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Notation 2026-09-04 Symbols used across the math and physics sheets, grouped the way appendix B of the
No bullshit guide groups them. See math fundamentals
Math notation
Symbol Meaning = = = ≠ \neq = equal, not equal ≈ \approx ≈ ∝ \propto ∝ approximately equal, proportional to ≡ \equiv ≡ identically equal, or defined as < < < ≤ \le ≤ > > > ≥ \ge ≥ less than (or equal), greater than (or equal) ± \pm ± ∓ \mp ∓ plus or minus; the opposite sign in the same formula ⇒ \Rightarrow ⇒ ⇔ \Leftrightarrow ⇔ implies; if and only if ∴ \therefore ∴ ∵ \because ∵ therefore, because ∞ \infty ∞ infinity ∣ x ∣ \lvert x \rvert ∣ x ∣ absolute value ⌊ x ⌋ \lfloor x \rfloor ⌊ x ⌋ ⌈ x ⌉ \lceil x \rceil ⌈ x ⌉ floor, ceiling n ! n! n ! factorial n ( n − 1 ) ⋯ 2 ⋅ 1 n(n-1)\cdots 2 \cdot 1 n ( n − 1 ) ⋯ 2 ⋅ 1 ( n k ) \binom{n}{k} ( k n ) binomial coefficient n ! k ! ( n − k ) ! \dfrac{n!}{k!(n-k)!} k ! ( n − k )! n ! ∑ i = 1 n a i \sum_{i=1}^{n} a_i ∑ i = 1 n a i sum a 1 + a 2 + ⋯ + a n a_1 + a_2 + \cdots + a_n a 1 + a 2 + ⋯ + a n ∏ i = 1 n a i \prod_{i=1}^{n} a_i ∏ i = 1 n a i product a 1 a 2 ⋯ a n a_1 a_2 \cdots a_n a 1 a 2 ⋯ a n x \sqrt{x} x x n \sqrt[n]{x} n x square root, n n n e e e π \pi π i i i 2.71828 … 2.71828\ldots 2.71828 … 3.14159 … 3.14159\ldots 3.14159 … − 1 \sqrt{-1} − 1 f ( x ) f(x) f ( x ) f − 1 ( x ) f^{-1}(x) f − 1 ( x ) function of x x x ln x \ln x ln x log b x \log_b x log b x natural log, log base b b b deg \deg deg r a d \mathrm{rad} rad degrees, radians Q.E.D. \text{Q.E.D.} Q.E.D. ■ \blacksquare ■ end of proof
Set notation
Symbol Meaning N , Z , Q , R , C \mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C} N , Z , Q , R , C naturals, integers, rationals, reals, complex { a , b } \{a, b\} { a , b } { x : P ( x ) } \{x : P(x)\} { x : P ( x )} set by listing, set by condition ∈ \in ∈ ∉ \notin ∈ / is (not) an element of ⊆ \subseteq ⊆ ⊂ \subset ⊂ subset, proper subset ∪ \cup ∪ ∩ \cap ∩ ∖ \setminus ∖ union, intersection, difference ∅ \varnothing ∅ empty set ∣ A ∣ \lvert A \rvert ∣ A ∣ cardinality A × B A \times B A × B Cartesian product [ a , b ] [a, b] [ a , b ] ( a , b ) (a, b) ( a , b ) closed, open interval ∀ \forall ∀ ∃ \exists ∃ for all, there exists R n \mathbb{R}^n R n n n n
Complex numbers notation
Symbol Meaning i i i imaginary unit, i 2 = − 1 i^2 = -1 i 2 = − 1 z = a + b i z = a + bi z = a + bi Cartesian form; Re ( z ) = a \operatorname{Re}(z) = a Re ( z ) = a Im ( z ) = b \operatorname{Im}(z) = b Im ( z ) = b z ˉ \bar{z} z ˉ z ∗ z^* z ∗ conjugate a − b i a - bi a − bi ∣ z ∣ \lvert z \rvert ∣ z ∣ modulus a 2 + b 2 = z z ˉ \sqrt{a^2 + b^2} = \sqrt{z\bar{z}} a 2 + b 2 = z z ˉ arg z \arg z arg z φ \varphi φ argument, the angle from the positive real axis z = r ( cos φ + i sin φ ) z = r(\cos\varphi + i\sin\varphi) z = r ( cos φ + i sin φ ) polar form, r = ∣ z ∣ r = \lvert z \rvert r = ∣ z ∣ z = r e i φ z = re^{i\varphi} z = r e i φ exponential form e i θ = cos θ + i sin θ e^{i\theta} = \cos\theta + i\sin\theta e i θ = cos θ + i sin θ Euler's formula z n = r n e i n φ z^n = r^n e^{in\varphi} z n = r n e in φ de Moivre
Vectors notation
Symbol Meaning v ⃗ \vec{v} v v \mathbf{v} v a vector ( v x , v y , v z ) (v_x, v_y, v_z) ( v x , v y , v z ) components ı ^ , ȷ ^ , k ^ \hat{\imath}, \hat{\jmath}, \hat{k} ^ , ^ , k ^ unit vectors along x x x y y y z z z v ⃗ = v x ı ^ + v y ȷ ^ + v z k ^ \vec{v} = v_x\hat{\imath} + v_y\hat{\jmath} + v_z\hat{k} v = v x ^ + v y ^ + v z k ^ component form ∥ v ⃗ ∥ \lVert \vec{v} \rVert ∥ v ∥ ∣ v ⃗ ∣ \lvert \vec{v} \rvert ∣ v ∣ magnitude v x 2 + v y 2 + v z 2 \sqrt{v_x^2 + v_y^2 + v_z^2} v x 2 + v y 2 + v z 2 v ^ \hat{v} v ^ unit vector v ⃗ / ∥ v ⃗ ∥ \vec{v} / \lVert \vec{v} \rVert v / ∥ v ∥ u ⃗ ⋅ v ⃗ \vec{u} \cdot \vec{v} u ⋅ v dot product u x v x + u y v y + u z v z = ∥ u ⃗ ∥ ∥ v ⃗ ∥ cos θ u_x v_x + u_y v_y + u_z v_z = \lVert\vec{u}\rVert \lVert\vec{v}\rVert \cos\theta u x v x + u y v y + u z v z = ∥ u ∥ ∥ v ∥ cos θ u ⃗ × v ⃗ \vec{u} \times \vec{v} u × v cross product, magnitude ∥ u ⃗ ∥ ∥ v ⃗ ∥ sin θ \lVert\vec{u}\rVert \lVert\vec{v}\rVert \sin\theta ∥ u ∥ ∥ v ∥ sin θ 0 ⃗ \vec{0} 0 zero vector v ⃗ ∥ \vec{v}_\parallel v ∥ v ⃗ ⊥ \vec{v}_\perp v ⊥ components parallel and perpendicular to a reference
Mechanics notation
Symbol Quantity SI unit t t t time s x x x r ⃗ \vec{r} r position m v ⃗ \vec{v} v v v v velocity, speed m/s a ⃗ \vec{a} a g g g acceleration, gravitational acceleration m/s² m m m mass kg F ⃗ \vec{F} F N ⃗ \vec{N} N T ⃗ \vec{T} T force, normal force, tension N μ s \mu_s μ s μ k \mu_k μ k static, kinetic friction coefficient none p ⃗ \vec{p} p momentum m v ⃗ m\vec{v} m v kg·m/s W W W K K K U U U E E E work, kinetic, potential, total energy J P P P power W θ \theta θ ω \omega ω α \alpha α angle, angular velocity, angular acceleration rad, rad/s, rad/s² τ \tau τ L L L I I I torque, angular momentum, moment of inertia N·m, kg·m²/s, kg·m² k k k T T T f f f A A A spring constant, period, frequency, amplitude N/m, s, Hz, m
Calculus notation
Symbol Meaning lim x → a f ( x ) \lim_{x \to a} f(x) lim x → a f ( x ) limit of f f f x x x a a a x → a + x \to a^+ x → a + x → a − x \to a^- x → a − from the right, from the left Δ x \Delta x Δ x d x dx d x finite change, infinitesimal change f ′ ( x ) f'(x) f ′ ( x ) d f d x \dfrac{df}{dx} d x df x ˙ \dot{x} x ˙ derivative (Lagrange, Leibniz, Newton for time) f ′ ′ ( x ) f''(x) f ′′ ( x ) d 2 f d x 2 \dfrac{d^2 f}{dx^2} d x 2 d 2 f x ¨ \ddot{x} x ¨ second derivative ∂ f ∂ x \dfrac{\partial f}{\partial x} ∂ x ∂ f partial derivative ∫ f ( x ) d x \int f(x)\,dx ∫ f ( x ) d x antiderivative, indefinite integral ∫ a b f ( x ) d x \int_a^b f(x)\,dx ∫ a b f ( x ) d x definite integral, area under f f f a a a b b b F ( x ) ∣ a b F(x) \big\vert_a^b F ( x ) a b F ( b ) − F ( a ) F(b) - F(a) F ( b ) − F ( a ) ∑ n = 0 ∞ a n \sum_{n=0}^{\infty} a_n ∑ n = 0 ∞ a n infinite series ∇ f \nabla f ∇ f gradient
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