Unicode Scientific Symbols Cheatsheet
This page lists commonly used scientific symbols you can paste directly into Markdown/HTML. Rendering depends on the font; for best coverage use modern fonts (e.g., Noto Sans/Serif).
Table of Contents
- Table of Contents
- 1. Greek alphabet
- 2. Unicode subscripts and superscripts
- 3. Core constants and special symbols
- 4. Arithmetic operators and basic math
- 5. Relational operators
- 6. Set theory and common math objects
- 7. Logic symbols
- 8. Calculus and analysis operators
- 9. Linear algebra, vectors, and matrices
- 10. Probability and statistics
- 11. Common arrows
- 12. Operator catalog
- 13. Physics and engineering symbols
- 14. Quantum information staples
- 15. Quick copy block
- 16. Notes for Markdown users
1. Greek alphabet
1.1 Full alphabet table
| Upper | Lower | English name | Notes / Common use |
|---|---|---|---|
| Α | α | alpha | angles, fine-structure (α), coefficients |
| Β | β | beta | coefficients, beta distribution, velocity fraction |
| Γ | γ | gamma | Euler gamma, photons (γ), Lorentz factor (γ) |
| Δ | δ | delta | change (Δ), small change (δ), Dirac delta (δ) |
| Ε | ε | epsilon | permittivity (ε), small quantity |
| Ζ | ζ | zeta | Riemann zeta (ζ) |
| Η | η | eta | efficiency (η), viscosity (η) |
| Θ | θ | theta | angles |
| Ι | ι | iota | index, small quantity |
| Κ | κ | kappa | curvature (κ), dielectric constant, kappa statistic |
| Λ | λ | lambda | wavelength (λ), eigenvalues, rate parameter |
| Μ | μ | mu | micro (µ in SI contexts), mean (μ), reduced mass |
| Ν | ν | nu | frequency (ν), degrees of freedom |
| Ξ | ξ | xi | random variable, correlation length |
| Ο | ο | omicron | rarely used as a symbol |
| Π | π | pi | circle constant (π) |
| Ρ | ρ | rho | density (ρ), correlation coefficient |
| Σ | σ | sigma | summation (Σ), std dev (σ), conductivity (σ) |
| Τ | τ | tau | time constant (τ), torque, optical depth |
| Υ | υ | upsilon | rarely used; sometimes fluid velocity |
| Φ | φ | phi | phase (φ) |
| Χ | χ | chi | chi-square (χ²) |
| Ψ | ψ | psi | wavefunction (ψ) |
| Ω | ω | omega | angular frequency (ω), ohm (Ω) |
1.2 Variant Greek forms
- epsilon: ε and ϵ
- theta: θ and ϑ
- phi: φ and ϕ
- rho: ρ and ϱ
- pi: π and ϖ
- sigma (final form): σ and ς
2. Unicode subscripts and superscripts
2.1 Digits
- Subscripts: ₀ ₁ ₂ ₃ ₄ ₅ ₆ ₇ ₈ ₉
- Superscripts: ⁰ ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹
Examples: x₀, a₁, r², 10⁻³
2.2 Signs and parentheses
- Subscript: ₊ ₋ ₌ ₍ ₎
- Superscript: ⁺ ⁻ ⁼ ⁽ ⁾
Examples: x₍ₙ₊₁₎, (n+1)⁻¹
2.3 Common superscript letters
ᵃ ᵇ ᶜ ᵈ ᵉ ᶠ ᵍ ʰ ⁱ ʲ ᵏ ˡ ᵐ ⁿ ᵒ ᵖ ʳ ˢ ᵗ ᵘ ᵛ ʷ ˣ ʸ ᶻ
2.4 Common subscript letters
- ₐ ₑ ₕ ₖ ₗ ₘ ₙ ₒ ₚ ₛ ₜ ₓ
- (Coverage varies by font; Unicode has fewer subscript letters than superscripts.)
Examples: Vₛ, Iₓ, kₙ
2.5 Combining-mark tip (fallback)
Not true subscripts, but can help for emphasis when you cannot use LaTeX/MathJax:
- combining low line: ̲ (U+0332)
- combining dot below: ̣ (U+0323)
If you need full math-quality subscripts for arbitrary text, prefer LaTeX/MathJax (e.g., x_{n+1}).
3. Core constants and special symbols
| Symbol | Name | Meaning / Typical use |
|---|---|---|
| ∞ | infinity | unbounded limit |
| π | pi | circle constant |
| ℯ | Euler’s number (alt) | sometimes used for e |
| ℏ | h-bar | reduced Planck constant |
| ° | degree | angles, temperature |
| ′ ″ | prime, double prime | derivatives, arcminutes/arcseconds |
| Å | ångström | 10⁻¹⁰ m (materials/optics) |
| µ | micro sign | 10⁻⁶ (SI prefix) |
| Ω | ohm | electrical resistance unit |
4. Arithmetic operators and basic math
| Symbol | Name | Meaning |
|---|---|---|
| + | plus | addition |
| − | minus (true minus) | subtraction (prefer over hyphen -) |
| ± | plus–minus | two possible signs |
| ∓ | minus–plus | paired with ± |
| × | multiplication | scalar multiply (often) |
| · | dot | multiplication, dot product context |
| ÷ | division | division (often avoided in formal math) |
| / | solidus | division / ratio |
| √ | square root | principal root |
| ∛ ∜ | cube/fourth root | roots |
| ! | factorial | n! |
| ‖x‖ | norm bars | norm / magnitude |
| x |
5. Relational operators
| Symbol | Name | Meaning |
|---|---|---|
| = | equals | equality |
| ≠ | not equal | inequality |
| <, > | less/greater | ordering |
| ≤, ≥ | less/greater or equal | ordering with equality |
| ≪, ≫ | much less/greater | asymptotic comparisons |
| ≈ | approximately equal | approximation |
| ≃ | asymptotically equal | used in analysis/physics |
| ≅ | congruent / approximately equal | context dependent |
| ∼ | similarity / distributed as | “similar”, or “X ∼ …” |
| ≡ | identically equal / congruent | equivalence, modular equality |
| ∝ | proportional to | proportionality |
| ≲, ≳ | less/greater or approx | approximate inequalities |
6. Set theory and common math objects
| Symbol | Name | Meaning |
|---|---|---|
| ∈ | element of | membership |
| ∉ | not element of | non-membership |
| ⊂, ⊃ | subset/superset (strict) | proper inclusion |
| ⊆, ⊇ | subset/superset | inclusion allowing equality |
| ∪ | union | set union |
| ∩ | intersection | set intersection |
| \ | set difference | A \ B |
| ∅ | empty set | no elements |
| ℕ ℤ ℚ ℝ ℂ | number sets | naturals, integers, rationals, reals, complex |
7. Logic symbols
| Symbol | Name | Meaning |
|---|---|---|
| ¬ | not | logical negation |
| ∧ | and | conjunction |
| ∨ | or | disjunction |
| ⊕ | xor | exclusive OR (also “direct sum” in algebra) |
| ⊼ | nand | Sheffer stroke variant |
| ⊽ | nor | Peirce arrow variant |
| → | implies | implication |
| ↔ | iff | biconditional |
| ⇒ | implies (double) | often used in proofs |
| ⇔ | iff (double) | equivalence |
| ∀ | for all | universal quantifier |
| ∃ | there exists | existential quantifier |
| ∄ | there does not exist | negated existential |
| ⊤ | true | truth constant |
| ⊥ | false | falsity / contradiction |
| ⊢ | proves | syntactic entailment |
| ⊨ | models | semantic entailment |
| ∴ | therefore | conclusion marker |
| ∵ | because | reason marker |
8. Calculus and analysis operators
| Symbol | Name | Meaning |
|---|---|---|
| d/dx | derivative | ordinary derivative |
| ∂ | partial derivative | multivariate derivative |
| ∇ | nabla / del | gradient operator |
| ∇· | divergence | divergence |
| ∇× | curl | curl |
| ∇² | Laplacian | Laplace operator |
| ∫ | integral | integration |
| ∬ ∬ ∬ | multiple integrals | double/triple (font support varies) |
| ∮ | contour integral | closed-path integral |
| ∑ | summation | sum over index |
| ∏ | product | product over index |
| lim | limit | limiting process |
| sup / inf | supremum / infimum | bounds |
| ∘ | composition | function composition |
9. Linear algebra, vectors, and matrices
| Symbol | Name | Meaning |
|---|---|---|
| 𝟙 or I | identity | identity matrix/operator |
| Aᵀ | transpose | transpose |
| Aᴴ | Hermitian transpose | conjugate transpose (often †) |
| A† | dagger | adjoint operator |
| det(A) | determinant | determinant |
| tr(A) | trace | trace |
| ⟨x, y⟩ | inner product | dot/inner product |
| x·y | dot product | Euclidean inner product |
| x×y | cross product | 3D cross product |
| ⊗ | tensor (Kronecker) product | tensor product |
| ⊕ | direct sum | block sum (also XOR in logic) |
| ≽, ≼ | semidefinite ordering | A ≽ 0 |
10. Probability and statistics
| Symbol | Name | Meaning |
|---|---|---|
| P(A) | probability | probability of event A |
| 𝔼[X] | expectation | mean value |
| Var(X) | variance | dispersion |
| Cov(X,Y) | covariance | joint variability |
| σ | standard deviation | spread |
| μ | mean | average |
| ⊥⊥ | independent | statistical independence |
| ~ | distributed as | X ~ Normal(0,1) |
11. Common arrows
| Symbol | Name | Meaning |
|---|---|---|
| ← → ↔ | arrows | direction / relation |
| ↑ ↓ | up/down | limits, monotonicity |
| ⇐ ⇒ ⇔ | double arrows | implication/equivalence |
| ↦ | maps to | function mapping |
| ⟶ | long right arrow | transformations |
| ⟵ | long left arrow | reverse mapping |
| ↪ | hook arrow | injection / embedding |
| ↠ | twohead arrow | surjection |
12. Operator catalog
12.1 Algebraic and binary operators
- addition/subtraction: + − ± ∓
- multiplication forms: × · ∗
- division forms: ÷ /
- composition: ∘
- convolution (common notation): ∗
- direct sum: ⊕
- tensor product: ⊗
- wedge / exterior: ∧
12.2 Bitwise and logic-like operators
- AND: ∧ (or
&in code) - OR: ∨ (or
|in code) - XOR: ⊕ (or
^in code) - NOT: ¬ (or
~/!in code) - NAND/NOR: ⊼ ⊽
12.3 Set operators
- union/intersection: ∪ ∩
- set difference: \
- Cartesian product: ×
- membership: ∈ ∉
12.4 Comparison and ordering operators
- equal/unequal: = ≠
- approximate: ≈ ≃ ≅
- inequalities: < > ≤ ≥
- much less/greater: ≪ ≫
- proportional: ∝
- equivalence: ≡
12.5 Calculus operators
- derivative/partial: d/dx, ∂
- gradient/div/curl: ∇, ∇·, ∇×
- integral/sum/product: ∫ ∮ ∑ ∏
- Laplacian: ∇²
13. Physics and engineering symbols
| Symbol | Typical meaning |
|---|---|
| ℏ | reduced Planck constant |
| λ | wavelength |
| ω | angular frequency |
| k | wavenumber / spring constant |
| ε, ε₀ | permittivity, vacuum permittivity |
| μ, μ₀ | permeability (or mean), vacuum permeability |
| σ | conductivity / standard deviation |
| ρ | density |
| Φ, φ | flux / phase |
| Δ | change (finite) |
| δ | small change / variation |
| ∇ | spatial differential operator |
| ⟂ ∥ | perpendicular / parallel |
Maxwell’s equations (differential form)
Gauss’s law (electric): ∇·E = ρ/ε₀
Gauss’s law (magnetism): ∇·B = 0
Faraday’s law of induction: ∇×E = −∂B/∂t
Ampère–Maxwell law: ∇×B = μ₀J + μ₀ε₀ ∂E/∂t
Maxwell’s equations (integral form)
Gauss’s law (electric): ∯_S E·dA = Q_enc/ε₀
Gauss’s law (magnetism): ∯_S B·dA = 0
Faraday’s law of induction: ∮_C E·dl = − d/dt ( ∯_S B·dA )
Ampère–Maxwell law: ∮_C B·dl = μ₀ I_enc + μ₀ε₀ d/dt ( ∯_S E·dA )
Schrödinger equation
Time-dependent Schrödinger equation: iℏ ∂ψ/∂t = Ĥψ
Common single-particle form (potential V): iℏ ∂ψ/∂t = [ −(ℏ²/2m) ∇² + V ] ψ
Poisson’s equation
Electrostatics (for potential φ): ∇²φ = −ρ/ε₀
General Poisson form: ∇²u = f
14. Quantum information staples
| Symbol | Name | Meaning |
|---|---|---|
| |ψ⟩ | ket | quantum state vector |
| ⟨ψ| | bra | dual vector |
| ⟨ψ|φ⟩ | inner product | amplitude / overlap |
| ρ | density matrix | mixed state |
| Tr(ρ) | trace | normalization and expectation |
| ⊗ | tensor product | composite systems |
| 𝟙 | identity | identity operator |
| σₓ σᵧ σ_z | Pauli operators | X, Y, Z |
15. Quick copy block
Greek: α β γ δ ε θ λ μ ν π ρ σ τ φ χ ψ ω
Logic: ¬ ∧ ∨ ⊕ → ↔ ∀ ∃ ⊤ ⊥
Calc: ∂ ∇ ∫ ∮ ∑ ∏ √ ∞
Linear algebra: ⟨ ⟩ ‖ ⊗ ⊕ †
Relations: ≤ ≥ ≠ ≈ ∝ ≡
16. Notes for Markdown users
- Prefer the true minus − over hyphen - in equations.
- Some symbols (blackboard-bold sets, multiple integrals) depend heavily on font support.
- If you need exact spacing and typography, use LaTeX/MathJax instead of raw Unicode.