CSET Mathematics (211/212/213) key terms

Any of the rules that addition and multiplication must obey in a number system: closure, commutativity, associativity, identity elements, inverses, and distributivity. Together they let you justify every algebraic manipulation used in solving equations.

A rule describing how the less-than relation interacts with addition and multiplication on the real numbers, including trichotomy and closure of positivity under sums and products. It underlies every inequality argument and sign analysis.

The property that every nonempty set of real numbers bounded above has a least upper bound in the reals. It distinguishes the reals from the rationals and guarantees limits of convergent increasing sequences exist.

The smallest real number that is greater than or equal to every element of a given set. Unlike a maximum, it need not belong to the set, as with the supremum 1 of all numbers strictly below 1.

The greatest real number that is less than or equal to every element of a set. It is the mirror image of the supremum and likewise need not be an element of the set itself.

The fact that between any two distinct real numbers there lies a rational number, and likewise an irrational one. It shows the rationals are spread everywhere on the number line yet still leave gaps.

A set whose elements can be listed in a sequence indexed by the natural numbers, either finitely or infinitely. The integers and rationals are countable; the real numbers are not.

A proof technique that builds a new object differing from every entry on a proposed list at one position each. It establishes that the real numbers cannot be placed in a one-to-one correspondence with the naturals.

A measure of the size of a set, defined by whether a one-to-one correspondence exists between two sets. Finite sets are compared by counting; infinite sets can share size with proper subsets.

The property that applying an operation to any two elements of a set always yields another element of that same set. For example, the integers are closed under addition but not under division.

The distance of a real number from zero, written with vertical bars and equal to the number itself when nonnegative and its opposite when negative. It converts statements about magnitude into inequalities.

The statement that the absolute value of a sum is at most the sum of the absolute values. It generalizes to distances in the plane and underlies epsilon-delta estimates throughout analysis.

A number of the form a + bi where a and b are real and i squared equals negative one. Complex numbers form a field closed under all four operations except division by zero.

The number obtained by reversing the sign of the imaginary part, so a + bi becomes a - bi. Multiplying a complex number by its conjugate yields the real value a squared plus b squared.

The distance from the origin to the point representing a complex number in the plane, equal to the square root of a squared plus b squared. It multiplies when complex numbers are multiplied.

Writing a complex number as r times the quantity cosine theta plus i sine theta, where r is the modulus and theta the argument. It turns multiplication into multiplying moduli and adding angles.

The rule that raising a complex number in polar form to the nth power raises the modulus to n and multiplies the angle by n. It also generates all n distinct nth roots.

A set of objects called vectors together with addition and scalar multiplication satisfying closure, associativity, commutativity, and distributive laws. It provides the setting for linear combinations, spans, and bases.

A property of a set of vectors whereby the only way a linear combination of them equals the zero vector is with all coefficients zero. It guarantees each vector adds a genuinely new direction.

A set of vectors that both spans the space and is linearly independent, so every vector has exactly one representation as a combination of them. The number of basis vectors is the dimension.

A scalar computed from a square matrix that is nonzero exactly when the matrix is invertible. Geometrically its absolute value gives the factor by which the associated transformation scales area or volume.

A square matrix for which another matrix exists whose product with it is the identity matrix. Such a matrix has nonzero determinant and represents a transformation that can be undone.

A rectangular array of numbers used to represent linear transformations and systems of equations. Rows and columns are combined by the row-times-column rule when matrices are multiplied.

A rule assigning to each element of a domain exactly one element of a codomain. The single-output requirement is what distinguishes a function from a general relation.

The domain is the full set of inputs for which a function is defined; the range is the set of outputs actually produced. Both matter when solving equations and graphing.

The function that undoes another, existing only when the original is one-to-one on its domain. Its graph is the reflection of the original graph across the line y equals x.

Applying one function to the output of another, written f of g of x. Composition is generally not commutative, and its domain excludes inputs where the inner function is undefined.

The expression giving the roots of a quadratic equation as negative b plus or minus the square root of b squared minus four a c, all divided by two a. It solves every quadratic with real or complex roots.

The quantity b squared minus four a c in a quadratic equation. Its sign tells whether the roots are two real, one repeated real, or a complex conjugate pair.

Rewriting a quadratic expression as a perfect square plus a constant, which reveals the vertex of the parabola and derives the quadratic formula. It is also the standard route to circle equations.

Writing a quadratic as a times the square of x minus h, plus k, where the point (h, k) is the parabola's vertex. It makes the maximum or minimum value immediate.

The result that every nonconstant polynomial with complex coefficients has at least one complex root, hence factors completely into linear factors. It guarantees a degree-n polynomial has exactly n roots counted with multiplicity.

A test listing candidate rational zeros of a polynomial as factors of the constant term divided by factors of the leading coefficient. It narrows the search when factoring by hand.

The statement that a polynomial has a factor x minus c exactly when evaluating the polynomial at c gives zero. It links factoring directly to locating zeros and intercepts.

The result that dividing a polynomial by x minus c leaves a remainder equal to the polynomial evaluated at c. Synthetic division applies it quickly to test candidate zeros.

A line that the graph of a function approaches arbitrarily closely as the input or output grows without bound. Vertical asymptotes arise from zeros of denominators; horizontal ones from end behavior.

A function given as one polynomial divided by another. Its graph features vertical asymptotes where the denominator vanishes and holes where common factors cancel.

The exponent to which a fixed base must be raised to produce a given number; it is the inverse of exponentiation. Logarithms convert products into sums and solve exponential equations.

A function of the form a times b to the x with positive base b, growing or decaying at a rate proportional to its current value. Its inverse is a logarithm with the same base.