominal number in the displayed unit and removes its unit information. nominal(q) keeps the units but removes uncertainty. sigma(q) returns the numerical standard uncertainty in the displayed unit; stdunc(q) keeps those units. Converting a display unit does not change the underlying quantity.

Angles use rad or deg; trigonometric quantity functions convert to their dimensionless SI argument. Offset temperatures such as Celsius and Fahrenheit are intentionally rejected. Use K; automatic offset-versus-interval temperature arithmetic is not implemented. Parentheses can group unit products; implicit products are not supported.

A measured area

length = measure(2,0.01,"m")
width = measure(30,0.2,"cm")
convert(length*width,"m^2")

This gives 0.6 +/- 0.005 m^2. The two measurements are independent here. Reusing the same variable is different from entering a second measurement with the same numerical value.

Keep the dependency, not just the printed answer

Assignments, Out[n], normal relaunch, and Save/Open preserve source identity and therefore correlation. Copying a measurement to plain text and pasting it creates a new independent measure call. Use an Out[n] reference when the original measurement is intended.

correlated(value,sigma,reference,rho,"unit") constructs a measurement with the specified correlation to a reference, for -1 <= rho <= 1. covariance(a,b) retains product units; correlation(a,b) is dimensionless. The source model supports up to 128 independent sources, not arbitrary covariance-matrix input.

Know where linearization stops

Uncertainty is approximated by a local first-order expansion, including covariance. It can be misleading for strongly nonlinear functions, large uncertainties, or near singularities. Warnings are safeguards, not complete validity tests. No Monte Carlo, second-order correction, or distribution-specific coverage probability is computed.

Supported measured operations include arithmetic, fixed real powers, sqrt, abs, trigonometric and hyperbolic functions, exp, and logarithms where their real domains and derivatives are defined. Unsupported functions or nondifferentiable operating points report an error instead of silently dropping uncertainty.

Build, substitute, evaluate

s = sym("x^2 + y")
s = substitute(s,"y",3)
evaluate(s,"x",2)

The final result is 7. sym("x") is a reusable symbolic value. Numerical evaluation accepts scalar or array bindings and the existing complex conventions. A symbolic value cannot recursively serve as its own numerical binding.

Exact coefficients, bounded scope

Decimal coefficients typed inside a quoted symbolic expression become exact rational coefficients when they fit checked 64-bit integers. This is not arbitrary precision. Passing an already evaluated numerical scalar preserves its floating-point representation instead. Overflow and expression-size limits report errors.

The subset supports combining polynomial terms, expansion, restricted univariate factoring, substitution, and differentiation of supported elementary functions. factor extracts integer roots from -100 through 100 for polynomials of degree up to 8; it can leave an unfactored residual. roots solves linear or quadratic polynomials numerically, including complex roots, not general equations.

The system avoids unconditional x/x cancellation and does not replace sqrt(x^2) with x. Original domains still apply after a transformation, and displayed warnings flag domain-sensitive expressions. This is not a full algebra system, assumption solver, symbolic matrix package, or integration engine.

Explicit rules, no hidden tensor mode

An einsum expression requires ->, up to eight inputs, and one ASCII letter per axis. Repeated labels must have equal sizes, including repeated labels on the same operand. There is no ellipsis, implicit index broadcasting, automatic conjugation, or implicit metric. Write conj explicitly for a Hermitian contraction.

Elementwise arithmetic and supported scientific functions broadcast along trailing dimensions in the usual array sense. shape(Tn) inspects dimensions; flatten(Tn) returns a vector. permute(Tn,[2,0,1]) reorders axes. reshape accepts an explicit shape vector. A full index such as Tn[1,0,1] returns one entry; a prefix index returns the remaining subarray. Negative indices are supported; slices are not.

Numerical arrays, not geometric tensors

These untyped commands contract stored numbers without assigning index variance or barred kind. For explicit geometric index types, metrics, raising/lowering and changes of frame, use the indexed commands on pages 20-22. General abstract tensor calculus remains outside this version.

Use einsum for higher-rank contractions, not @. For a reduction unsupported directly on a higher-rank tensor, flatten it explicitly. Contractions are work-budgeted and may reject large operations even when the individual arrays fit in memory.

Plot arrays you have already calculated

t = linspace(0,10,500)
y = exp(-0.3*t)*cos(2*pi*t)
plot(t,y,"Damped oscillation")

plot draws lines; scatter draws points; logplot starts with a logarithmic y axis. A matrix y supplies one curve per row, with up to eight curves. The x vector and each row must have matching lengths, from 2 through 10,000 samples.

Sample a symbolic expression

plot(sym("sin(x)"),"x",-pi,pi,401)

This form samples a supported symbolic expression on a uniform grid. The count is optional (401 by default, 2-2,000 allowed). Nonreal or domain-invalid samples become gaps. Unknown variables, unsupported functions, and exhausted work limits remain errors.

Complex plane and axis controls

theta = linspace(0,2*pi,500)
z = exp(i*theta)
plot(real(z),imag(z),"Unit circle")

Enable Equal scale to give equal distances on the linear axes equal screen lengths. Log X and Log Y are separate controls; equal scale is unavailable while either is logarithmic. Nonpositive log samples are omitted with a visible count. Complex values are never silently reduced to their real parts: choose real, imag, abs, or arg explicitly.

Copy or export the figure

Copy SVG places the figure on the clipboard. Export saves an SVG file in the location you choose. SVG is a vector format suitable for resizing; the current plot data and selected axes are used. Exported files are not erased by New.

Sampling cannot guarantee that a narrow feature or discontinuity is detected. A pole exactly on a sample can create a gap, but a pole between samples can still be connected. Inspect the function and choose a suitable domain and sampling density. There is no adaptive plotting, 3D plotting, unit-aware axis conversion, or uncertainty-band plot in this version.

What is preserved

Saved JSON contains variables, full retained result values, expressions, result identifiers, plot data, unit metadata, uncertainty source identities, and the unfinished draft. Opening restores these values without executing their source expressions. A malformed, oversized, or incompatible file reports an error without replacing the active chain. The current open-file limit is 16 MB.

Window position, pinning, and appearance belong to the app, not the chain file. New resets variables, ans, worksheet entries, result numbering, errors, and the input draft; the next result is Out[1]. It does not reset the window or delete saved files. There is no undo for New, so save important work first.

Privacy and portability

Saved chains are readable JSON, not encrypted documents. Their calculations may be sensitive. Share them deliberately and keep your own backups. File access is limited to the app container and files you choose through the system panels; no account or network service is needed.

Candidate 07 restores earlier compatible sessions. Chains with indexed tensors and metrics require Candidate 06 or a later compatible version. Older candidates cannot read those new value types. Complete-command history is preserved; older sessions recover the individual history entries they actually saved.

A few deliberate limits

Symbolic expressions are limited to 512 nodes; expansion is limited to 256 terms. General symbolic integer powers are bounded to magnitude 32, and expansion to nonnegative powers through 16. Measurement propagation supports 128 independent sources. Stored worksheet entries share the existing aggregate limits, so save important intermediate work before old entries are evicted.

Calculations are transactional: an error in a batch commits no assignments, values, or plots from that batch. This is a reproducible numerical scratchpad with explicit advanced commands, not a general scripting language or a complete replacement for a numerical programming environment.

Keep multiline editing natural

Within a multiline draft, ordinary Up and Down move the caret. At the first logical line, Up recalls an older command; at the last logical line, Down recalls a newer command. A whole-field selection also permits recall. A partial selection retains ordinary selection movement. Soft-wrapped segments belong to the same logical line.

What history remembers

We keep up to 256 successful command submissions, subject to a combined text limit. Failed submissions do not become history. New clears it. Save/Open preserves it. Older saved sessions did not retain original submission boundaries, so they restore only the individual entries they actually contain.

A Lorentz-signature example

Axes are numbered from zero. A symmetric bilinear metric may be indefinite. The constructor checks symmetry, dimensions and nonsingularity; it does not impose positive definiteness.

Contractions do not guess

contract pairs one upper and one lower occurrence of each dummy label, in the same space/frame and bar kind. Free labels occur once and must appear in the output. An extra or mismatched label reports an error. No metric is silently inserted. The argument labels select axes in order; slot metadata supplies the geometric types.

For raw component operations, use components(T) explicitly and then the ordinary array functions or einsum. Tensor multiplication uses tproduct; typed addition requires identical ordered slot types. Nonlinear elementwise functions do not silently preserve a geometric tensor type.

State the metric convention explicitly

A Hermitian metric in our syntax has ordered slots (_~a, _b), corresponding to the matrix H in conjugate(u)^T H v. Its inverse has upper slots (^a, ^~b). H must equal its conjugate transpose and be nonsingular, but may be indefinite.

Metric duality is not conjugation

With a bilinear metric, raising or lowering changes variance but preserves bar kind. The conjugate metric acts on barred slots. With our Hermitian convention, raising or lowering changes variance and bar kind; it does not independently conjugate the input components. tinner explicitly conjugates its first operand for a Hermitian metric, and not for a bilinear metric.

Neither a component transpose nor a conjugate transpose should be mistaken for raising or lowering an index. Use permute or transpose to reorder typed slots, conj to conjugate, and raise/lower for metric duality.

Command reference

indexed, components, signature, metric, metrictensor, inversemetric, raise, lower, contract, tproduct, tinner, changebasis, symmetrize and antisymmetrize. conj, transpose and permute also preserve the appropriate typed metadata.

Scope and saved work

Typed tensors support at most six slots and share the numerical engine limits. Metrics and basis maps use the dense linear-algebra limits. Values and types survive normal session restoration and Save/Open. Full abstract dummy-index canonicalization, symbolic tensor fields, covariant derivatives, connection coefficients and curvature are not yet commands in this version.