Notation cheat sheet
Every symbol, operator, decoration and constant used in this guide, grouped
by topic. Each entry says how to read it aloud, what it means, gives a small
example, and names the chapter where it does the most work. When a symbol
has several meanings (λ, σ, π, e), all of them are listed. The
chapter you're reading always says which one applies.
Arithmetic, comparison and rounding
| Symbol | Read it as | Meaning | Example | Ch. |
|---|---|---|---|---|
+, - / − |
"plus", "minus" | addition, subtraction (or a negative sign) | 5 − 3 = 2 |
all |
· or × |
"times" | multiplication (· is used inside formulas, × in prose) |
3·4 = 12 |
all |
/ or ÷ |
"divided by" | division | 12/4 = 3 |
all |
± |
"plus or minus" | a range, or both signs | 50 ± 9.8 ms = 40.2 to 59.8 |
14, 22 |
= |
"equals" | exactly equal | 2 + 2 = 4 |
all |
≠ |
"is not equal to" | different | AB ≠ BA |
10 |
≈ |
"approximately" | close enough for the purpose | π ≈ 3.14 |
all |
≡ |
"is congruent to" (or "is defined as") | equal after reducing mod n |
17 ≡ 2 (mod 5) |
2, 19 |
∝ |
"is proportional to" | equal up to a constant factor | P(spam|words) ∝ … |
16 |
< > ≤ ≥ |
"less than", "greater than or equal" … | ordering | ρ < 1 |
all |
≪ / << |
"much less than" | orders of magnitude smaller | r ≪ n in LoRA |
20 |
|x| |
"absolute value of x" | drop the sign | |-3| = 3 |
9, 10 |
⌊x⌋ |
"floor of x" | round down | ⌊2.7⌋ = 2 |
2 |
ceil(x) / ⌈x⌉ |
"ceiling of x" | round up | ceil(6.1) = 7 |
1, 29 |
√x |
"square root of x" | the number that squared gives x |
√144 = 12 |
9, 14 |
xⁿ, x^n |
"x to the n" | x multiplied by itself n times |
2¹⁰ = 1024 |
5 |
x⁻¹ |
"x inverse" | 1/x, or the modular / matrix inverse |
3⁻¹ ≡ 4 (mod 11) |
10, 19 |
n! |
"n factorial" | n·(n-1)·…·1 |
4! = 24 |
14, 15 |
% |
"percent" or, in code, "mod" | per hundred; in Python a % n is the remainder |
17 % 5 = 2 |
2 |
∞ |
"infinity" | grows without bound | W → ∞ as ρ → 1 |
12, 25 |
½, ¼ |
"a half", "a quarter" | fractions | ½λw² |
13, 21 |
→ |
"goes to" / "maps to" | a limit, or a function's input → output | h → 0 |
4, 12 |
° |
"degrees" | angle; 360° = 2π radians |
90° rotation |
7, 10 |
↔ |
"swaps with" / "if and only if" | two-way correspondence | rows ↔ columns |
10 |
∎ |
"end of proof" | the argument is complete | 17 |
Decorations on letters: subscripts, superscripts, hats, bars
| Notation | Read it as | Meaning | Example | Ch. |
|---|---|---|---|---|
xᵢ, x_i, xₖ, x₁ |
"x sub i" | a subscript is an index or label: the i-th item, or a named variant (β₁, f_max, d_k) |
x₃ = third value |
all |
x_{n+1} |
"x sub n plus one" | the next value in a sequence or iteration | Newton's x_{n+1} |
12 |
A[i][j], Aᵢⱼ |
"A i j" | the entry in row i, column j |
A²[R1][R4] = 1 |
10 |
x², xᵏ, xᵗ, gᵉ |
"x squared", "x to the k" | a superscript is a power (not an index), except ᵀ (transpose) and ⁻¹ (inverse) |
σ² = variance |
all |
x̄, ȳ |
"x bar", "y bar" | the average of the x (or y) values |
x̄ = 3 |
9, 14 |
p̂, θ̂ |
"p hat" | an estimate computed from data | p̂ = k/n |
21, 22 |
x*, θ* |
"x star" | the optimal value | n* = 20 servers |
12, 13 |
f'(x), f''(x) |
"f prime", "f double prime" | first and second derivatives | (x²)' = 2x |
12 |
Δx |
"delta x" | a (finite) change in x |
slice width in a Riemann sum | 12 |
x_new ← … |
"x becomes …" | assignment / an update step | w ← w - η·g |
9, 21 |
[i = c] |
"one if i equals c, else zero" | indicator (Iverson bracket) | used in the softmax gradient | 13 |
Sets and logic
| Symbol | Read it as | Meaning | Example | Ch. |
|---|---|---|---|---|
{…} |
"the set of …" | a collection with no duplicates | {2, 3, 5, 7} |
4 |
∈ / ∉ |
"is in" / "is not in" | membership | 3 ∈ {1,2,3} |
4 |
⊆ |
"is a subset of" | every element is also in the other set | {1} ⊆ {1,2} |
4 |
∪ |
"union" | in either set | A ∪ B |
4, 8 |
∩ |
"intersection" | in both sets | A ∩ B |
4, 8 |
∅ |
"the empty set" | no elements, an impossible event | A ∩ B = ∅ |
8 |
Ω |
"omega" | sample space: every possible outcome | coin: {H, T} |
8 |
ℝ |
"the reals" | all real numbers | x ∈ ℝ |
10 |
ℝⁿ |
"R n" | lists (vectors) of n real numbers |
an embedding ∈ ℝ⁷⁶⁸ |
10, 20 |
ℤ, ℤₙ |
"the integers", "Z mod n" | whole numbers; {0, …, n-1} with wrap-around |
ℤ₁₇ |
19, 24 |
ℤₙ* |
"the units mod n" | elements of ℤₙ that have an inverse; there are φ(n) |
ℤ₁₀* = {1,3,7,9} |
19 |
∀ / ∃ |
"for all" / "there exists" | quantifiers | ∀x: x² ≥ 0 |
4 |
¬, NOT |
"not" | negation | ¬true = false |
3 |
∧, AND, & |
"and" | both | 1 ∧ 0 = 0 |
3 |
∨, OR, | |
"or" | either (or both) | 1 ∨ 0 = 1 |
3 |
⊕, XOR, ^ |
"x-or" | exactly one of the two | 1 ⊕ 1 = 0 |
3 |
<<, >> |
"shift left / right" | multiply / divide by powers of 2 (in code) | 1 << 4 = 16 |
1 |
f: A → B |
"f maps A to B" | a function from inputs A to outputs B |
hash: str → int |
4 |
Number theory and cryptography
| Symbol | Read it as | Meaning | Example | Ch. |
|---|---|---|---|---|
a mod n |
"a mod n" | remainder of a ÷ n |
17 mod 5 = 2 |
2 |
a ≡ b (mod n) |
"a is congruent to b mod n" | n divides a - b |
38 ≡ 2 (mod 12) |
2, 19 |
a | b / a ∤ b |
"a divides b" / "does not divide" | b is (not) a multiple of a |
5 | 15, p ∤ a |
19 |
gcd(a, b) |
"greatest common divisor" | largest number dividing both | gcd(48, 18) = 6 |
19 |
a⁻¹ (mod n) |
"inverse of a mod n" | the x with a·x ≡ 1; exists iff gcd(a,n)=1 |
17⁻¹ ≡ 2753 (mod 3120) |
19, 24 |
φ(n) |
"phi of n", Euler's totient | count of 1…n coprime to n |
φ(61·53) = 3120 |
19, 24 |
π(N) |
"pi of N", prime-counting function | number of primes ≤ N (not 3.14…) |
π(100) = 25 |
19 |
p, q |
large secret primes (RSA) / a public prime (DH) | 61, 53 | 24 | |
n = p·q |
"the modulus" | RSA public modulus; also a curve's group order in ECDSA | 3233 | 24 |
e, d |
"public / private exponent" | RSA keys, e·d ≡ 1 (mod φ(n)) (this e is not 2.718…) |
17, 2753 | 24 |
g, a, b, A, B |
DH generator, private exponents, public values | A = gᵃ mod p |
24 | |
G, k·G |
"G", "k times G" | elliptic-curve base point; adding G to itself k times |
3G = (10, 6) |
24 |
O |
"the point at infinity" | the identity ("zero") of an elliptic-curve group | 19G = O |
24 |
(r, s), k, z |
ECDSA signature, per-signature nonce, message hash | 24 | ||
H(·) |
"hash of" | a cryptographic hash function | SHA-256(m) |
24 |
2ᵇ "bits of security" |
an attack needs about 2ᵇ operations |
AES-128: 2¹²⁸ |
24 |
Functions, sums and calculus
| Symbol | Read it as | Meaning | Example | Ch. |
|---|---|---|---|---|
f(x) |
"f of x" | function f applied to input x |
f(3) = 9 for x² |
4 |
f(g(x)), f∘g |
"f of g of x", "f composed with g" | composition: apply g, then f |
layers of a network | 13 |
Σᵢ₌₁ⁿ xᵢ |
"sum from i equals 1 to n of x i" | add up a sequence (a for loop with +=) |
Σ₁³ i = 6 |
6, all |
∏ᵢ xᵢ |
"product over i" | multiply a sequence | ∏(1 - i/N) |
24 |
lim(h→0) |
"the limit as h goes to 0" | the value approached as h shrinks |
slope of a tangent | 12 |
d/dx, dy/dx |
"d by d x", "d y d x" | derivative: rate of change of y per unit x |
d/dx x² = 2x |
12 |
∂f/∂x |
"partial f partial x" | derivative w.r.t. x, others held fixed |
∂(x²y)/∂x = 2xy |
13 |
∂²f/∂xᵢ∂xⱼ |
"second partial" | curvature; an entry of the Hessian | 13 | |
∇f |
"grad f" / "nabla f" | vector of all partial derivatives | ∇f = (4, 13) |
13, 21 |
∇ₓL |
"gradient of L with respect to x" | gradient taken w.r.t. one specific variable | input gradient in FGSM | 28 |
D_u f |
"directional derivative along u" | slope in direction u, = ∇f·u |
12.8 |
13 |
J |
"the Jacobian" | matrix of derivatives of a vector function | 13 | |
H (calculus) |
"the Hessian" | matrix of second derivatives | [[2,1],[1,2]] |
13, 21 |
∫ₐᵇ f(x) dx |
"integral from a to b of f of x d x" | area under f; the total accumulated |
∫₀³ x² dx = 9 |
12 |
dx |
"d x" | an infinitely thin slice of x |
12 | |
F(x) |
"big F" | an antiderivative (F' = f), or a CDF |
F(x) = x³/3 |
12, 14 |
C (integration) |
"plus C" | the constant of integration | ∫2x dx = x² + C |
12 |
e^x, exp(x) |
"e to the x" | the exponential function | e⁰ = 1 |
5, 12 |
ln x |
"natural log" | log base e |
ln e = 1 |
5 |
log₂ x, log₁₀ x |
"log base 2 / base 10" | bits; decibels and orders of magnitude | log₂ 1024 = 10 |
5, 23, 27 |
sin θ, cos θ, tanh x |
"sine", "cosine", "tanch" | trigonometric / hyperbolic functions | cos 0 = 1 |
7, 12 |
σ(x) |
"sigmoid of x" | 1/(1 + e⁻ˣ), squashes to (0, 1) |
σ(0) = 0.5 |
12 |
ReLU(z) |
"rel-u" | max(0, z) |
ReLU(-1.4) = 0 |
10, 21 |
softmax(z) |
"softmax" | eᶻⁱ / Σ eᶻʲ, turns scores into probabilities |
(0.66, 0.24, 0.10) |
13, 20 |
sign(x) |
"sign of x" | +1, 0 or -1 |
sign(-0.3) = -1 |
28 |
max, min |
"max", "min" | largest / smallest value | max(0, -2) = 0 |
all |
argmax_θ f(θ) |
"arg max over theta" | the θ that makes f largest (not the max value) |
MLE, greedy decoding | 14, 22 |
O(f(n)) |
"big O of f of n" | grows no faster than f(n) |
binary search O(log n) |
18 |
i (complex) |
"i" | the imaginary unit, i² = -1 |
e^(iπ) = -1 |
27 |
X_k, x_n |
"big X sub k", "little x sub n" | DFT output (frequency) and input (time samples) | 27 | |
f * g |
"f convolved with g" | slide, multiply, sum | a blur filter | 27 |
z (control) |
"z" | the root of a characteristic equation; stable iff |z| < 1 |
z² - z + K = 0 |
29 |
Linear algebra
| Symbol | Read it as | Meaning | Example | Ch. |
|---|---|---|---|---|
x, v, w (lower-case) |
"vector x" | an ordered list of numbers | x = (1, 2) |
10 |
A, W, M (upper-case) |
"matrix A" | a grid of numbers / a linear transformation | [[0.5,-1],[2,0.25]] |
10 |
m × n |
"m by n" | shape: m rows, n columns |
(2×3)·(3×4) = 2×4 |
10 |
A·B, A @ B |
"A times B" | matrix product (row · column) | A@B in NumPy |
10 |
a · b (vectors) |
"a dot b" | dot product, Σ aᵢbᵢ |
(3,4)·(4,3) = 24 |
10 |
⊙, * (NumPy) |
"element-wise product" | multiply matching entries | [1,2]⊙[3,4] = [3,8] |
10 |
Aᵀ, xᵀ |
"A transpose" | swap rows and columns | [1 2]ᵀ = a column |
10, 13 |
A⁻¹ |
"A inverse" | undoes A: A⁻¹A = I |
10 | |
I |
"the identity" | 1s on the diagonal, "do nothing" | I·x = x |
10, 11 |
det(A) |
"determinant of A" | volume scaling; 0 means not invertible |
det([[1,2],[3,4]]) = -2 |
11 |
trace(A) |
"trace of A" | sum of the diagonal = sum of eigenvalues | trace = 4 |
11 |
rank(A) |
"rank of A" | number of independent rows/columns | low-rank LoRA update | 20 |
‖x‖₁, ‖x‖₂, ‖x‖∞ |
"L1 / L2 / L-infinity norm" | sum of abs values / length / largest abs value | (3,4): 7, 5, 4 |
10, 28 |
‖A‖_F |
"Frobenius norm" | √(Σ all entries²) |
√50 |
20 |
λ, v (eigen) |
"eigenvalue", "eigenvector" | A·v = λ·v |
λ = 1, 3 |
11 |
P, D |
eigenvector matrix and diagonal eigenvalue matrix, A = PDP⁻¹ |
11 | ||
U, Σ, V |
"U, sigma, V" | SVD factors, A = UΣVᵀ; σᵢ = singular values |
σ = 6.71, 2.24 |
20 |
L = D - A (graphs) |
"the Laplacian" | degree matrix minus adjacency matrix | 11, 17 | |
λ₂ |
"lambda two", Fiedler value | algebraic connectivity of a graph | ring of 4: λ₂ = 2 |
11 |
κ |
"kappa", condition number | λ_max / λ_min: how ill-conditioned a problem is |
21 | |
Q, K, V, d_k |
"queries, keys, values, key dimension" | attention inputs | d_k = 64 |
20 |
Probability and statistics
| Symbol | Read it as | Meaning | Example | Ch. |
|---|---|---|---|---|
P(A) |
"probability of A" | a number in [0, 1] |
P(heads) = 0.5 |
8 |
P(A | B) |
"probability of A given B" | probability once B is known |
P(fail | checkout) = 0.25 |
8, 16 |
P(A ∩ B), P(A ∪ B) |
"A and B", "A or B" | joint / either | 8 | |
X, Y (capitals) |
"random variable X" | a number decided by chance | latency of a request | 8, 14 |
X ~ D |
"X is distributed as D" | X is drawn from distribution D |
X ~ N(100, 20²) |
14 |
p(x), f(x), F(x) |
"PMF / PDF / CDF" | mass, density, cumulative probability | F(250 ms) = 0.99 |
14 |
E[X] |
"expected value of X" | the probability-weighted average | E[attempts] = 1/p |
8 |
Var(X), σ² |
"variance" | average squared distance from the mean | p(1-p) |
8, 9 |
σ, s |
"sigma", "s" | standard deviation (population / sample) | s ≈ 22.17 |
9 |
μ, x̄ |
"mu", "x bar" | mean (population / sample) | 9 | |
Cov(X,Y), r |
"covariance", "correlation" | how two variables move together; r ∈ [-1, 1] |
r ≈ 0.77 |
9 |
z |
"z-score" | distance from the mean in standard deviations | z = 4.5 |
9, 22 |
Φ(z) |
"capital phi of z" | the standard normal CDF | Φ(1.96) ≈ 0.975 |
14 |
N(μ, σ²) |
"normal with mean mu, variance sigma squared" | the bell curve | N(0, 1) |
14 |
Bin(n,p), Poisson(λ), Beta(α,β) |
named distributions | Beta(14, 88) |
14, 16 | |
C(n, k) / P(n, k) |
"n choose k" / "permutations" | number of combinations / ordered arrangements | C(5,2) = 10 |
15 |
λ (probability) |
"lambda" | an average rate of events | 100 req/s | 14, 25 |
π, πᵢ (Markov) |
"pi" | stationary distribution: long-run share of time per state | π_B = 0.032 |
11, 25 |
π (security) |
"pi", base rate | fraction of events that are malicious | 10⁻⁴ |
26 |
H(X) |
"entropy of X" | average surprise in bits | fair coin: 1 bit | 23 |
H∞(X) |
"min-entropy" | -log₂(max p), the guessing-attack measure |
1 bit | 23 |
H(p, q), D_KL(p‖q) |
"cross-entropy", "KL divergence" | cost of using q when the truth is p |
22, 23 | |
I(X; Y) |
"mutual information" | bits about X revealed by Y |
23 | |
ℒ(θ), ℓ(θ) |
"likelihood", "log-likelihood" | probability of the data as a function of θ (ℒ is also the Lagrangian in Ch. 21) |
ℓ'(p) = 0 |
21, 22 |
α, β (testing) |
"alpha", "beta" | false-positive rate; false-negative rate (power = 1-β) |
α = 0.05 |
22 |
χ² |
"chi-squared" | a test statistic for counts; equals z² for 2×2 tables |
22 | |
ε (privacy) |
"epsilon" | differential-privacy budget | ε = 0.5 |
26 |
δ |
"delta" | a small failure probability (Hoeffding), an effect size (A/B), or a perturbation (FGSM) | δ = 0.05 |
14, 22, 28 |
| TP, FP, TN, FN | "true/false positive/negative" | confusion-matrix counts | 26 | |
| TPR, FPR | "true / false positive rate" | recall; false-alarm rate | 26 |
Queueing, networking and control
| Symbol | Read it as | Meaning | Example | Ch. |
|---|---|---|---|---|
λ, μ |
"lambda", "mu" | arrival rate, service rate | 90 and 100 req/s | 25 |
ρ |
"rho" | utilization λ/μ |
0.9 | 25 |
L, W, W_q |
average number in system, time in system, time waiting | L = λW |
25 | |
c, a |
number of servers; offered load in Erlangs | a = 9 |
25 | |
c_a, c_s, τ |
"c a", "c s", "tau" | variability of arrivals / service; mean service time | 25 | |
B, S/N, C |
bandwidth (Hz), signal-to-noise ratio, capacity (bits/s) | C = B·log₂(1 + S/N) |
23, 27 | |
f, f_s, f_max |
frequency, sampling rate, highest frequency present | f_s > 2f_max |
27 | |
d, M, N (PageRank) |
damping factor, link matrix, number of pages | d = 0.85 |
11 | |
e_t, K, D |
control error at tick t, loop gain, disturbance |
0 < K < 2 |
29 | |
Kp, Ki, Kd |
PID gains (proportional, integral, derivative) | 29 | ||
SRTT, RTTVAR, RTO |
smoothed RTT, RTT variation, retransmission timeout | RTO = SRTT + 4·RTTVAR |
9 |
The Greek alphabet, as used here
| Letter | Name | Meanings in this guide |
|---|---|---|
α |
alpha | significance level; EWMA smoothing factor; Beta prior count; LoRA scale; Pareto tail index |
β |
beta | momentum/decay factor (β₁, β₂ in Adam); Type II error; Beta prior count; RTTVAR gain |
δ Δ |
delta | small change (Δx); failure probability; effect size; perturbation; error signal in backprop |
ε |
epsilon | tiny tolerance or error; perturbation budget; privacy budget; Adam's divide-by-zero guard |
η |
eta | learning rate |
θ |
theta | model parameters; an angle; an unknown rate |
κ |
kappa | condition number |
λ |
lambda | eigenvalue; arrival/event rate; Lagrange multiplier (price); curvature; regularization strength |
μ |
mu | mean; service rate |
π |
pi | 3.14159…; stationary distribution; base rate; prime-counting function π(N) |
ρ |
rho | utilization; correlation |
σ / Σ |
sigma | standard deviation; sigmoid; singular value / summation; SVD's diagonal matrix |
τ |
tau | mean service time |
φ / Φ |
phi | Euler's totient φ(n) / the standard normal CDF Φ(z) |
χ |
chi | χ², the chi-squared statistic |
ω / Ω |
omega | angular frequency / the sample space |
∇ |
nabla (not a letter) | the gradient operator |
∂ |
"partial" (not a letter) | partial derivative |
Constants and "magic numbers"
| Constant | Value | Where it comes from | Ch. |
|---|---|---|---|
e |
2.71828… | the base whose exponential is its own derivative; lim (1 + 1/n)ⁿ |
5, 12 |
π |
3.14159… | circle circumference ÷ diameter; appears in the normal PDF via √π |
7, 12 |
i |
√-1 |
imaginary unit; rotations in the Fourier transform | 27 |
ln 2 |
0.693 | doubling time ≈ 0.693/r (Rule of 72) |
6, 12 |
√(2π) |
2.5066 | the normal distribution's normalizing constant | 14 |
1.96 |
z for a two-sided 95% interval | 14, 22 | |
0.84 |
z for 80% power | 22 | |
68 / 95 / 99.7 % |
share of a normal within 1, 2, 3 σ | 9 | |
1.18 (√(2 ln 2)) |
birthday bound: 50% collision after 1.18·√N |
24 | |
3 (rule of three) |
-ln 0.05 ≈ 3.0 |
0 failures in n trials ⇒ rate < 3/n at 95% |
22 |
0.25 |
maximum slope of the sigmoid | 12 | |
0.85 |
PageRank damping factor | 11 | |
1.22 (√(3/2)) |
constant in the Mathis TCP throughput formula | 12 | |
1/8, 1/4, 4 |
TCP RTO gains α, β and the "4 × RTTVAR" margin (RFC 6298) |
9 | |
0.9, 0.999, 10⁻⁸ |
Adam's default β₁, β₂, ε |
21 | |
65537 |
2¹⁶ + 1 |
the standard RSA public exponent | 24 |
3329 |
ML-KEM's modulus q |
24 | |
-147.55 |
20·log₁₀(4π/c) |
free-space path loss constant (metres, Hz) | 27 |
c |
3 × 10⁸ m/s |
speed of light | 27 |
2¹²⁸ |
3.4 × 10³⁸ |
"128-bit security": infeasible work | 24 |
16 |
RIP's "infinity" hop count | 17 |
Units and abbreviations
| Unit | Meaning | Conversion |
|---|---|---|
| bit / byte | binary digit / 8 bits | 1 B = 8 b |
| Mbit/s, Gbit/s | megabits / gigabits per second (network rates use powers of 10) | 1 Gbit/s = 10⁹ bit/s |
| ms, µs, ns | milli-, micro-, nanoseconds | 1 ms = 1000 µs |
| Hz, MHz, GHz | cycles per second | 2.4 GHz = 2.4 × 10⁹ Hz |
| dB | ten times the log₁₀ of a power ratio | +3 dB ≈ ×2, +10 dB = ×10 |
| dBm | power relative to 1 milliwatt | 20 dBm = 100 mW |
| Erlang | one server kept continuously busy | a = λ/μ |
| RTT, MSS, BDP | round-trip time, max segment size, bandwidth-delay product | BDP = bandwidth × RTT |
| SNR | signal-to-noise ratio (linear or dB) | 30 dB = 1000 |
| MTBF | mean time between failures | = 1/λ |
| p50, p95, p99 | 50th, 95th, 99th percentiles | p99 = F⁻¹(0.99) |
For every formula written out in full, see the Formula cheat sheet in Part 4. For every named theorem, law and inequality, see the Theorem index that follows it.