Random Facts
definition:
* Every vector space V over a field has a basis.
* Let R be a ring. Every ideal I != R is contained in a maximal ideal.
* Every group of order 4 is abelian.
* The general linear group GL2(R) is an open subset of the space R^4 of all
2 x 2 matrices, because it is the set {det P != 0}, and det is continuous.
-----
Algebra / Laws
definition:
* Reflexive: a == a.
* Transitive: (a == c) ^ (b == c) -> a == b.
* Symmetric: a == b -> b == a.
* a == b -> a + c == b + c.
* a == b -> a - c == b - c.
* a == b -> a * c == b * c.
* a == b -> a / c == b / c, for c != 0.
* The whole is equal to the sum of its parts.
* Trichotomy: exactly one of the relations a < b or a == b or a > b is true.
-----
Algebra / Relations
definition:
* For a relation R on a set A:
+ R is reflexive if aRa for all a in A.
+ R is symmetric if aRb -> bRa.
+ R is antisymmetric if aRb ^ bRa -> a == b.
+ R is transitive if aRb ^ bRc -> aRc.
+ R is an equivalence relation if it is reflexive, transitive, and symmetric.
-----
Algebra / Groups, Fields, Rings, Spaces, and Modules
definition:
* A semigroup is a set S together with a law of composition which is
associative and has an identity element.
* A group is a semigroup in which every element has an inverse; that is,
a set G together with a law of composition which is associative and has
an identity element, and such that every element of G has an inverse.
* An abelian group is a group whose law of composition is also commutative.
* Customarily, the group and its set share the same symbol, though the group
may use a notation to indicate the type of operation used (G+, for example,
for addition over the set G). Abelian groups often use the notation +, since
addition is commutative.
* A field is a set F together with two laws of composition, addition and
multiplication, where F+ and Fx (= F - {0}) are abelian groups of addition
and multiplication, respectively, and the two laws together obey the
distributive laws as usual.
* A ring is like a field, except that multiplication does not need to form
a group; in fact, it is only required to be a semigroup (associative, with
an identity), and obey the distributive laws with addition (which is still
an abelian group) as usual. Some algebraists do not even require the
multiplication law to have an identity.
* A commutative ring is a ring that has a commutative multiplication.
* A vector space V over a field F (for example, the real number field, which
forms a real vector space) is a set V together with two laws of composition,
an addition law that forms an abelian group V+, and a scalar multiplication
law that is associative with multiplication in F, has a scalar identity
(which is F's identity), and obeys the distributive laws in the special case
of scalar multiplication and vector addition.
* A module is a generalization of vector spaces using rings instead of just
fields for the scalar multiplication.
-----
Algebra / Proportion / Definitions
definition:
* a/b == c/d is called a proportion (the equality of two ratios); a and d are
the extremes, and b and c are the means. d is the fourth proportional.
-----
Algebra / Proportion / Identities
definition:
* If a/b == c/d, and none of a, b, c, or d are zero, then:
+ b*c == a*d (product of means equals product of extremes).
+ b/a == d/c (inversion).
+ a/c == b/d (exchange of means).
+ d/b == c/a (exchange of extremes).
+ (a+b)/b == (c+d)/d (addition).
+ (a-b)/b == (c-d)/d (subtraction).
+ (a+c)/(b+d) == a/b == c/d (ratio of sums).
-----
Combinatorics / Binomial Coefficient
definition:
* The binomial coefficent (n k) (shown vertically), usually read "n choose k",
is the number of subsets of order k in the set {1, 2, ..., n}. k and n are
natural numbers.
* For every n and every k <= n, (n k) = (n-1 k) + (n-1 k-1).
* (n k) = n! / k! * (n-k)!.
-----
Functions / Linear / Classifications
definition:
* A linear function (or transformation) of a set of variables u1 ... uk
takes the form a1 * u1 + ... + ak * uk + c, where a1 ... ak and c are
all scalars.
* A homogenous linear function is as above but has c = 0.
* Linearity means that the function applied to a sum is equal to the sum
of the function applied to the terms, f(a + b) = f(a) + f(b), and the
function applied to a scaled value is equal to the scalar times the
function applied to the original value, f(c * a) = c * f(a).
>>>
hint:
* Homogenous linear is missing a term.
* Linearity pertains to summing and scaling.
-----
Geometry / Cartesion Coordinates / Definitions
definition:
* A x B is the Cartesion product of sets A and B, and is the set of all
ordered pairs {(a,b) E A x B | a E A ^ b E B}.
* For a given Cartesian pair (a,b), a is the abscissa, and b is the ordinate.
* The four regions of the Cartesian plane delimited by the axes and the origin
are numbered counterclockwise from top right, I (+,+), II (-,+), III (-,-),
and IV (+,-).
-----
Geometry / Lines / Identities
definition:
* The distance between a given point (x0, y0) and a line given by
a * x + b * y + c = 0, is:
|(a * x0 + b * y0 + c) / sqrt(a^2 + b^2)|
* The equation of a line with y-intercept b and x-intercept a is:
(x / a) + (y / b) = 1
* The equation of a line passing through two points (x1, y1) and (x2, y2) is:
(y - y1) / (y1 - y2) = (x - x1) / (x1 - x2)
* The equation of a line with slope m passing through (x0, y0) is:
y - y0 = m * (x - x0)
-----
Geometry / Lines and Planes / Definitions
definition:
* The intersection of a straight line and a plane is a point called the foot of
the line.
* Skew lines are noncoplanar lines.
* A dihedral angle is the simple extension of a two dimensional angle
(two rays meeting at a vertex) directly to three dimensions (two half-planes
meeting at a line). It can be named by four points, two on the common edge,
and one unique to each half-plane.
* A plane angle is the embedding of a two dimensional angle into three
dimensions; it is formed by two rays, one in each face of a dihedral angle,
and both perpendicular to the edge of the dihedral angle.
* The measure of a dihedral angle is the measure of its plane angle.
* All usual classifications, such as acute, obtuse, right, straight, adjacent,
congruent, complementary, supplementary, perpendicular, etc. apply directly
to dihedral angles as if applied to the corresponding plane angles. When
more than one dihedral angle is involved, assume the plane angles meet at
the same vertex if necessary.
* The projection of a point onto a given plane is the foot of the perpendicular
drawn from the point to the plane.
* The projection of a line or line segment onto a plane is the set of points
which are the projections of the points of the line or line segment.
* The angle a line makes to a plane is the angle it makes to its projection
onto the plane.
* A plane can be determined by:
+ Two intersecting lines.
+ Two parallel lines.
+ A line and a non-collinear point.
-----
Geometry / Lines and Planes / Identities
definition:
* Two intersecting lines determine a plane, and a line perpendicular to both
is perpendicular to the plane.
* If two parallel planes are cut by a third plane, the lines of intersection
are parallel.
* If each of three non-collinear points of a plane is equidistant from two
points outside the plane, then every point of the plane is equidistant from
those two points.
-----
Geometry / Conic Sections / Identities
definition:
* The equation of a circle centered at (x0, y0) with radius r is:
(x - x0)^2 + (y - y0)^2 = r^2
* The equation of a parabola with vertex at (x0, y0) and a vertical directrix
at x = x0 - p (so focus is at (x0 + p, y0)) is:
(y - y0)^2 = 4 * p * (x - x0)
* The equation of a parabola with vertex at (x0, y0) and a horizontal
directrix at y = y0 - p (so focus is at (x0, y0 + p)) is:
(x - x0)^2 = 4 * p * (y - y0)
* The equation of an ellipse with a center at (x0, y0), horizontal major axis
length 2 * a, and vertical minor axis with length 2 * b (so a and b are in
effect radii of sorts) is:
(x - x0)^2 / a^2 + (y - y0)^2 / b^2 = 1
* Swap a and b if the major axis is vertical and the minor axis is horizontal.
* The distance c between the center and either focus in an ellipse is:
c = sqrt(a^2 - b^2)
* The equation of a hyperbola with center at (x0, y0) and horizontal
transverse axis of length 2 * a and vertical conjugate axis of length 2 * b
is:
(x - x0)^2 / a^2 - (y - y0)^2 / b^2 = 1
* Swap a and b if the transverse axis is vertical and the conjugate axis is
horizontal.
* The distance c between the center and either focus of a hyperbola is:
c = sqrt(a^2 + b^2)
-----
Geometry / Conic Sections / Eccentricity
definition:
* The eccentricity e for conic sections is c / a, where c is the distance from
one focus to the midpoint between the foci, and a is the length of the
"radius" along the major/transverse axis.
* A circle has e = 0 (only one focus).
* An ellipse has 0 < e < 1.
* A parabola has e = 1 (effectively one focus is at infinity).
* A hyperbola has e > 1.
-----
Geometry / Locus / Definitions
definition:
* A locus is exactly the set of points satisfying a given set of conditions.
* To prove a locus is correct, it is necessary to prove both of the following:
+ If a point is on the locus, it satisfies the given conditions.
+ If a point satisfies the given conditions, it is on the locus.
-----
Geometry / Locus / Classification
definition:
* The locus of points at a given distance from a fixed point is a circle with
the point at its center and radius equal to that distance.
* The locus of points at a given distance from a line is a pair of lines, one
on each side, parallel to the first line, and at the given distance from it.
* The locus of points equidistant from the sides of an angle is the angle
bisector.
* The locus of points equidistant from two points is the perpendicular bisector
of the segment joining the two points.
* The locus of the vertex of the right angle of a right triangle with a fixed
hypotenuse is a circle with the hypotenuse as a diameter.
* The locus of the points equidistant from two parallel lines is a line
parallel to both and midway between them.
* The locus of the points equally distant from a fixed line (the directrix) and
a point (the focus) is a parabola. The point along the line perpendicular
to the directrix through the focus (the point of closest approach to the
line) is the vertex of the parabola.
* The locus of points for which the sum of their distances from two fixed
points (the foci) is constant is an ellipse. The line segment crossing the
ellipse through the foci is the major axis, and its perpendicular bisector
is the minor axis.
* The locus of points for which the difference of their distances from two
fixed points (the foci) is a constant is a hyperbola. It has made of two
curves! The line segment crossing the curves through the foci is the
transverse axis; its perpendicular bisector is the conjugate axis. The
points at which the transverse axis intersects the curves are the vertices
(the points of closest approach of the curves).
-----
Geometry / Line Segments and Rays
definition:
* Line segments contain their endpoints.
* A half line does not contain its starting point.
* A ray contains its starting point.
* For three collinear ordered points X, Y, and Z, the rays YX and YZ are
opposite rays.
* Congruent segments have the same length.
* The midpoint of the segment bisects that segment.
* The line dividing a segment at the midpoint is its bisector.
-----
Geometry / Regions
definition:
* The union of a object and its interior is called a region; for a circle, this
is a circular region, and for a polygon, it is a polygonal region.
-----
Geometry / Angles / Classification
definition:
* Angles measuring between 0 and 90 degrees are acute.
* Angles measuring 90 degrees are right.
* Angles measuring between 90 and 180 degrees are obtuse.
* Angles measuring 180 degrees are straight (and also lines).
* Angles measuring between 180 and 360 degrees are reflex.
-----
Geometry / Angles / Definitions
definition:
* Adjacent angles share the same vertex and a common side, but have no common
interior points.
* Vertical angles share a common vertex and have sides that are two pairs of
opposite rays (in other words, two intersecting lines form two pairs of
vertical angles, arranged like bridge teams).
* Congruent angles have equal measure.
* Complementary angles are a pair that sum to 90 degrees.
* Supplementary angles are a pair that sum to 180 degrees.
* A ray bisects (or is the bisector of) an angle if the ray divides the
angle into two angles of equal measure.
* If the two non-common sides of adjacent angles form opposite rays, then
the rays are a linear pair (and the angles are supplementary).
-----
Geometry / Intersections
definition:
* Parallel lines are both coplanar and do not intersect.
* If two distinct points of a line lie in a given plane, so does the line.
* Three or more lines, rays, or segments are concurrent if they all intersect
at the same point.
-----
Geometry / Intersections / Perpendicular
definition:
* A line, ray, or line segment that both bisects and is perpendicular to
another line segment is called the perpendicular bisector of that segment.
* The distance of a point to a line is the measure of the perpendicular line
segment from the point to the line.
* The foot of a perpendicular from a point to a line is the point where the
perpendicular meets the line.
* The perpendicular bisector of a line segment is unique within a given plane.
* Any point on the perpendicular bisector is equidistant from the endpoints.
The converse is also true.
* Two points equidistant from the endpoints of a line segment determine its
perpendicular bisector.
-----
Geometry / Angles / Identities
definition:
* Complements of the same or equal angles are equal.
* Supplements of the same or equal angles are equal.
* Vertical angles are equal.
* Two supplementary angles are right if they have the same measure.
* If two lines intersect to form one right angle, they actually form four.
-----
Geometry / Congruency
definition:
* Two or more geometric figures are congruent when they have the same shape
and size.
* For angles, this means the same measure; for line segments, this means the
same length.
* Congruency is an equivalence relation (reflexive, symmetric, and transitive).
-----
Geometry / Triangles / Classification
definition:
* A triangle with no equal sides is scalene.
* A triangle with two equal sides is isosceles. The third side is called the
base, and the angle opposite the base is the vertex angle.
* A triangle with three equal sides is equilateral.
* A triangle with three equal interior angles (of 60 degrees) is equiangular.
* Equilateral and equiangular are equivalent.
* A triangle with one obtuse angle is obtuse.
* A triangle with three acute angles is acute.
* A triangle with one right angle is right. The side opposite the right angle
is the hypotenuse, and the others are legs.
-----
Geometry / Triangles / Definitions
definition:
* The perimeter of a triangle is the sum of the measures of the sides.
* An interior angle is formed by two sides and includes the third side in its
collection of points.
* An exterior angles is formed by one side and the extension of an adjacent
side.
-----
Geometry / Triangles / Special Lines / Definitions
definition:
* An altitude of a triangle is a line segment from a vertex, perpendicular to
the opposite side. An altitude may lie outside the triangle!
* A median of a triangle is a line segment connecting a vertex to the midpoint
of the opposite side.
* An angle bisector is a line segment bisecting an angle and extending to the
opposite side.
* A midline is a line segment joining the midpoints of two sides of the
triangle.
-----
Geometry / Triangles / Special Lines / Concurrency
definition:
* The three lines containing the altitudes are concurrent.
* The medians are concurrent at a point which is 2/3 the distance from any
vertex to the midpoint of the opposite side. This point is called the
centroid of the triangle.
* The angle bisectors are concurrent at a point equidistant from each side of
the triangle, the incenter of the inscribed circle.
* The perpendicular bisectors of the sides of a triangle are concurrent at a
point equidistant from each vertex of the triangle, the circumcenter of the
circumscribed circle, and may lie outside the triangle.
-----
Geometry / Triangles / Identities / General
definition:
* The midline of a triangle is parallel to, and half as long as, the third
side of the triangle.
* Any line parallel to one side of a triangle divides the other two lines
proportionally.
* An angle bisector divides the opposite side in the same ratio as the lengths
of the other two sides, in the same order.
* The measure of an exterior angle of a triangle is equal to the sum of the
measures of the two nonadjacent interior angles of that triangle.
* The sum of the measures of the interior angles is 180 degrees.
* The sum of the measures of the exterior angles (using only one per vertex)
is 360 degrees.
* If two angles of two triangles are respectively equal, so is the third angle.
* A triangle has at most one angle >= 90 degrees.
* If two angles of a triangle are equal, than the opposite sides are equal.
The converse is also true.
* A line bisecting one side of a triangle, and parallel to a second, bisects
the third side.
-----
Geometry / Triangles / Law of Cosines and Law of Sines
definition:
* Law of Cosines: For any triangle with sides of length a, b, and c, and
angle A opposite the side with length a:
a^2 = b^2 + c^2 - 2 * b * c * cos A
* This becomes the Pythagorean theorem for measure A = 90 degrees, as the last
term drops out.
* Law of Sines: For any triangle with sides of length a, b, and c, and opposite
angles of measure A, B, and C, and the circumscribed circle of radius R:
(a / sin A) * (b / sin B) * (c / sin C) = 2 * R
-----
Geometry / Triangles / Identities / Right Triangles
definition:
* The length of the median to the hypotenuse of a right triangle is equal to
1/2 the length of the hypotenuse.
* If a triangle has sides of length a, b, and c, and c^2 = a^2 + b^2, the
triangle is a right triangle.
* In a 30-60 right triangle, the sides have lengths in the ratio 1::sqrt(3)::2.
* In a 45-45 right triangle, the sides have lengths in the ratio 1::1::sqrt(2).
* Acute angles of a right triangle are complementary.
-----
Geometry / Triangles / Identities / Isosceles Triangles
definition:
* The altitude of an equilateral triangle is sqrt(3)/2 times as long as a side.
* The bisector of the vertex angle of an isosceles triangle is also the
perpendicular bisector of the base.
-----
Geometry / Triangles / Congruency
definition:
* Methods of proving congruency of triangles are named by the sides and angles
that are compared; these must be three angles or sides in sequence around
the perimeter of the triangle. The following methods are valid:
+ SSS: Three sides are equal.
+ SAS: Two sides and an included angle are equal.
+ ASA: Two angles and an included side are equal.
+ AAS: Two angles and a not-included side are equal.
* ASS and AAA are *not* valid congruency tests. ASS may lead to two possible
triangles (based on the value of the included angle), and AAA does nothing
to control scaling, so proves only similarity, not congruence.
* ASS may be used in the special case that the angle is a right angle.
-----
Geometry / Triangles / Area / Formulas
definition:
* 1/2 * b * h, where b is the length of a base (any side of the triangle),
and h is the length of an altitude to that base.
* 1/2 * a * c * sin B, where a and c are the lengths of two sides, and B is
the measure of the included angle.
* 1/2 * a * c for right triangles, with a and c the lengths of the two legs.
* sqrt(s * (s - a) * (s - b) * (s - c)), where s = 1/2 * (a + b + c). Known as
Heron's formula, this avoids numeric imprecision for certain shapes of
triangles (and avoids the use of trigonometric transcendental functions).
* sqrt(3)/4 * x^2 for equilateral triangles, where x is the length of a side.
* sqrt(3)/3 * h^2 for equilateral triangles, where h is the length of the
altitude.
* 1/2 * h^2 * tan(alpha/2) for isosceles triangles, where h is the length of
the altitude to the side opposite the angle alpha.
-----
Geometry / Triangles / Area / Identities
definition:
* Triangles sharing the same base, and have their opposite vertices on a line
parallel to that base, have equal areas, because they all have the same
altitude.
* A median drawn to a side of a triangle divides the triangle into two new
triangles of equal area, because the base has been bisected, but the
respective altitude is unchanged.
-----
Geometry / Triangles / Inequalities
definition:
* The sum of the lengths of two sides of a triangle is greater than the length
of the third side.
* The measure of an exterior angle of a triangle is greater than the measure of
either nonadjacent enterior angle.
* If two sides of a triangle are unequal, so are the angles opposite them, and
the greater angle lies opposite the greater side. The converse is also true.
* If two sides of one triangle are equal to two sides of a second triangle, and
the included angle of the first is greater than the included angle of the
second, the third side of the first is greater than the third side of the
second. The same is true, switching third angle and third side.
-----
Geometry / Polygons / General / Definitions
definition:
* A polygon is a figure with the same number of sides and angles.
* A convex polygon is one whose interior angles measure less than 180 degrees,
or alternately that a line segment between any two points in the interior of
the polygon never crosses to the exterior of the polygon.
* A concave polygon has at least one interior angle more than 180 degrees.
-----
Geometry / Polygons / Areas
definition:
* Polygons having the same area are called equivalent polygons.
* The area of a regular polygon of n sides, each of length s, is equal to
one-half the product of the polygon's apothem and its perimeter:
1/2 * a * n * S.
* The area of a regular polygon of n sides, inscribed in a circle of radius r,
is:
1/2 * n * r^2 * sin(360/n)
* The area of a regular polygon of n sides, circumscribed about a circle of
radius r, is:
n * r^2 * tan(180/n)
* The area of a parallelogram is the product of its base and height.
* The area of a trapezoid is the product of its height and the average of the
bases.
* The area of a rhombus is one-half the product of its diagonals.
-----
Geometry / Polygons / Regular / Definitions
definition:
* A regular polygon is both equilateral and equiangular.
* The center of a regular polygon is the center of the inscribed or
circumscribed circle.
* A radius of a regular polygon is a line segment drawn from the center of
the polygon to one of its vertices, and is also the radius of the
circumscribed circle.
* A central angle of a regular polygon is the angle formed by radii drawn to
two consecutive vertices.
* An apothem of a regular polygon is a radius of its inscribed circle drawn to
one if its sides.
* If a circle is divided into three or more equal arcs, then you can construct
two similar regular polygons:
+ The inscribed polygon is formed from the chords of the arcs.
+ The circumscribed polygon is formed by the tangents through the endpoints
of the arcs.
-----
Geometry / Polygons / Regular / Identities
definition:
* As the number of sides of a regular polygon inscribed in a circle increases
without bound, the length of the apothem approaches the length of the radius,
and the area of the polygon approaches the area of the circle.
* An apothem of a regular polygon is the perpendicular bisector of the
respective side.
* A radius of a regular polygon bisects the angle of its respective vertex.
* Each central angle of a regular polygon has 360 / n degrees, where n is the
number of sides of the polygon.
* Each interior angle of a regular polygon has 180 * (n-2)/n degrees.
* Each exterior angle of a regular polygon has 360 / n degrees.
* The sum of the interior angles is (n-2) * 180 degrees.
* The sum of the exterior angles is 360 degrees.
* The perimeter of a regular polygon of n sides inscribed in a circle of radius
r is given by:
P = 2 * n * r * sin(180/n degrees)
* The perimeter of a regular polygon of n sides sircumscribed about a circle
of radius r is given by:
P = 2 * n * r * tan(180/n degrees)
* An equilateral polygon inscribed in a circle is regular.
-----
Geometry / Circles / Definitions
definition:
* A major arc is an arc greater than a semicircle.
* A minor arc is an arc less than a semicircle; its measure is the measure of
the central angle that intercepts that arc.
* A sector is the set of points between two radii and their intercepted arc.
* A secant is a line intersecting a circle at two points.
* A chord is a line segment joining two points on a circle.
* A line of centers is a line passing through the centers of two or more
circles.
* A central angle is an angle whose vertex is at the center of a circle and
whose sides are radii.
* A tangent is A line with exactly one point of intersection with a circle; the
intersection is the point of tangency.
* A segment of a circle is a region bounded by an arc and the chord containing
the endpoints of the arc. If the chord is not a diameter, the smaller
segment produced by the chord is the minor segment, and the larger is the
major segment.
* The midpoint of an arc is the point on the circle that divides the arc into
two congruent arcs.
-----
Geometry / Circles / Identities
definition:
* The distance from a point to a circle is the distance from the point to the
intersection of the circle and a line between the point and the circle's
center.
* The radius to the point of tangency is perpendicular to the tangent.
* In a circle, parallel lines intercept equal arcs.
* In the same circle or congruent circles, equal arcs have equal chords.
* In the same circle or congruent circles, equal chords are equidistant from
the center.
* A diameter perpendicular to a chord of a circle bisects the chord and its
arcs.
* If two chords intersect within a circle, the product of the two segments
(separated by the point of intersection) of one chord is equal to the
product of the segments of the other.
* If two secants are drawn to a circle from a point outside the circle, the
products of the full secants and just their external segments are equal.
* If a tangent and a secant are drawn to a circle from the same point outside
that circle, then the length of the tangent is the mean proportional
(geometric mean) between the length of the full secant and the length of
its external segment.
* If two tangents are drawn to a circle from the same external point, the
tangents have equal length.
* If two circles intersect in two points, then their line of centers is the
perpendicular bisector of their common chord.
* If two circles are tangential, their line of centers is perpendicular to the
common tangent, and passes through the point of tangency.
* A circle can be inscribed in, or circumscribed about, any regular polygon.
* The area of a minor segment of a circle is equal to the area of its sector
minus the area of the triangle formed by its radii and the chord.
* The radius of circle circumscribed about an equilateral triangle has length
2/3 * h, where h is the length of the altitude.
* The radius of a circle inscribed in an equilateral triangle has length
1/3 * h, where h is the length of the altitude.
-----
Geometry / Circles / Identities / Angles
definition:
* The measure of an inscribed angle is equal to one-half the measure of its
intercepted arc.
* An angle inscribed in a semicircle is a right angle.
* An angle formed by the intersection of two secants outside a circle is equal
to one-half the difference of the arcs intercepted inside the angle.
* An angle formed by a tangent and a chord is equal to one-half the measure of
the intercepted arc.
* If two chords intersect within a circle, each angle formed is equal to
one-half the sum if its intercepted arc and the intercepted arc of its
vertical angle.
* The angle formed by the intersection of a tangent and a secant outside a
circle is equal to one-half the difference of the internally intercepted
arcs.
* The angle formed by two tangents drawn to a circle from an outside point is
equal to one-half the difference of the measures of the intercepted arcs.
* In the same circle or congruent circles, equal inscribed angles have equal
intercepted arcs.
* All inscribed angles inscribed in the same or equal arcs have equal measures.
* Opposite angles of an inscribed quadrilateral are supplementary.
* If two tangents are drawn to a given circle from the same external point,
then the line passing through that point and the circle's center is the
angle bisector of the angle between the two tangents.
-----
Geometry / Circles / Congruency
definition:
* Congruency of circles is congruency of radii.
* Congruency of arcs is congruency of central angle and either radii or arc
length.
-----
Geometry / Circles / Inequalities
definition:
* In the same or equal circles, if two chords are unequal, then they are not at
equal distances from the center, and the longer chord is at a smaller
distance from the center.
* In the same or equal circles, greater goes with greater for each of these:
+ central angle and arc
+ minor arc and chord
-----
Geometry / Circles / Tangents
definition:
* A common tangent of two circles is a line tangent to both.
* A common internal tangent is a common tangent which intersects the line of
centers between the two centers.
* A common external tangent is a common tangent which *does not* intersect the
line of centers between the centers of the two circles.
* Tangent circles are circles in a plane tangent to the same line at the same
point.
* Tangent circles on the same side of the tangent are internally tangent;
otherwise, they are externally tangent.
-----
Geometry / Circles / Inscribed and Circumscribed
definition:
* A circle is inscribed in a polygon if all the sides of the polygon are
tangential to the circle.
* A circumscribed circle passes through all of the vertices of a polygon.
* Concentric circles have the same center and unequal radii.
* An inscribed angle is an angle whose vertex is on the circle and whose sides
are chords of the circle.
* An angle is inscribed in an arc of a circle if its vertex lies on the arc,
and its sides are chords joining the vertex and ends of the arc.
* The incenter of a triangle is the center of the inscribed circle of that
triangle.
* The circumcenter of a triangle is the center of the circumscribed circle of
that triangele.
-----
Geometry / Transversal / Definitions
definition:
* A transversal of two or more lines is a line that cuts across these lines
at one point per line.
* If two lines are cut by a transversal, nonadjacent angles on opposite sides
of the transversal, but on the INterior of the two lines are called
alternate INterior angles.
* If two lines are cut by a transversal, nonadjacent angles on opposite sides
of the transversal, but on the EXterior of the two lines are called
alternate EXterior angles.
* If two lines are cut by a transversal, angles on the same side of the
transversal, and in corresponding positions with respect to the crossed
lines, are called corresponding angles.
-----
Geometry / Transversal / Parallelism Identities
definition:
* For parallel lines cut by a transversal (and conversely, these indicate
parallelism):
+ alternate interior angles are congruent.
+ alternate exterior angles are congruent.
+ corresponding angles are congruent.
+ consecutive interior angles are supplementary
* If three or more parallel lines intercept congruent segments on one
transversal, then they intercept congruent segments on any transversal.
* Three or more parallel lines intercept proportional segments on any two
transversals.
-----
Geometry / Quadrilaterals / Parallelograms
definition:
* A parallelogram is a quadrilateral whose opposite sides are parallel.
* The altitude is the perpendicular segment connecting any point of a line
containing one side of a parallelogram to the line containing the opposite
side.
-----
Geometry / Quadrilaterals / Parallelograms / Identities
definition:
* A diagonal divides the parallelogram into two congruent triangles.
* The diagonals of a parallelogram bisect each other.
* Consecutive angles of a parallelogram are supplementary.
* If both pairs of opposite sides, or both pairs of opposite angles, of a
quadrilateral are congruent, it is a parallelogram.
* If two opposite sides of a quad are both parallel and equal, it is a
parallelogram.
* If one angle is congruent to the opposite angle, and one side is parallel to
the opposite side, then the quad is a parallelogram.
* If one angle is congruent to its opposite, and one side is congruent to its
opposide, then the quad is a parallelogram.
-----
Geometry / Quadrilaterals / Rectangles / Identities
definition:
* If a parallelogram has one right angle, it has four, and is a rectangle.
* If a parallelogram has equal diagonals, it is a rectangle.
* If a parallelogram is inscribed within a circle, it is a rectangle.
-----
Geometry / Quadrilaterals / Rhombi / Identities
definition:
* If a quadrilateral has four equal sides, it is a rhombus.
* If a parallelogram has two adjacent sides equal, it has four, and is a
rhombus.
* If a parallelogram has perpendicular diagonals, it is a rhombus.
* If a diagonal of a parallelogram bisects the angles of the vertices it joins,
the figure is a rhombus.
* A square has all the features of a rhombus and a rectangle.
-----
Geometry / Quadrilaterals / Trapezoids / Definitions
definition:
* A trapezoid is a quad with exactly two sides parallel, which are called
bases.
* The median of a trapezoid is the line joining the midpoints of the
non-parallel sides. It is parallel to the bases, and has a length that
is the average of the base lengths.
* The altitude of trapezoid connects any point in the line containing one
base perpendicularly to the line containing the other base.
* An isosceles trapezoid is a trapezoid with equal nonparallel sides.
* A pair of angles including only one of the parallel sides is called a pair
of base angles.
* The base angles of an isosceles trapezoid are equal.
* The diagonals of an isosceles trapezoid are equal.
* The opposite angles of an isosceles trapezoid are supplementary.
* If the diagonals of a trapezoid are congruent, or one pair of base angles
of a trapezoid are congruent, or one pair of opposite angles is
supplementary, then it is isosceles.
-----
Geometry / Triangles / Similarity
definition:
* Two general triangles are similar if:
+ The three sides of one are in proportion to the three corresponding sides
of the second.
+ Two angles of one are congruent to two corresponding angles of the second.
+ Two sides are proportionate, and the included angle is congruent, between
the two triangles.
+ The corresponding sides are all parallel or all perpendicular to each
other.
* Two right triangles are similar if:
+ The ratio of the hypotenuses is proportional to the the ratio of either
pair of corresponding legs.
+ One acute angle (and therefore both) is congruent between the triangles.
-----
Geometry / Triangles / Similarity / Identities
definition:
* The ratio of the measures of any two corresponding sides, line segments,
or perimeter measures are equal to all other linear ratios (this is the
ratio of similitude)
* The ratio of similitude of any pair of similar triangles equals the square
root of the ratio of their areas.
* The ratio of areas of two similar triangles is equal to the ratio of the
squares of the lengths of any two corresponding sides or line segments of
those triangles.
* In a right triangle, the altitude to the hypotenuse separates the triangle
into two triangles that are similar to each other and to the original
triangle.
-----
Geometry / Trigonometry / Definitions
definition:
* SOHCAHTOA:
+ Sin = Opposite / Hypotenuse.
+ Cos = Adjacent / Hypotenuse.
+ Tan = Opposite / Adjacent.
* Cot (Cotangent) = 1 / Tan = Adjacent / Opposite.
-----
Geometry / Trigonometry / Identities
definition:
*MISSING*
-----
Geometry / Trigonometry / Identities / Special Angles
definition:
* The side ratios of 45-45 right triangles are 1:1:sqrt(2).
* The side ratios of 30-60 right triangles are 1:sqrt(3):2.
* Therefore:
x sin x cos x tan x cot x
0 0 1 0 inf
30 1/2 sqrt(3)/2 sqrt(3)/3 sqrt(3)
45 sqrt(2)/2 sqrt(2)/2 1 1
60 sqrt(3)/2 1/2 sqrt(3) sqrt(3)/3
90 1 0 inf 0
-----
Logic / Notation
definition:
* A statement is a sentence that is true or false, but not both.
* a ^ b denotes (a and b).
* a v b denotes (a or b).
* ~a denotes (not a).
* a -> b denotes (if a, then b).
* a -> b is also referred to as a condition statement or implication.
* "If a" is the hypothesis or premise, and "then b" is the conclusion.
* a is the antecedent of the implication, and b is the consequent.
* The converse of a -> b is b -> a.
* The contrapositive of a -> b is ~b -> ~a.
* The inverse of a -> b is ~a -> ~b.
* a <-> b denotes (a iff b) or (a if and only if b).
* a <-> b is also referred to as a biconditional statement.
-----
Logic / Principles
definition:
* An argument is valid if the truth of the premises means that the conclusion
must also be true.
* The contrapositive of a true statement is true, and of a false statement is
false (contrapositive preserves truth).
* The inverse or converse of a true statement is not necessarily true.
* If the converse of a true statement is true, then the inverse is true.
Likewise, if the converse is false, then the inverse is false.
* Statements that are either both true or both false are logically equivalent.
* If a statement and its converse are both true, the conditions in the
hypothesis are both necessary and sufficient for the conclusion. If the
statement is true, but its converse is false, the conditions are sufficient
but not necessary. If the statement and converse are both false, the
conditions are neither sufficient nor necessary.
-----
Logic / Syllogism
definition:
* A syllogism is an arrangement of three statements that allow the third to
be deduced from the first two, and this is called deductive reasoning.
* The first statement, the major premise, is a general statement about a group.
* The second statement, the minor premise, indicates that an individual is a
member of that group.
* The third statement, the deduction, states that the general statement must
then apply to the individual.
-----
Logic / Indirect Proof
definition:
* Indirect proof is proof that under a given hypothesis, all conclusions
other than the one we wish to prove are necessarily false, leaving the
remaining possibility as true.
* First list all possible conclusions.
* Then prove all but one of these conclusions must be false, because they
contradict something known to be true (a prior theorem, postulate, or
definition, for example).
* The only remaining possible conclusion must then be true.
-----
Logic / Inductive Reasoning
definition:
* Inductive reasoning draws conclusions or generalizations from several known
particular cases.
* First, prove that the conditions of the statement are valid for the smallest
possible value of n.
* Next, assume the conditions are true for a general case n = k.
* Show that the conditions are true for n = k + 1 given that assumption.
* The statement is then valid for all n >= min(n).
-----
Logic / Definition
definition:
* In a logical system, certain things are assumed to be known and true, and
other things are derived from these in some way.
* For terminology, the assumed terms are called undefined terms, and the
derived terms are called defined terms.
* A definition is considered a good one under the following conditions:
+ It names the term being defined.
+ It uses only known terms or accepted undefined terms.
+ It places the term into the smallest set to which it belongs.
+ It states the characteristics of the defined term which distinguish it from
other members of the set.
+ It contains the least possible amount of information.
+ It is always reversible.
* For statements, the assumed statements are known as axioms, postulates, or
assumptions. The derived statements are known as theorems, proposititions,
corollaries, or lemmas. Among those, a corollary is any theorem that can
be deduced very easily from existing theorems or postulates.
-----
Matrices / Indices / Order of
definition:
An m x n matrix contains m rows and n columns.
An element a[i,j] refers to row i, column j.
>>>
hint:
Remember that vectors are m x 1 matrices, and
appear vertically in traditional layouts.
-----
Matrices / Classifications
definition:
* The zero matrix of some size has zeroes in every entry.
* Diagonal matrices have zeroes in every entry not on the diagonal.
* Identity matrices are square diagonal matrices with a one in every
diagonal entry.
* Matrix units e[i,j] have only one nonzero entry, which is a one, at some
location (i,j).
* Elementary matrices come in three forms, all square:
+ The identity matrix, plus one nonzero off-diagonal entry: I + a * e[i,j],
where i != j.
+ The identity matrix, with the (i,i) and (j,j) entries replaced by zero,
and ones placed at (i,j) and (j,i): I + e[i,j] + e[j,i] - e[i,i] - e[j,j].
+ The identity matrix, with one diagonal entry replaced by a nonzero
number c: I + (c - 1) * e[i,i].
>>>
hint:
* Some matrix types are named for the shape of their nonzero entries.
* Identity matrices and unit matrices are different.
* Elementary matrices represent fundamental operations on rows when left
multiplied with another matrix.
-----
Matrices / Algebra / Classifications
definition:
* The set of all invertible n x n matrices is called the
n-dimensional general linear group and is denoted GLn.
>>>
hint:
* GLn
-----
Matrices / Algebra / Algebraic Rules
definition:
* Addition as per vectors
* Scalar multiplication as per vectors
* Distributive law for + and * applies
* Multiplication associates
* Multiplication does NOT commute in general
* Multiplicative identities exist, are square, and may pre- or post-multiply
* Multiplicative inverses MAY exist for square matrices, and will commute
>>>
hint:
XXXX Missing hint -- perhaps:
The set of all matrices form a *foobar* algebraic system;
the set of all invertible n x n matrices forms GLn, the
general linear group of dimension n.
-----
Matrices / Algebra / Special Matrix Types
definition:
* Diagonal square matrices represent scaling (or collapse) of vector
dimensions; each diagonal entry affects a different dimension.
* Every matrix can be represented as a linear combination of the matrix units,
where each coefficient in the combination is the matrix entry matching the
nonzero entry in each matrix unit:
__
\
A = /_ a[i,j] * e[i,j]
i,j
* Elementary matrices represent the following operations on X in the
left multiplication E * X:
+ One off-diagonal nonzero entry a at (i,j) will add (a * row j) to row i.
+ Two ones at (i,i) and (j,j) moved to (i,j) and (j,i) will interchange
rows i and j.
+ A single diagonal one at (i,i) replaced by a nonzero scalar c will
multiply row i by c.
* For right multiplication, the elementary matrices operate on columns.
* Row echelon matrices (the result of using left multiplication by a series
of elementary matrices) are used to solve systems of linear equations. They
take the following form and are unique per starting matrix:
+ The first nonzero entry in every row (called a pivot) is 1.
+ The pivot of row i+1 is to the right of the pivot of row i.
+ All entries in the same column (and in particular, above) a pivot, other
than the pivot itself, are 0.
* Every square matrix has a REF (row echelon form) that is I or has the a
bottom row of zeroes. Only the former case indicates invertibility.
-----
Matrices / Algebra / Identities / Inversion
definition:
* For matrices A1 .. An, (A1 * ... * An)^-1 = An^-1 * ... * A1^-1;
in other words, the inversion of a product of matrices is the
product, in reverse order, of the inverses of each matrix.
* Elementary matrices are invertible, and their inverses are elementary. In
fact, the row swap matrices are their own inverse.
* The following are equivalent for a matrix A:
+ A can be reduced to I by a series of elementary row operations.
+ A is the product of elementary matrices.
+ A is invertible.
+ No rows of A are 0.
+ det A != 0.
+ The homogenous system A * X = 0 has only the trivial solution X = 0.
* Since the row operations that create I from A create A^-1 from I, you can
find A's inverse by applying row reductions to the n x 2n matrix [A | I].
>>>
hint:
* Products go back to back and one by one.
* You can undo any elementary row operation.
* A -> I == I -> A^-1
-----
Matrices / Algebra / Identities / Transpose
definition:
* (A + B)^t = A^t + B^t
* (c * A)^t = c * A^t
* (A * B)^t = B^t * A^t
* (A^t)^t = A
>>>
hint:
* Transpose of products is like inverse of products.
-----
Matrices / Algebra / Identities / Block Multiplication
definition:
Let M, M' be m x n and n x p matrices, and let r be an integer
less than n. Then we may decompose the matrices as follows:
M = [A|B] and M' = [A']
[--]
[B']
Where A has r columns and A' has r rows. Then the multiplication
can be decomposed into:
M * M' = A * A' + B * B'
This can be extended to any number of submatrices, as long as the
necessary products are defined. A decomposition into 2 x 2
submatrices is particularly common.
>>>
hint:
Subblocks of matrices can act like elements in a vector dot product.
-----
Matrices / Determinants
definition:
* Every square matrix A has a number associated with it called its determinant,
designated det A, mapping an n^2 dimensional real space to a single real
number (det: R^nxn --> R).
* Geometrically, this number is the volume of the parallelipiped which forms
the image of a unit hypercube under the operation of the matrix. The sign
of the determinant indicates whether the operation is orientation-reversing.
* det A = 0 iff the image collapses a dimension (in two dimensions, the
parallelogram becomes a line segment, and the two columns of A are
proportional).
* Conversely, A is invertible iff det A != 0.
>>>
hint:
* Determinants characterize geometric transforms.
-----
Matrices / Determinants / Formulas
definition:
* For a 1 x 1 matrix, det [a] = a.
* For a 2 x 2 matrix, det [a b][c d] = a * d - b * c.
* Beyond that, many different formulas are used. A recursive definition called
expansion by minors may be used, in which you sum up (using alternating
signs) the products of each entry in one of the columns or rows with the
recursive determinant of the matrix created if the row and column containing
that entry was removed.
* For expansion by minors on the first column, we get:
det A = + a[1,1] * det A[1,1] - a[2,1] * det A[2,1] +, -, ... +- a[n,1] * det A[n,1]
where each "det A[i,1]" denotes the determinant of the matrix formed by
removing row i and column 1 from A.
>>>
hint:
* Diagonals before corners for 2 x 2.
* Expansion by minors is a recursive definition of the determinant.
-----
Matrices / Determinants / Identities
definition:
* det I = 1.
* det (A * B) = det A * det B.
* det (A^-1) = (det A)^-1.
* det (A^t) = det A.
* det (triangular matrix) = Product(diagonal entries).
* For the elementary matrices, det e = 1 for row scale * add,
det e = -1 for row interchange, and det e = c for row scale.
* det A is linear in the rows of the matrix.
* det is a continuous function from any one of the classical groups to R or
C, the real or complex numbers of one dimension.
* If a row of A is 0, then det A = 0.
* If two rows of a matrix A are proportional, det A = 0.
* These all apply to columns as well, because of the transpose identity.
>>>
hint:
* Determinants of I, A * B, A^-1, A^t, e, triangular; and cases for det A = 0.
-----
Sets / Maps / Notation
definition:
* {s E S|foo}, where E denotes the rounded "element of" symbol, indicates
the subset s of S where foo is true.
* The map (aka function) description phi: S -> T indicates that the map phi
has the set S as its domain, and has a range that is a subset of T. If the
domain or range of a function are altered, by convention a new function is
defined.
* s ~~> t (where ~~> is a wiggly arrow) is the same as t = phi(s), meaning
that the element t in T is the image of s in S under phi.
* The image of a map phi is defined as:
im phi = {t E T|t=phi(s) for some s E S}.
* A map is surjective if the image of map phi on S is all of T.
* The map is injective if distinct elements of S have distinct images in T.
* A surjective injective map is called a bijective map.
* An injective map from a set to itself is a permutation.
* A map phi has an inverse psi iff (phi o psi: T -> T) and (psi o phi: S -> S)
are indentity maps, and a map has an inverse iff it is bijective.
* Whether or not a map has an inverse, it has an inverse image. If phi: S -> T
is a map, and U is a subset of T, then the inverse image of U is defined as:
phi^-1(U) = {s E S | phi(s) E U}
-----
Sets / Cardinality
definition:
* |S| is the cardinality of S, the number of its elements. The cardinality of
infinite sets is |S| = infinity.
* Trivially, for a map phi: S -> T, if phi is injective |S| <= |T|, if phi
is surjective, |S| >= T, and if |S| == |T|, phi is bijective iff it is
either injective or surjective.
* An infinite set is countable if there is a bijective map from the natural
number domain to the set.
-----
Sets / Partial Orderings / Notation
definition:
* A partial ordering of a set S is a relation s <= s' which *may* hold between
certain elements and which satisfies the following axioms for all s, s', s"
in S:
+ s <= s.
+ if s <= s' and s' <= s", then s <= s".
+ if s <= s' and s' <= s, then s == s'.
* A partial ordering is called a total ordering if in addition:
+ for all s, s' in S, s <= s' or s' <= s.
* Let S be a set whose elements are sets. If A and B are in S, then we can
define A <= B if A is a subset of B. This partial ordering on S is called
the ordering by inclusion.
* If A is a subset of a partially ordered set S, then an upper bound for A is
an element b in S such that for all a E A, a <= b. A partially ordered set
S is called inductive if every totally ordered subset T of S has an upper
bound in S.
* A maximal element m E S is any element such that S contains no larger one
(there is no element s E S with m <= s, except for m). This doesn't mean
that m is an upper bound for S; there may be many maximal elements.
* Zorn's Lemma (equivalent to the axiom of choice): An inductive partially
ordered set has a maximal element.
-----
Topology / Openness
definition:
* The open ball of radius r about a point X E R^k is the set of all points
whose distance to X is less than r:
B[X,r] = {X' E R^k| |X' - X| < r}.
* A subset U of R^k is open if the following is true for r dependent on X:
If X E U and r is sufficiently small, then B[x,r] is a subset of U.
* The union of an arbitrary family of open sets is open.
* The intersection of finitely many open sets is open.
* If f is a continuous function R^K -> R, then the sets {f > 0}, {f < 0}, and
{f != 0} are open.
* If V is a subset of S in R^k, either of the following indicates V is an open
subset of S:
+ V = U intersect S for some open set U of R^k.
+ For every point X E V, there is an r > 0 so that V contains the set
B[X,r] intersect S.
* An open subset V of S which contains a point p is the neighborhood of p in S.
-----
Topology / Closedness
definition:
* Closed is dual to open.
* A subset C of a set S is closed if its complement (S - C) is open.
* The intersection of an arbitrary family of closed sets is closed.
* The union of finitely many closed sets is closed.
* A subset C of R^k is called bounded if the coordinates of the points in C are
bounded, meaning that there is a positive real number b, a bound, such that
for X = (x1,...,xn) E C, |xi| <= b for all i = 1,...,n.
* If C is both closed and bounded, it is called a compact subset of R^k.
-----
Topology / Continuity
definition:
* A map f is continuous over S if for any s E S and every real number
epsilon > 0, there is a delta > 0 such that if s' E S and
|s' - s| < delta, then |f(s') - f(s)| < epsilon.
* A map f: S -> S' is called a homeomorphism if it is bijective if f and f^-1
are continuous.
* A path is a continuous map f: [0,1] -> R^k, and the path is said to lie in
S if f(t) E S for ever t E [0,1].
* A subset S of R^k is called path-connected if every pair of points p, q E S
can be joined by a path lying in S. In other words, for every pair of
points p, q E S, there is a path f such that:
+ f(t) E S for all t in the interval.
+ f(0) = p and f(1) = q.
* A path-connected set S is not the disjoint union of proper open subsets.
-----
Topology / Manifolds
definition:
* A subset S of R^n is called a manifold of dimension d if every point p of S
has a neighborhood in S which is homeomorphic to an open set in R^d.
-----