Precalculus

Notations
Arithmetic
Algebra
Trigonometry
Pre-calculus Problems and Solutions

I know Wikipedia says precalculus involves algebra and trigonometry, but I consider it to be everything before you take your first calculus class in college.

systems of equations

x-component = rcos(theta) y-component = rsin(theta)

arithmetic sequences geometric sequences

Increasing: 1, 2, 3, 4
Non-decreasing: 1, 1, 2, 3
The difference being in that an increasing sequence, for x(n) and x(n+1) > x(n) whereas in a non-decreasing sequence, x(n+1) >= x(n).

Notations

𝛢𝛼: Alpha
𝛣𝛽: Beta
𝛤𝛾: Gamma
𝛥𝛿: Delta
𝛦𝜀: Epsilon
𝛧𝜁: Zeta
𝛨𝜂: Eta
𝛩𝜃𝜗: Theta
𝛪𝜄: Iota
𝛫𝜅: Kappa
𝛬𝜆: Lambda
𝛭𝜇: Mu
𝛮𝜈: Nu
𝛯𝜉: Xi
𝛰𝜊: Omicron
𝛱𝜋: Pi
𝛲𝜌: Rho
𝛴𝜎: Sigma
𝛵𝜏: Tau
𝛶𝜐: Upsilon
𝛷𝜙𝜑: Phi
𝛸𝜒: Chi
𝛹𝜓: Psi
𝛺𝜔: Omega

Often used in set theory:

∅: Empty set. A set with no elements.
∁: Complement of a set. Includes everything not in the set.
↦: “Maps to” in function definition.
↣: “Implies” in logic or “maps to” in category theory.
∩: Intersection of sets. Contains all elements of all sets.
∪: Union of sets. Contains any element of any set.
⊆: Subset or equal to. All elements of the first set are in the second set.
⊂: Strict subset. All elements of the first set are in the second set, and the sets are not equal.
⊄: Not a subset. The first set is not a subset of the second set.
⊊: Strict subset. All elements of the first set are in the second set, and the sets are not equal.
⊇: Superset or equal to. All elements of the second set are in the first set.
⊃: Strict superset. All elements of the second set are in the first set, and the sets are not equal.
⊅: Not a superset. The first set is not a superset of the second set.
⊋: Strict superset. All elements of the second set are in the first set, and the sets are not equal.
⊖: Symmetric difference of sets. Elements which are in one of the sets, but not in their intersection.
∈: Element of a set.
∉: Not an element of a set.
∋: “Such that” in set builder notation or “Contains as member”.
∌: Does not contain as member.
ℵ (aleph numbers): Used to denote the cardinality (size) of infinite sets.
ℶ: Beth number, used in set theory.
ℷ: Gimel function, used in set theory.
ℸ: Daleth symbol, also used in set theory.

Often used in logic:

¬: Logical NOT.
∨: Logical OR.
∧: Logical AND.
⊕: Exclusive OR (XOR).
→: Implies; logical implication from left to right.
←: Implies; logical implication from right to left.
⇒: Implies; logical implication from left to right.
⇐: Implies; logical implication from right to left.
↔: Logical biconditional, ‘if and only if’.
⇔: Logical equivalence, ‘if and only if’.
∀: For all; universal quantification.
∃: There exists; existential quantification.
∄: There does not exist.
∴: Therefore.
∵: Because.
⊤: Tautology; a statement that is always true.
⊥: Contradiction; a statement that is always false.
⊢: Provable; in formal logic, “proves”.
⊨: Entails; logical consequence.
⫤: Does not entail; not a logical consequence.
⊣: Proof (end of proof).

Often used in quantative reasoning:

≠: Not equal to.
±: Plus-minus; indicates two possible values, one positive and one negative.
∓: Minus-plus; the opposite of the plus-minus symbol.
÷: Division.
√: Square root.
‰: Per mille (per thousand).
⊖: Circled minus, sometimes used to denote symmetric difference of sets.
⊘: Circled division symbol.
≤: Less than or equal to.
≥: Greater than or equal to.
≦: Less than or equal to.
≧: Greater than or equal to.
≨: Less than but not equal to.
≩: Greater than but not equal to.
⊏: Not a square image of.
⊐: Square original of.
⊑: Square image of or equal to.
⊒: Square original of or equal to.

Often used in geometry:

∠: Angle.
∟: Right angle.
≅: Congruent to. In geometry, it often refers to two figures or lengths that are the same size and shape.
~: Similar to. In geometry, it refers to figures that have the same shape but not necessarily the same size.
‖: Parallel to. In geometry, it refers to lines, line segments, or planes that never intersect, no matter how far they are extended.
⟂: Perpendicular to. It indicates that two lines meet at right angles.
⫛: Congruence modulo n; In geometry, it can represent equality of shapes regardless of their position or orientation.
≡: Identically equal to, congruence in modular arithmetic, or equivalence in logic.
≜: Defined as, or equivalent by definition to.
≈: Approximately equal to.
∝: Proportional to.
≪: Much less than.
≫: Much greater than.
⌊⌋: Floor function, which rounds a real number down to the greatest integer less than or equal to it.
⌈⌉: Ceiling function, which rounds a real number up to the least integer greater than or equal to it.
∏: Product operator in mathematics, often used in product notation, similar to how ∑ is used for summation.
∐: Coproduct symbol in category theory in mathematics.
∑: Summation symbol.
⋀: Logical AND in some contexts, or the infimum (greatest lower bound) in lattice theory. 13.⋁: Logical OR in some contexts, or the supremum (least upper bound) in lattice theory.
⋂: Intersection of sets.
⋃: Union of sets.
⨀: N-ary circled dot operator, often used to denote a general associative operation.
⨁: N-ary circled plus operator, often used to denote direct sum or coproduct.
⨂: N-ary tensor product in multilinear algebra.
𝖕, 𝖖, 𝖗: These are examples of “mathematical bold” script letters, often used in specific mathematical contexts to denote variables or other entities. They could stand for anything based on the context.

x := y means x “is defined to be” y

Arithmetic

radicals \(\sqrt[n]{m}\)

Euclidean GCD, gcd(48, 18) steps:

48/18 = 2 R 12
18/12 = 1 R 6 <— GCD
12/6 = 2 R 0

gcd(a, 0) = a
gcd(a, b) = gcd(b, a mod b)

Rational numbers

Irrational numbers \(\sqrt{2}, \pi, e, \phi\) are irrational

Fibonacci numbers 1, 1, 2, 3, 5, 8, …

Infinities …, -3, -2, -1, 0, 1, 2, 3, … \(\infty\)

Notations

Arithmetic

Algebra

Trigonometry

Pre-calculus Problems and Solutions