| symbol | latex code | explanation | |
|---|---|---|---|
aⁿ | a^n | Base a raised to exponent n — for a natural exponent, n factors of a multiplied together | |
a⁰ = 1 | a^0 = 1 | The zero power — equals 1 for every a ≠ 0; 0⁰ is left undefined in most conventions | |
a⁻ⁿ | a^{-n} | Negative exponent — the reciprocal 1/aⁿ; the minus flips, it never makes the result negative | |
a^(m/n) | a^{m/n} | Rational exponent — the n-th root of aᵐ; denominator is the root, numerator the power | |
aˣ | a^x | Exponential function — the exponent is the variable; contrast xⁿ, where the base varies | |
√a | \sqrt{a} | The principal square root — the non-negative solution only; the ± is supplied separately when solving equations | |
ⁿ√a | \sqrt[n]{a} | The n-th root — index n written in the crook of the radical; index 2 is left unwritten | |
√(a²) = |a| | \sqrt{a^2} = |a| | Simplifying an even root of a power produces an absolute value, not a bare a | |
a^(1/n) = ⁿ√a | a^{1/n} = \sqrt[n]{a} | The bridge between radical and rational-exponent notation — one object, two spellings | |
log_b x | \log_b x | The logarithm base b of x — the exponent b needs to reach x; defined for b > 0, b ≠ 1, x > 0 | |
ln x | \ln x | The natural logarithm — base e; the default in calculus and the sciences | |
log x | \log x | Base-10 in school texts and engineering; base-e in advanced mathematics — the convention depends on the field; European texts write lg for base-10 | |
log_b x = y ⇔ bʸ = x | \log_b x = y \iff b^y = x | The defining equivalence — every logarithmic statement is an exponential statement read backwards | |
P(x) = aₙxⁿ + … + a₀ | P(x) = a_n x^n + \cdots + a_1 x + a_0 | General polynomial — coefficients subscripted by the power they multiply; aₙ ≠ 0 is the leading coefficient | |
deg P | \deg P | The degree of the polynomial — the highest power with a nonzero coefficient | |
P(r) = 0 | P(r) = 0 | r is a root (zero) of P — evaluating at r returns zero | |
(x − r) | (x - r) | The linear factor paired to the root r — the factor theorem links the two notations | |
ax + b = 0 | ax + b = 0 | General linear equation — a, b are fixed coefficients, x the unknown; the letter roles are conventional, not intrinsic | |
ax² + bx + c = 0 | ax^2 + bx + c = 0 | General quadratic equation in standard form, a ≠ 0 | |
x = (−b ± √D)/2a | x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} | The quadratic formula — the ± packs both solutions into one line; expand it to two before substituting | |
D = b² − 4ac | D = b^2 - 4ac | The discriminant — also written Δ; its sign counts the real solutions (two, one, or none) | |
|x| = a | |x| = a | Absolute value equation — splits into x = a or x = −a when a > 0 | |
x ∈ {2, 5} | x \in \{2, 5\} | Solution set notation — the solutions listed as a set rather than as separate equations | |
< ≤ > ≥ | < \; \leq \; > \; \geq | Strict and inclusive inequality signs — the bar under the symbol admits equality | |
a < x < b | a < x < b | Chained inequality — two conditions at once; both signs must point the same way (linear inequalities) | |
|x| < a ⇔ −a < x < a | |x| < a \iff -a < x < a | Absolute value inequality — less-than gives a band; |x| > a gives the two outer rays instead | |
(−∞, −1) ∪ (2, ∞) | (-\infty, -1) \cup (2, \infty) | Solution written in interval notation — unions capture the disconnected solution sets typical of rational inequalities | |
aₙ | a_n | The n-th term of a sequence — the subscript is the position, not a multiplier | |
{aₙ} | \{a_n\} | The sequence as a whole object, braces around the general term | |
d | d | Common difference of an arithmetic sequence — aₙ₊₁ − aₙ, constant throughout | |
r | r | Common ratio of a geometric sequence — aₙ₊₁ / aₙ, constant throughout | |
Sₙ | S_n | Partial sum — the first n terms added; the subscript counts how many | |
Σ aₖ | \sum_{k=1}^{n} a_k | Summation notation — k runs from the lower bound to the upper, adding a term each step | |
Fₙ | F_n | The n-th Fibonacci number — capital F is reserved for this sequence by convention |