High School Math Notes

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    Geometry Axioms

    Point: an undefined term in geometry, a point can be thought of as a dot that represents a location on a plane or

    in space; just a location, no thickness or size

    Line: an undefined term in geometry, a line is understood to be straight, containing an infinite number of points

    extending infiting in two directions, and having no thickness

    Line Segment: a portion of a line with two end points

    Ray: consists of an intial point on a line and all of the points on that line on one side of it

    Plane: an undefined term in geometry, a plane is understood to be a flat surface that extends infinitely in al

    directions

    Angle: a figure formed by two rayss that have the same endpoint

    Collinear: three or more points not all of which lie on the same line

    Noncolinear: points that lie on the same line

    Perpendicular Lines: lines that cross at a 90 angleParallel Lines: Lines in the same plane that never meet and hav ethe same slope

    Skewed Lines: lines that are not parallel and do not cross; lines in different planes

    one and only one line passes through two points

    one and only one plane passes through two lines

    two points define an unique line

    three non-collinear points define an unique plane

    Logic

    Postulate: a statement that is accepted as true without proof

    Theorem: a statement that must be proved to be true

    Inductive Reasoning: forming conjectures on the basis of an observed pattern

    Deductive Reasoning: the process of drawing conclusion by using logical reasoning

    Conjecture: an educated guess based on an observation

    Hypothesis: the phrase in a conditional statement following the word if

    Conditional Statement: a statement that can be written in the form: if P then QConverse Statement: the statement formed by interchanging the hypothesis and conclusion of a conditiona

    statement

    Counter Example: an example which proves that a conditional statement is false in that the hypothesis is true but

    the conclusion is false

    Conclusion: the phrase in a conditional statement following the word then

    Lines

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    Ruler Postulate: The points on a line can be matched one to one with the set of real numbers. The real numbe

    that corresponds with a point is the coordinate of the point. The distance, AB between two points, A and B on a

    line is equal to the absolute value of the difference between the coordinates of A and B.

    Segment Addition Postulate: If B is between A and C then AB +BC= AC

    Parallel Postulate: IF there is a line and a point not on the line, THEN there is exactly on a line through the point

    parallel to the given line

    Perpendicular Postulate: IF there is a line and a point not on the line, THEN there is exactly on a line through the

    point perpendicular to the given line

    Transitive Property of Parallel Lines: IF two lines are parallel to the same line THEN they are parallel to each

    other Property of Perpendicular Lines: IF two coplanar lines are perpendicular to the same line THEN they are

    parallel to each other

    Perpendicular Bisector Theorem: IF a point is on the perpendicular bisector of a segment THEN it is equidistan

    from the endpoints of the segment

    Perpendicular Bisector Converse: IF a point is equidistant from the endpoints of a segment THEN it lies on the

    perpendicular bisector of the segment

    Angle Bisector Theorem: IF a point is on the bisector of an angle THEN it is equidistant from the two sides of the

    angle

    Angle Bisector Converse: IF a point is in the interior of an angle and equidistant from the sides of the angle

    THEN it lies on the bisector of the angle

    IF two distinct planes intersect, THEN their intersection is a line

    IF two distinct points lie in a plane, THEN the line containing them lies in the plane

    IF two distinct lines intersect, THEN their intersection is exactly one point

    IF three parallel lines intersect two transversals, THEN they divide the transversals proportionally

    IF two parallel lines intersect two other parallel lines and IF the distance between the first two lines is equa

    to the distance between the second two lines THEN their intersection forms four congruent segments

    Angles

    Protractor Postulate: Let OA be a ray and consider on of the half-plane P determined by the line OA. The rays

    of the form ray DO, where D is in a half-plane P can be put in one-to-one correspondence with the real numbers

    between 0 and 180, including 180. If C and D are in the half-plane P, then the measure of angle COD is equal to

    the absolute value of the difference between the real numbers for ray OCand ray OD.

    Angle Addition Postulate: If B is the interior of angle

    AOC, then

    AOB +

    BOC =

    AOCLinear Pair Postulate: IF two angles form a linear pair, THEN they are supplementary; that is, the sum of their

    measures is 180

    Corresponding Angles Postulate: IF two parallel lines are cut by a transversal, THEN the pairs of corresponding

    angles are congruent

    Corresponding Angles Postulate: IF two parallel lines are cut by a transversal so that corresponding angles are

    congruent, THEN the lines are parallel

    Arc Addition Postulate: The measure of an arc formed by two adjacent arcs is the sum of the measures of the

    two arcs.

    Exterior Angle Theorem: the measure of an exterior angle of a triangle is equal to ???????????

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    Interior Angle Sum: 180 (n2) n = # of sidesExterior Angle Sum: always = 360

    # of Diagonals:n (n3)

    2n = # of sides

    Slope:rise

    run=y

    x=

    y2y1x2x1 = m

    Area Congruence Postulate: IF two polygons are congruent THEN they have the same area

    Area Addition Postulate: the area of a region is the sum of the areas of all its non-overlapping partsCavalieris Principle: IF two solids have the same height and the same cross-sectional area at every level, THEN

    they have the same volume

    IF two polygons are similar THEN the ratio of their perimeters is equal to the ratio of their corresponding

    sides

    IF two polygons are similar with corresponding sides in the ratio of a : b THEN the ratio of their areas is

    a2 : b2

    Areas

    A =1

    2BH (triangle)

    A = s2 (square)

    A = BH (rectangle)

    A = BH (parallelogram)

    A =

    1

    2 (B1 +B2)H (trapezoid)

    A =1

    2d1d2 = BH (rhombus)

    A = r2 (circle)

    A = r1r2 (ellipse)

    A =2

    3BH (under symmetrical parabola)

    Surface Areas

    SA = 6s2 (cube)

    SA = 2 (rectangular prism)

    SA = (rectangle parallelpiped)

    Pyramid:

    Cone:

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    Sphere: SA = 4r2 (sphere)

    Ellipsoid:

    Volumes

    V = s3 (cube)

    V = HL2 (rectangular prism)

    V = LWH (rectangular parallelpiped) V = BH (irregular prism)

    V =1

    3BH (pyramid)

    V =1

    3r2h (cone)

    V = r2h (cylinder)

    V =4

    3r3 (sphere)

    V =4

    3r1r2r3 (ellipsoid)

    Triangles

    Similar:

    Congruent: AAS, HL

    Triangle Sum Theorem: the sum of the measures of the angles of a triangle is 180

    Exterior Angle Theorem: the measure of an exterior angle of a triangle is equal to the sum of the measures of

    the two remote (nonadjacent) interior angles

    Exterior Angle Inequality: the measure of an exterior angle of a triangle is greater than the measure of either of

    the two nonadjacent interior angles

    Triangle Proportionality Theorem: IF a line parallel to one side of a triangle intersects the other two sides THEN

    it divides the two sides proportionally

    Triangle Proportionally Theorem Converse: IF a line divides tow sides of a triangle proportionally THEN it isparallel to the third side

    Midsegment Theorem: The segment connection the midpoints of two dies of a triangle is parallel to the third side

    and is half its length

    Triangle Inequality: the sum of the lengths of any two sides of a triangle is greater than the length of the third side

    Third Angles Theorem: IF two angles of one triangle are congruent to two angles of a second triangle THEN the

    third angles are also congruent

    Base Angles Theorem: IF two sides of triangles are congruent THEN the angles opposite them are congruent

    Corollary: IF a triangle is equilateral THEN it is also equiangular

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    Hinge Theorem: IF two sides of one triangle are congruent to two sides of another triangle, and the included angle

    of the first is larger than the includes angle of the second THEN the third side of the first is longer than the third

    side of the second

    Converse of Hinge Theorem: IF two sides of one triangle are congruent to two sides of another triangle, and the

    third side of the first is longer than the third side of the second THEN the included angle of the first is larger than

    the included angle of the second

    The Pythagorean Theorem: in a right triangle, the square of the length of the hypotenuse is equal to the sum of

    the squares of the lengths of the legsThe Converse of The Pythagorean Theorem: IF the square of the length of the longest side of a triangle is equa

    to the sum of the squares of the lengths of the two shorter sides, THEN the triangle is a right triangle

    Concurrency Properties:

    the lines containing the perpendicular bisectors of a triangle are concurrent

    their common point is the circumcenter of the triangle

    the circumcenter is equidistant from the three vertices of the triangle

    the angle bisectors of a triangle are concurrent their common point is the incenter of the triangle

    the incenter is equidistant from the three sides of the triangle

    the medians of a triangle are concurrent

    their common points is the centroid of the triangle

    the centroid is two thirds of the distance from each vertex to the midpoint of the opposite side

    the lines containing the altitudes of a triangle are concurrent

    their common point is the orthocenter

    IF two angles of a triangle are congruent THEN the sides opposite them are congruent

    the acute angles of a right triangle are complementary

    every triangle is congruent to itself

    IF one side of a triangle is longer than another side THEN the angle opposite the longer side is larger than

    the angle opposite the shorter side

    IF one angle of a triangle is larger than another angle THEN the side opposite the larger angle is longer than

    the side opposite the smaller angle

    IF a ray bisects an angle of a triangle THEN it divides the opposite side into segments whose lengths are

    proportional to the lengths of the other two sides

    IF the altitude is drawn to the hypotenuse of a right triangle THEN the two triangles formed are similar to the

    original triangle and to each other

    in a right triangle, the length of the altitude from the right angle to the hypotenuse is the geometric mean of

    the lengths of the two segments of the hypotenuse

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    Right Triangles

    hc =ab

    c(altitude of c)

    ta = 2bccos

    A2

    (b + c)

    (angle bisector of a)

    tb = 2accos

    B2

    (a + c)

    (angle bisector of b)

    tc = ab

    2

    (a + b)(angle bisector of c)

    ma =

    4b2 + a2

    2(median of a)

    mb =

    4a2 + b2

    2(median of b)

    mc =c2

    (median of c)

    r=ab

    a + b + c(inscribed circle radius)

    R =c

    2(circumscribed circle radius)

    Equilateral Triangles

    K= a2

    3

    4(area)

    ha = hb = hc = a

    3

    2(altitude)

    ma = mb = mc = a

    3

    2(median)

    ta = tb = tc = a

    3

    2(angle bisector)

    R = a

    3

    3(circumscribed circle radius)

    r= a

    3

    6(inscribed circle radius)

    Quadrilaterals

    IF both pairs of opposite sides of a quadrilateral are congruent THEN the quadrilateral is a parallelogram

    IF both pairs of opposite angles of a quadrilateral are congruent THEN the quadrilateral is a parallelogram

    IF an angle of a quadrilateral is supplementary to both of its consecutive angles THEN the quadrilateral is a

    parallelogram

    IF the diagonals of a quadrilateral bisect each other THEN the quadrilateral is a parallelogram

    IF the diagonals of a quadrilateral are perpendicular THEN the area of the quadrilateral is half the sum of the

    lengths of the diagonals or A = 0.5 (d1 + d2)

    IF one pair of opposite sides of a quadrilateral are congruent and parallel THEN the quadrilateral is a paral-

    lelogram

    a parallelogram is a rhombus IF AND ONLY IF its diagonals are perpendicular

    a parallelogram is a rhombus IF AND ONLY IF each diagonal bisects a pair of opposite angles

    a parallelogram is a rectangle IF AND ONLY IF its diagonals are congruent

    a quadrilateral is a rhombus IF AND ONLY IF it has four congruent sides

    a quadrilateral is a rectangle IF AND ONLY IF it has four right angles

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    SASAS Congruence Theorem: IF three sides and the included angles of one quadrilateral are congruent to the

    corresponding three sides and included angles of another quadrilateral THEN the quadrilateral are congruent

    ASASA Congruence Theorem: IF three angles and the included side of one quadrilateral are congruent to the

    corresponding three angles and the included sides of another quadrilateral THEN the quadrilateral are congruent

    Parallelograms

    opposite sides are congruent

    opposite angles are congruent

    consecutive angles are supplementary

    diagonals bisect each other

    Trapezoids

    Trapezoid Base Angles Theorem: IF a trapezoid is isosceles THEN each pair of base angles is congruent

    Trapezoid Diagonals Theorem: IF a trapezoid is isosceles THEN its diagonals are congruent

    Midsegment Theorem for Trapezoids: the midsegment of a trapezoid is parallel to each base and its length is

    half the sum of the lengths of its bases

    IF a trapezoid has one pair of congruent base angles THEN it is an isosceles trapezoid

    IF a trapezoid has congruent diagonals THEN it is an isosceles trapezoid

    Kites

    diagonals are perpendicular

    exactly one pair of opposite angles are congruent

    Circles

    Standard Form: (xh)2 + (y k)2 = r2radius = r center = (h, k)

    L = (arc length)

    A = (arc sector area)

    Major Arc: an arc of more than 180; represented by three lettersMinor Arc: an arc of less than 180; represented by two lettersInscribed Angle of a Circle: an angle whose vertex is on the circle and its sides are chords of the circle

    Intercepted Arc: the arc that lies in the interior of an inscribed angle

    Tangent of a Circle: a line that intersects the circle at exactly one point

    Secant of a Circle: a line that intersects the circle in two points

    IF a line is tangent to a circle THEN it is perpendicular to the radius drawn to the point of tangency in a plane

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    IF a line is perpendicular to a radius of a circle at its endpoint on the circle THEN the line is tangent to the

    circle

    IF two segments from the same exterior point are tangent to a circle THEN the segments are congruent

    in the same circle or in congruent circles, two arcs are congruent IF AND ONLY IF their central angles are

    congruent

    in the same circle or in congruent circles, two minor arcs are congruent IF AND ONLY IF their corresponding

    chords are congruent

    IF a diameter of a circle is perpendicular to a chord THEN the diameter bisects the chord and its arc

    IF a chord AB is perpendicular bisector of another chord THEN AB is a diameter

    in the same circle or in congruent circles, two chords are congruent IF AND ONLY IF they are equidistant

    from the center

    IF an angle is inscribed in a circle THEN its measure is half the measure of its intercepted arc

    IF two inscribed angles of a circle intercept the same arc THEN the angles are congruent

    an angle that is inscribed in a circle is a right angle IF AND ONLY IF its corresponding arc is a semicircle

    a quadrilateral can be inscribed in a circle IF AND ONLY IF its opposite angles are congruent

    IF a tangent and a chord intersect at a point on a circle THEN the measure of each angle from it is half the

    measure of its intercepted arc

    IF two chord intersect in the interior of a circle THEN the measure of each angle is half the sum of the

    measure of the arcs intercepted by the angle and its vertical angle

    IF a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle THEN the measure

    of the angle from is half the difference of the measure of the intercepted arcs

    Trigonometry

    A2 +B2 = C2 (Pythagorian theorem)

    A sin () +B cos () =

    A2 +B2 sin+

    4

    =

    A2 +B2 cos

    4

    (sum of sin and cos)

    Right Angle Trigonometry:

    Sine = oppositehypotenuse

    Cosine = adjacenthypotenuse

    Tangent = oppositeadjacent

    Secant = hypotenuseopposite

    Cosecant = hypotenuseadjacent

    Cotangent = oppositeadjacent

    sinA

    a=

    sinB

    b=

    sinC

    c(law of sines)

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    Law of Cosines:

    A2 = B2 +C22BCcosAB2 = A2 +C22ACcosBC2 = A2 +B22AB cosCLaw of Tangents: cosA =cosB cosC+ sinB sinCcosA

    Linear Equations

    Standard Form: Ax +By = C A,B,Care integers and A is postive

    Slope-Intercept Form: y = mx + b m = slope, b = y-intercept

    Point Slope Form: yy0 = m (xx0) (y0,x0) = a point on the line m = slope

    m = tan =y2y1

    x2x1 (slope of a line)

    = tan1 m2m11 + m1m2

    (angle between intersecting lines)

    (x2x1)2 + (y2y1)2 (distance between two points)

    x1 +x2

    2,

    y1 +y22

    (midpoint of a line segment P1P2)

    m1x2 + m2x1

    m1 + m2,

    m1y2 + m2y1m1 + m2

    (point dividing a line segment P1P2 in a ratio of m1 : m2)

    Quadratic Equations

    y = Ax2 +Bx +C (quadratic equation; standard form)

    y = a (xh)2 + k (quadratic equation; vertex form)

    x =b2a

    or x = h (axis of symmetry)

    b2a

    , f

    22a

    or (h, k) (vertex)

    (0,C) or (0, k) (Y intercept)

    x = b

    b2

    4ac

    2a (quadratic formula)

    Completing The Square:

    ax2 + bx + c = 0 (standard form)

    h = b2a

    k= c b2

    4a

    a (xh)2 + k= 0 (vertex form)

    x = hk

    a(roots)

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    Polynomials

    Factoring:

    x2y2 = (x +y)(xy) (difference of squares) x2 + 2xy +y2 = (x +y)2 (trinomial square sums)

    x22xy +y2 = (xy)2 (trinomial square difference) (x +y)3 = x3 + 3x2y + 3xy2 +y3 (cubed binomial sum)

    (xy)3 = x33x2y + 3xy2y3 (cubed binomial difference) x3 +y3 = (x +y)

    x2xy +y2 (sum of cubes)

    x3y3 = (xy)x2 +xy +y2 (difference of cubes)Fundamental Theorem of Algebra: every non-zero single-variable polynomial with complex coefficients has ex-

    actly as many complex roots as its degree, if each root is counted up to its multiplicityIntegral Root Theorem: all possible rational roots of the polynomial xn + an1xn1 + + a0 = 0 are of the form

    x =p where:

    p is an integer factor of the constant term a0

    Rational Root Theorem: all possible rational roots of the polynomial anxn + an1xn1 + + a0 = 0 are of the

    form: x =pq

    where:

    p is an integer factor of the constant term a0

    q is an integer factor of the leading coefficient an

    Radical Conjugate Root Theorem:

    if a polynomial P (x) with rational coefficients has a +

    b as a zero, where a, b are rational and

    b is

    irrational, then a

    b is also a zero

    Complex Conjugate Root Theorem:

    if P is a polynomial in one variable with real coefficients, and a + bi is a root of P with a and b real numbersthen its complex conjugate a

    bi is also a root of P

    Descartes Rule of Signs:

    the number of positive roots of the polynomial anxn + an1xn1 + + a0 = 0 is either equal to the numbe

    of sign differences between consecutive nonzero coefficients, or less than it by a multiple of 2

    multiple roots of the same value are counted separately

    the number of negative roots is the number of sign changes after negating the coefficients of odd-power

    terms (otherwise seen as substituting the negation of the variable for the variable itself), or fewer than it by a

    multiple of 2

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    Exponents

    the variables m and n are integers

    no denominater equals zero

    a

    0

    = 1(a = 0

    )

    am an = am+n

    (ab)n = anbn

    (am)n = amn

    an =1

    an

    am

    an = am

    n

    ab

    n=

    an

    bn

    a1n = n

    a

    amn =

    n

    am =

    n

    am

    Radicals

    even roots require if you divid by a variable in an equation, you mus

    check for zero as a root

    n

    an

    = |a| (nis even)

    n

    a n

    b =n

    ab

    n

    an

    b= n

    a

    b(b = 0)

    a1n = n

    a

    amn =

    n

    am =

    n

    am

    x mn nm = |x|

    n

    m

    x = mn

    x

    a n

    x + b n

    x = ab n

    x (like radicals)

    a +

    b

    a

    b

    = a2b2 (conjugates)

    conjugates can be used to rationalize a binomia

    denominator

    Simplest Form

    Cant Have: negative exponent

    Fix: make fraction and rationalize denominator or multiply by another negative

    Cant Have: rational exponent

    Fix: turn base into number raised to a power equal to the denominator

    Cant Have: radical in the denominator

    Fix: use conjugate for binomials multiply radical by itself number of times equal to the index

    Rationalize denominator

    make radical a fraction and make denominator perfect square, etc.

    multiply fraction by the denominator to make denominator which is a radical into an integer

    Logarithms

    the variables M, N, and b are positive numbers

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    the variable b = 1 IF y = bx TH E N logby = x

    loga ax = x

    loga 1 = 0

    loga a = 1

    alogax = x

    logb (MN) = logbM+ logbN

    logb

    M

    N

    = logbM logbN

    logb (Mx) = x logbM

    logbM=logcM

    logc b(change of base; b = 1, c = 1, M > 0)

    Complex Fractions

    Complex Numbers

    x = ix, x > 0

    in =

    r= 0 1

    r= 1 i

    r= 2 1r= 3 i

    (value of i after dividing exponent by 4)

    Re [Z] = A = rcos (real part)

    Im [Z] = B = rsin (imaginary part)

    r= |z| = |A +Bi| =

    A2 +B2 (modulus; absolute value)

    = tan

    1BA

    (argument)

    arguments have 2 periodicity

    z = r(cos+ i sin) (polar form)

    z = rei (alternate polar form)

    ei = cos+ i sin (Euler relation)

    z1z2 = r1r2[cos (1 +2) + i sin (1 +2)] (complex product; polar form)

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    z1

    z2=

    r1

    r2[cos (12) + i sin (12)] , z2 = 0 (complex quotient; polar form)

    zn = rn (cos n+ i sin n) , n = integer (complex to power; polar form)

    wk = r1n

    cos

    + 2k

    n

    + i sin

    + 2k

    n

    , k= {0, 1, 2, . . . , n1} (root of complex; polar form)

    d

    dtei(t) = iei(t) d

    dt(complex derivative)

    d

    dt(Re [Z]) = Re

    dZ

    dt

    (commutativity)

    Inequalities

    Trichotomy Property: given any two real numbers a and b, then only one of the following statements must hold

    true

    a < b a = b a > b

    for a, b, and c are real numbers:

    all of the following properties hold for and in addition to < and > remember to change the direction of the inequality when multiplying or dividing by a negative number

    IF: a < b THEN: a + c < b + c (inequality addition property)

    IF: a > b THEN: a + c > b + c (inequality addition property)

    IF: a < b THEN: a

    c < b

    c (inequality subtraction property)

    IF: a > b THEN: ac > b c (inequality subtraction property) IF: a < b THEN: ac < bc (inequality multiplication property; c > 0)

    IF: a > b THEN: ac > bc (inequality multiplication property; c > 0)

    IF: a > b THEN: ac < bc (inequality multiplication property; c < 0)

    IF: a < b THEN: ac > bc (inequality multiplication property; c < 0)

    IF: a < b THEN:a

    c 0)

    IF: a > b THEN:a

    c>

    b

    c(inequality division property; c > 0)

    IF: a > b THEN:a

    c

    b

    c(inequality division property; c < 0)

    Absolute Values

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    |a| 0 |a| = 0 a = 0 |ab| = |a| |b| |a + b| |a|+ |b| |a| = |a|

    |ab| = 0 a = b

    |ab| |a c|+ |cb|

    ab

    = |a||b| if b = 0

    |a

    b| ||

    a| |

    b||

    Number Theory

    Number Divisibility Condition

    1 automatic

    2 last digit is even

    3 sum of the digits is divisible by 3

    4 the last two digits are divisible by 4

    5 the last digit is a 0 or a 5

    6 the number is divisible by 2 AND 3

    7 the sum of [three times the first digit] and [the second digit] is divisible by 7

    8 the last three digits are divisible by 8

    9 the sum of the digits is divisible by 9

    10 the last digit is zero

    limk

    n

    A = xk+1 =1

    n

    (n1)xk+ A

    xn1k

    (nth root numerical approximation algorithm)

    Greatest Common Divisor (Greatest Common Denominator) (GCD) Properties:

    every common divisor of a and b is a divisor of gcd (a, b)

    if a divides the product bc, and gcd (a, b) = d, then a/d divides c if m is a non-negative integer, then gcd (m a, m b) = m gcd (a, b) if m is any integer, then gcd (a + m b, b) = gcd (a, b)

    if m is a nonzero common divisor of a and b, then gcdam

    ,bm=

    gcd(a, b)m

    the gcd is a multiplicative function in the following sense: if a1 and a2 are relatively prime, then gcd (a1 a2, b) gcd (a1, b) gcd (a2, b)

    the gcd is a commutative function: gcd (a, b) = gcd (b, a)

    the gcd is an associative function: gcd (a, gcd (b, c)) = gcd (gcd (a, b) , c)

    Fundamental Theorem of Artihmetic (Unique Prime Factorization Theorem): any integer greater than 1 can

    be written as a unique product (up to ordering of the factors) of prime numbers

    there are infinitely many prime numbers

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    Mihailescus Theorem (Catalans Conjecture): the only solution in the natural numbers ofxayb = 1 forx, a, y,is x = 3, a = 2, y = 2, b = 3

    Combinatorics

    Artihmetic Series: constant adding or subtracting

    Geometric Series: constant multiplying or dividing

    an = a1 + r(n1) (nth term of arithmetic series)

    Sn =n

    2(a1 + an) (sum of first n terms of arithmetic series)

    an = a1rn1 (nth term of geometric series)

    Sn =a1 (1 rn)

    1 r (sum of first n terms of geometric series)

    S =a1

    1

    r, |r| < 1 (infinite sum of geometric series)

    Factorials:

    n! =n

    1

    (n > 0)

    0! = 1

    Permutation: n things ordered in the permutation number of ways taking r at a time

    nPr =n!

    (n r)!(1

    r

    n)

    Combination: n things grouped in the combination number of ways taking r at a time

    nCr =n

    r

    =

    n!

    r! (n r)! (0 r n)

    Binomial Theorem:

    faster than Pascals triangle and distribution after (a + b)3

    the series contains n + 1 terms

    once the combinations start to repeat, just go back down the pattern

    if a or b is a variable with a nonzero coefficient, both the variable and the coefficient must be raised to the

    appropiate power

    to find a single term, realize that the sum of the exponents of a and b is equal to n

    (a + b)n =n

    r=0

    nCr anrbr (Compact Formal)

    (a + b)n = nC0 anb0 + nC1 a

    n1b1 + nC2 an2b2

    + nCn1 a1bn1 + nCn a0bn (Formal)

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    (a + b)n = an + n an1b + nC2 an2b2 + n abn1 + bn (Simplified)

    nCr =(nCr1) (n r)

    r1 (coefficient of the rterm using the previous term)

    Probability

    Disjoint: cant occur togetherConditional: if A occurs then P (B)

    P (A and B) = P (A) P (B) (independent) P (A or B) = P (A) + P (B) (disjoint)

    P (B |A) = P (A and B)P (A)

    (conditional)

    P (A and B) = P (A) P (B |A) (general)

    P (A or B) = P (A) + P (B)P (A and B) (general) P (x = k) =

    n

    k

    Pk(1P)nk (binomial)

    Statistics

    Median: in a set with an odd number of values, the value exactly in the middle of the set ordered from smallest to

    largest; with an even number of values, the average of the two in the middle

    Mode: the most frequently occuring value or values in a set

    Range:

    Interquartile Range (IQR):

    A = x =1

    N

    N

    i=1

    xi (arithmetic mean)

    G =

    n

    i=1

    ai

    1n

    = n

    a1a2 . . .an (geometric mean)

    H =1

    1n ni=1 1ai

    =n

    1a1 + 1a2 + + 1an(harmonic mean)

    H G A (relation of means) the means are only equal when every element of the data set are equal

    =

    1

    N

    N

    i=1

    (xi x)2 (population standard deviation)

    =

    1

    n

    1

    n

    i=1

    (xi x)2 (sample standard deviation)

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    Conics

    Conic Section: a geometric figure created by the intersection of a plane and a hollow double napped cone

    Locus: the set of points sharing a given property

    Focus:

    Directrix:

    Focal Parameter: the distance between a focus and the nearest directrix

    Latus Rectum:

    Eccentricity (e):

    Linear Eccentricity: the distance between the center and one of the foci

    e =c

    a=

    [distance between foci]

    [distance between vertices](eccentricity)

    e = 0 (circle)

    0 < e < 1 (ellipse)

    e = 1 (parabola)

    e > 1 (hyperbola)

    r=ke

    1 + e cos(polar conic)

    Discriminant Test: Ax2 +Bxy +Cy2 +Dx +Ey + F = 0 (general quadratic curve)

    B24AC< 0 (ellipse) B24AC= 0 (parabola) B24AC> 0 (hyperbola)

    Degenerate Cases:

    No Real Graph: circle or ellipse with negative right hand side

    Point: circle or ellipse with right hand side equal to zero

    Single Line: neither variable squared or only one variable present and equal to zero

    Parallel Lines: one variable squared and the other absent and the right hand side positive

    Intersecting Lines: hyperbola with a negative right hand side

    Rotation of Axes:

    x = x cosy sin y = x siny cos

    Angle of Rotation:

    tan (2) =B

    A

    C()

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    Parabola: the locus of points in a plane where each point is equidistant from a focus and a directrix

    (xh)2 = 4p (yk) (parabola standard form; opens up)

    (h, p + k) (focus)

    y = kp (directrix) (h, k) (vertex)

    (xh)2 =4p (y k) (parabola standard form; opens down)

    (h, kp) (focus) y = k+ p (directrix)

    (h, k) (vertex)

    (y k)2 = 4p (xh) (parabola standard form; opens to the right)

    (p + h, k) (focus)

    y = hp (directrix) (h, k) (vertex)

    (y k)2 =4p (xh) (parabola standard form; opens to the left)

    (hp, k) (focus) y = h + p (directrix)

    (h, k) (vertex)

    Circle: the locus of points in a plane that are equidistance from a focus

    (xh)2 + (y k)2 = a2 (circle standard form)

    (h, k) (center)

    a (radius)

    Ellipse: the locus of points in a plane where the sum of the distances between two focii are equal to a constant

    (xh)2

    a2+

    (yk)2b2

    = 1 (ellipse standard form; foci on the x-axis)

    a = semimajor axis b = semiminor axis c =

    a2b2 (center to focus distance)

    (hc, k) (foci) (ha, k) (vertices) (h, k) (center)

    (xh)2

    b2+

    (yk)2a2

    = 1 (ellipse standard form; foci on the y-axis)

    a = semimajor axis b = semiminor axis

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    Negation:

    sin (x) = sin (x)

    cos (x) = cos (x)

    tan (

    x) =

    tan (x)

    csc (x) = csc (x)

    sec (x) = sec (x)

    cot (

    x) =

    cot (x)

    Reciprocal:

    sin (x) =1

    csc (x)

    cos (x) =1

    sec (x)

    tan (x) =1

    cot (x) =sin (x)

    cos (x)

    Periodicity:

    sin

    x +

    2

    = cos (x)

    cos

    x +

    2

    =sin (x)

    tan

    x +

    2

    =cot (x)

    csc

    x + 2

    = sec (x)

    sec

    x +

    2

    =csc (x)

    cot

    x +

    2

    = tan (x)

    sin (x +) = sin (x)

    cos (x +) = cos (x)

    tan (x +) = tan (x)

    csc (x +) =

    csc (x)

    sec (x +) = sec (x)

    cot (x +) = cot (x)

    sin (x + 2) = sin (x)

    cos (x + 2) = cos (x)

    tan (x + 2) = tan (x)

    csc (x + 2) = csc (x)

    sec (x + 2) = sec (x)

    cot (x + 2) = cot (x)

    Cofunction:

    sin

    2x

    = cos (x)

    tan2x= cot (x)

    sec

    2x

    = csc (x)

    cos

    2x

    = sin (x)

    cot2x= tan (x)

    csc

    2x

    = sec (x)

    Addition and Subtraction:

    sin (x +y) = sin (x) cos (y) + cos (x) sin (y)

    sin (xy) = sin (x) cos (y)cos (x) sin (y) cos (x +y) = cos (x) cos (y)

    sin (x) sin (y)

    cos (xy) = cos (x) cos (y) + sin (x) sin (y)

    tan (x +y) =tan (x) + tan (y)

    1 tan (x) tan (y)

    tan (xy) = tan (x) tan (y)1 + tan (x) tan (y)

    Double Angle:

    sin (2x) = 2sin (x) cos (x)

    tan (2x) =2tan (x)

    1 tan2 (x) cos (2x) = cos2 (x) sin2 (x) = 2cos2 (x)1 = 12sin2 (x)

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    Half Angle:

    sinx

    2

    =

    1 cosx

    2

    cosx

    2

    =

    1 + cosx

    2

    tan

    x2

    = 1 cosx

    sinx= sinx

    1 + cosx

    Reducing Powers:

    sin2 (x) =1cos (2x)

    2

    cos2 (x) =1 + cos (2x)

    2

    tan2 (x) =1

    cos(

    2x)1 + cos (2x)

    Product to Sum:

    sin (x) cos (y) =1

    2[sin (x +y) + sin (xy)]

    cos (x) sin (y) =1

    2[sin (x +y) sin (xy)]

    cos (x) cos (y) =1

    2[cos (x +y) + cos (xy)]

    sin (x) sin (y) =1

    2[cos (xy) cos (x +y)]

    Sum to Product:

    sinx + siny = 2sin

    x +y

    2

    cos

    xy

    2

    sinx siny = 2cos

    x +y

    2

    sin

    xy

    2

    cosx + cosy = 2cosx +y2cosxy

    2

    cosxcosy =2sin

    x +y

    2

    sin

    xy

    2