Notes on Basic Math by Serge Lang Contents Part I Algebra Chapter 1 Numbers The integers Rules for addition Rules for multiplication Even and odd integers; divisibility Rational numbers Multiplicative inverses Chapter 2 Linear Equations Equations in two unknowns Equations in three unknowns Chapter 3 Real Numbers Addition and multiplication Real numbers: positivity Powers and roots Inequalities Chapter 4 Quadratic Equations Interlude On Logic and Mathematical Expressions On reading books Logic Sets and elements Notation Part II Intuitive Geometry Chapter 5 Distance and Angles Distance Angles The Pythagorean theorem Chapter 6 Isometries Some standard mappings of the plane Isometries Composition of Isometries Congruences Chapter 7 Area and Application Area of a disc of radius r Circumference of a circle of radius r Part III Coordinate Geometry Chapter 8 Coordinates and Geometry Coordinate systems Distance between points Equations of a circle Rational points on a circle Chapter 9 Operations on Points Dilations and reflections Addition, subtraction, and the parallelogram law Chapter 10 Segments, Rays, and Lines Segments Rays Lines Ordinary equation for a line Chapter 11 Trigonometry Radian measure Sine and cosine The graphs The tangent Addition Formulas Rotations Chapter 12 Some Analytic Geometry The straight line again The parabola The ellipse The hyperbola Rotation of hyperbolas Part IV Miscellaneous Chapter 13 Functions Definition of a function Polynomial functions Graphs of functions Exponential function Logarithms Chapter 14 Mappings Definition Formalism of mappings Permutations Chapter 15 Complex Numbers The complex plane Polar form Chapter 16 Induction and Summations Induction Summation Geometric Series Chapter 17 Determinants Matrices Determinants of order 2 Properties of 2 by 2 determinants Determinants of order 3 Properties of 3 by 3 determinants Cramer's Rule Numbers The Integers (Z. *. 0 <) n means n is a positive integer e.g. 1 2 3 4 5 6 7 8 9 10 11 0 = n means n is zero N. n means n is a natural number i.e. zero or positive integer natural number line with origin labeled 0 (Z. *. 0 >) n means n is a negative integer e.g. _1 _2 _3 _4 _5 _6 .. Z. n means n is an integer (zero, positive integer, negative integer) integer number line as iterated measurement from 0 addition as iterated motion on the number line (Z. n) implies (n = n + 0) and n = 0 + n n - ~ as (- n) + subtraction as adding a negative (Z. n) implies (0 = n + - n) and 0 = (- n) + n n and - n are on opposite sides of 0 on the standard number line read - n as "minus n" or "the additive inverse of n" Rules For Addition (n + m) = m + n commutative ((n + m) + k)=n + m + k associative 0 = n + - n right inverse 0 = (- n) + n left inverse n = - - n idempotent (- n + m) = (- n) - m negation distributes over addition (*. / 0 < n) implies 0 < + / n positive additivity (*. / 0 > n) implies 0 > + / n negative additivity (n = m + k) implies m = n - k left solvable (n = m + k) implies k = n - m right solvable ((n + m) = n + k) implies m = k cancelation rule (n = n + m) implies m = 0 unique right identity (n = m + n) implies m = 0 unique left identity Rules For Multiplication (n * m) = m * n commutative ((n * m) * k) = n * m * k associative n = 1 * n identity 0 = 0 * n annihilator (n * (m + k)) = (n * m) + n * k left-distributive ((n + m) * k) = (n * k) + m * k right-distributive (- n) = _1 * n minus is multiplication by negative one (- n * m)=(- n) * m minus permutes over multiplication (- n * m) = n * - m minus permutes over multiplication (n * m) = (- n) * - m (n ^ k) = * / k #: n exponentiation is iterated multiplication (n ^ m + k) = (n ^ m) * n ^ k (* / n ^ m) = n ^ + / m (n ^ m ^ k) = n ^ m * k (n ^ * / m) = ^ / n , m ((n + m) ^ 2) = (n ^ 2) + (2 * n * m) + m ^ 2 (*: n + m) = (*: n) + (+: n * m) + *: m ((n - m) ^ 2) = (n ^ 2) - (2 * n * m) + m ^ 2 (*: n - m) = (*: n) - (+: n * m) + *: m ((n + m) * n - m) = (n ^ 2) - m ^ 2 ((n + m) * n - m) =(*: n) - *: m n ((+ * -) = (*: [) - (*: ])) m Even And Odd Integers; Divisibility odd integers: 1 3 5 7 9 11 13 .. even integers: 2 4 6 8 10 12 14 .. 'n is even' means n = 2 * m for some m with Z. m 'n is odd' means n = 1 + 2 * m for some m with Z. m if E means even and I means odd then E = E + E and E = I + I I = E + I and I = I + E E = E * E and I = I * I E = I * E and E = E * I E = E ^ 2 and I = I ^ 2 1 = _1 ^ E and _1 = _1 ^ I n (-. |) m means "n divides m" if n = m * k for some integer k n (-. |) n and 1 (-. |) n "a is congruent to b modulo d" if a - b is divisible by d if (a - b) | d and (x - y) | d then ((a + x) - b + y) | d if (a - b) | d and (x - y) | d then ((a * x) - b * y) | d Rational Numbers fractions: mrn with m , n integer numerals and -. n = 0 e.g. 0r1 _2r3 3r4 ... dividing by zero does not give meaningful information rational number line (m % n) = s % t if *. / (-. 0 = n , t) , (m * t) = n * s m = m % 1 (-. 0 = a , n) implies (m % n) = (a * m) % a * n cancellation rule (- m % n) = (- m) % n (- m % n) = m % - n (*. / (Q. r) , 0 < r) iff *. / (r = n % m) , (Z. , 0 <) n , m "d is a common divisor of a and b" if d divides both a and b the lowest form of a is mrn where 1 is the only common divisor of m and n every positive rational has a lowest form if -. n = 1 and the only common divisor of m and n is 1 then mrn = m % n ((a % d) + b % d) = (a + b) % d ((m % n) + a % b) = ((m * b) + a * n) % n * b (0 = 0 % 1) and 0 = 0 % n (a = 0 + a) and a = a + 0 negative rational numbers have the form _mrn _mrn = - mrn and mrn = - _mrn rational addition is commutative and associative ((m % n) * a % b) = (m * a) % n * b ((m % n) ^ k) = (m ^ k) % n ^ k (Q. r) <: -. 2 = r ^ 2 a real number that is not rational is called irrational rational * is associative, commutative, and distributes over + (Q. r) <: (a = 1 * a) *. 0 = 0 * a ! = (* / 1 + i.) i.e. (! n) = 1 * 2 * 3 * ... * n ! = ] * (! <:) i.e. (! 1 + n) = (1 + n) * ! n (n ! m) = (! n + m) % (! n) * ! m binomial coefficients (n ! m) = ((! + /) % (* / !)) n , m multinomial coefficients (n ! m) = m ! n (n ! m + 1) = (n ! m) + (n - 1) ! m decimals Multiplicative Inverses (*. / (Q. a) , -. a = 0) implies *. / (Q. b) , 1 = a * b "b is a multiplicative inverse of a" if *. / 1 = a (* ~ , *) b (b = c) if *. / (-. 0 = a) , 1 = (a * b) , (b * a) , (a * c) , c * a (-. 0 = a) implies *. / (1 = a * % a) , 1 = (% a) * a (-. 0 = a =: n % m) implies *. / ((% a) = m % n) , (% a) = (n % m) ^ _1 (1 = a * b) implies b = a ^ _1 (0 = a * b) implies +. / 0 = a , b ((a % b) = c % d) if *. / (-. 0 = b , d) , (a * d) = b * c (b = c) if *. / (-. 0 = a) , (a * b) = a * c times cancellation law (*. / -. 0 = b , c) implies ((a * b) % a * c) = b % c quotient cancellation law ((a % b) + c % d) = ((a * d) + b * c) % b * d ((x ^ n) - 1) % x - 1) = (x ^ n - 1) + (x ^ n - 2) + ... + x + 1 if n is odd then (((x ^ n) + 1) % x + 1) = - ` + / x ^ n - 1 + i. n Linear Equations Equations In Two Unknowns assuming c = (a * x) + b * y and u = (v * x) + w * y yields x = ((w * c) - u * b) % (w * a) - v * b y = ((v * c) - w * u) % (v * b) - w * a elimination method: common multiples Equations In Three Unknowns iterate elimination method Real Numbers Addition And Multiplication the real number line addition of real numbers is commutative, associative, a = 0 + a , 0 = a + - a (0 = a + b) implies b = - a unique additive inverse * is commutative,associative,distributes over +, a = 1 * a, 0 = 0 * a ((a + b) ^ 2) = (a ^ 2) + (2 * a * b) + b ^ 2 ((a - b) ^ 2) = (a ^ 2) - (2 * a * b) + b ^ 2 ((a + b) * a - b) = (a ^ 2) - b ^ 2 every nonzero real number has a unique multiplicative inverse the E , I system satisfies the addition and multiplication properties Real Numbers: Positivity positivity as being on a side of 0 on the number line a > 0 means "a is positive" (*. / 0 < a , b) implies *. / 0 < (a * b) , a + b (*. / 0 < a) implies (*. / 0 < * / , + /) a ~: / (0 = a) , (0 < a) , 0 > - a a < 0 means -. *. / (0 = a) , (- a) > 0 "a is negative" means a<0 (a < 0) iff 0 < - a (0 < 1) and 0 > _1 every positive integer is positive (0 > a * b) if (0 < a) and 0 > b (0 > a * b) if (0 > a) and 0 < b (0 < a) implies 0 < 1 % a (0 > a) implies 0 > 1 % a assume completeness: (a > 0) implies *. / (0 < %: a) , a = (%: a) ^ 2 "the square root of a" means %: a an irrational number is a real number that is not rational e.g. %: 2 Assuming *. / a = *: b , x yields 0 = - / *: b , x 0 = x (+ * -) b +. / 0 = x (+ , -) b +. / x = (- , ]) b ((x ^ 2) = y ^ 2) implies (x = y) or x = - y (| x) = %: *: x absolute value (% (%: x + h) + %: x) = ((%: x + h) - %: x) % h rationalize 0 < a ^ 2 (%: a % b) = (%: a) % %: b alternatively ((%: % /) = (% / %:)) a , b (*. / (Q. x , y , z , w) , (N. *. 0 <) n) implies ( *. / (Q. c , d) , (c + (d * %: n)) = (x + y * %: n) * z + w * %: n (| a - b) = | b - a Powers And Roots assume *. / (0 < a) , (N. , 0 < ) n implies a = (n %: a) ^ n for a unique n %: a "the nth-root of a" means n %: a (a ^ 1 % n) = n %: a (0 < a , b) implies ((n %: a) * n %: b) = n %: a * b fractional powers: *. / (Q. x) , 0 < a implies there exists a ^ x such that ((a ^ x) = a ^ n) if x = n ((a ^ x) = n %: a) if x = 1 % n (a ^ x + y) = (a ^ x) * a ^ y (a ^ x * y) = (a ^ x) ^ y ((a * b) ^ x) = (a ^ x) * b ^ x *. / (1 = a ^ 0) , 1 = * / #: 0 (a ^ - x) = 1 % a ^ x (a ^ m % n) = (a ^ m) ^ 1 % n (a ^ m % n) = (a ^ 1 % n) ^ m Inequalities a < b means 0 < b - a a < 0 means 0 < - a a < b means b > a inequalities on the numberline a <: b means a < b or a = b a >: b means a > b or a = b (*. / (a < b) , b < c) implies a < c (*. / (a < b) , 0 < c) implies (a * c) < b * c (*. / (a < b) , c < 0) implies (b * c) < a * c x is in the open interval a , b if (a < *. b >) x x is in the closed interval a,b if (a <: *. b >:) x x is in a clopen interval a,b if +. / ((a < *. b >:) , (a <: *. b >)) x (a <),(a <:) , (a >) , a >: infinite intervals intervals and the numberline (*. / (0 < a) , (a < b) , (0 < c) , c < d) implies (a * c) < b * d (*. / (a < b) , (b < 0) , (c < d) , d < 0) implies (a * c) > b * d (*. / (0 < x) , x < y) implies (1 % y) < 1 % x (*. / (0 < b) , (0 < d) , (a % b) < c % d) implies (a * d) < b * c (a < c) implies ((a + c) < b + c) and (a - c) < b - c (*. / (0 < a) , a < b) implies (a ^ n) < b ^ n (*. / (0 < a) , a < b) implies (a ^ 1 % n) < b ^ 1 % n (*. / (0 < b , d) , (a % b) < c % d) implies ((a % b) < (a + c) % b + d) (*. / (0 < b , d) , (a % b) < c % d) implies ((a + c) % b + d) < c % d) (*. / (0 < b , d , r) , (a % b) < c % d) implies ( (a % b) < (a + r * c) % b + r * d) (*. / (0 < b , d , r) , (a % b) < c % d) implies ( ((a + r * c) % b + r * d) < c % d) (*. / (0 < b , d , r) , (r < s) , (a % b) < c % d) implies ( ((a + r * c) % b + r * d) < (a + s * c) % b + s * d) Quadratic Equations ((*. / (-. a = 0) , (0 = (a * x ^ 2) + (b * x) + c) , (0 <: (b ^ 2) - 4 * a * c)) implies +. / (x = (- b + %: (b ^ 2) - 4 * a * c) % 2 * a) , (x = (- b - %: (b ^ 2) - 4 * a * c) % 2 * a)) (0 > (b ^ 2) - 4 * a * c) implies -. *. / (R. x) , 0 = (a * x ^ 2) + (b * x) + c On Logic And Mathematical Expressions Logic proof as list of statements each either assumed or derived from a deduction rule converse: the converse of "if A, then B" is "if B, then A" "A iff B" means "if A, then B" and "if B, then A" proof by contradiction: take A false, derive a contradiction, conclude A true equations are not complete sentences logical equivalence as A iff B Sets And Elements set: a collection of objects element: an object in a set subset: s0 is a subset of s1 if every element of s0 is an element of s1 empty set: a set that does not have any elements set equality: s0 equals s1 if s0 is a subset of s1 and s1 is a subset of s0. Indices "let x,y be something" includes the possibility that x=y "let x,y be distinct somethings" excludes the possibility that x=y x0 x1 x2 x3 .. xn is a finite sequence Distance And Angles Distances assume p0 d p1 gives the distance between the points p0 , p1 assume that for any points p0,p1,p2 0 <: p0 d p1 nonnegative (0 = p0 d p1) iff p0 = p1 nondegenerate (p0 d p1) = p1 d p0 symmetric (p0 d p1) <: (p0 d p2) + p2 d p1 triangle inequality note the geometric meaning of the triangle inequality the length of a side of a triangle is at most the sum of the others assume that two distinct points lie on one and only one line (-. p0 = p1) implies *. / (p0 p1 i p0 , p1), (*. / p2 p3 i p0 , p1) implies p2 p3 i = p0 p1 i define betweenness as equality case of the triangle inequality (p0 p1 B p2) iff (p0 d p1) = (p0 d p2) + p1 d p2 define segment as the points between a pair of endpoints (p0 p1 W p2) iff p0 p1 B p2 (by definition of B we have p0 p1 i p2) assume the length of a segment is the distance between its endpoints (mW p0 p1) = p0 d p1 assume rulers pick out unique points (*./(0<:a),a<:p0 d p1) implies *./(p0 p1 W p2),a=p0 d p2 for some p2 ((*./(p0 p1 W),(= p0 d))p2,p3) implies p2=p3 define circle as the points equidistant from a common point (p0 p1 o p2) if (p0 d p1)=p0 d p2 geometric circle from metric define (p0 r bdB) as the circle with center p0 and radius r (p0 r bdB p1) if r=p0 d p1 metric circle as boundary of a ball prove two points uniquely define a circles (p0 p1 o p2) implies (p0 p1 o = p0 p2) prove a point and radius uniquely define a circle (p0 r bdB p1) implies (p0 r bdB p2) iff p0 p1 o p2 define (p0 r clB p1) as the disc with center p0 and radius r (p0 r clB p1) if r>:p0 d p1 Angles assume distinct points lie on a unique line (-.p0=p1) implies *./(p0 p1 i p0,p1), (*./p2 p3 i p0,p1) implies (p2 p3 i = p0 p1 i) assume a pair of nonparallel lines share a unique point (-.p0 p1 p2 H p3) implies (p0 p1 i *. p2 p3 i)p4 for some p4 (*./(-.p0 p1 p2 H p3),(p0 p1 i *. p2 p3 i)p4,p5) implies p4=p5 assume a point belongs to a unique parallel to a line p0 p1 p2 H p2 (*./(p0 p1 p2 H p3),p2 p3 i p4) implies p0 p1 p2 H p4 (*./p0 p1 p2 H p3,p4) implies (p2 p3 i = p2 p4 i) assume "parallel to" is an equivalence relation p0 p1 p0 H p1 (p0 p1 p2 H p3) implies p2 p3 p0 H p1 (*./(p0 p1 p2 H p3),p0 p1 p4 H p5) implies p2 p3 p4 H p5 assume a point belongs to a unique perpendicular to a line (*./(p0 p1 p2 L p3),p2 p3 i p4) implies p0 p1 p2 L p4 (*./p0 p1 p2 L p3,p4) implies (p2 p3 i = p2 p4 i) assume a parallel to a perpendicular is perpendicular (*./(p0 p1 p2 L p3),p2 p3 p4 H p5) implies p0 p1 p4 L p5 assume a perpendicular to a perpendicular is parallel (*./(p0 p1 p2 L p3),p2 p3 p4 L p5) implies p0 p1 p4 H p5 define a halfline as points on the same side of a line relative to a vertex (p0 p1 R p2) if (p2 B p0 p1)+.p1 B p0 p2 assume a halfline is determined by its vertex and any other point on it ((p0 p1 R p2)*.-.p0=p2) implies p0 p1 R = p0 p2 R define (p0 p1 R) as the halfline with vertex p0 to which p1 is incident assume a pair of distinct points determine two distinct rays (-.p0=p1)<:p0 p1 R (-.=) p1 p0 R assume a point on a line divides it into two distinct halflines (p0 p1 i p2)<: (p0 p1 R p2)+.(p0 p1 i p3) implies (p0 p1 R p3)+.p0 p2 R p3 assume two distinct halflines sharing a vertex separate the plane into two parts define angle as one of the parts of the plane separated by such halflines assume two points on a circle divide it into two distinct arcs note Lang uses counterclockwise oriented angles rather than neutral angles assume p0 p1 p2 c is the counterclockwise arc of (p1 p0 o) from p0 to (p1 p2 R) define (p0 p1 p2 V) as the angle from p1 p0 R to p1 p2 R containing p0 p1 p2 c define the vertex of (p0 p1 p2 V) as p1 define (p0 p1 p2 V) is a zero angle as (p1 p0 R = p1 p2 R) define (p0 p1 p2 V) is a full angle as (p2 p1 p0 V) is a zero angle note special notation to distinguish a full angle from a zero angle define (p0 p1 p2 V) is a straight angle as (p0 p1 i p2) prove if (p0 p1 p2 V) is a straight angle then so is (p2 p1 p0V) define (p0 p1 p2 r clBV) as the sector of (p1 r clB) determined by (p0 p1 p2 V) (p0 p1 p2 r clBV p3) if (p1 r clB p3)*.(p0 p1 p2 V p3) define mclB p0 r as the measure of the area of (p0 r clB) define mclBV p0 p1 p2 r as the the measure of the area of (p0 p1 p2 r clBV) define (mV p0 p1 p2) using the ratio (mclBV p0 p1 p2 r) to mclB p1 r (mV p0 p1 p2)=x deg if *./(0<:x),(x<:360),((mclBV p0 p1 p2 r)%mclB p0 r)=x%360 define "x deg" as "x degrees" prove the measure of a full angle is 360 deg (p0 p1 R p2) implies (360 deg)= mV p2 p1 p0 prove the measure of a zero angle is 0 deg prove the measure of a straight angle is 180 deg define a right angle as one whose measure is half a straight angle i.e. 90 deg (p0 p1 p2 V) is right iff 90=mV p0 p1 p2 assume the area of a disc of radius r is pi*r^2 where pi is near 3.14159 prove that the measure of an angle is independent of r Pythagorean Theorem define p W p0 as +. / 2 (p0 W ~) \ p define noncolinear points p0,p1,p2 as -. p0 p1 i p2 define triangle as segments between three points (p0 p1 p2 A p3) if p0 p1 p2 p0 W p3 define the triangle with vertices p0 , p1 , p2 as (p0 p1 p2 A) define the sides of (p0 p1 p2 A) as (p0 p1 W), (p1 p2 W), and (p2 p0 W) define triangular region as the points bounded by and having a triangle define area of a triangle as area of a triangular region define mA p0 p1 p2 as the measure of the area of (p0 p1 p2 A) note triangular regions are also called simplexes note pairs of sides of a triangle determine angles define a right triangle as one having a right angle (p0 p1 p1 p2 Z p3) if *./ (p0 p1 p2 A p3) , 90 = mV p1 p2 p0 define the legs of a right triangle as the sides of its right angle define the hypotenuse of a right triangle as the non-leg side assume right triangles with corresponding legs of equal length are congruent (*./(p0 p1 p2 Z),(p3 p4 p5 Z),((p1 d p2)=p4 d p5),(p2 d p0)=p5 d p3) implies *./((mV p0 p1 p2)=mV p3 p4 p5),((mV p1 p0 p2)=mV p4 p3 p5), ((p0 d p1)=p3 d p4),(mA p0 p1 p2)=mA p3 p4 p5 assume parallels perpendicular to parallels cut corresponding segments equally (*./(p0 p1 p2 H p3),(p0 p1 p0 L p2),p0 p1 p1 L p3) implies *./((p0 d p1)=p2 d p3),(p1 d p2)= p3 d p0 define (0=mH p0 p1 p2 p3) if -.(p0 p1 p2 H p3) define ((p0 d p1)=mH p2 p0 p3 p1) if p2 p0 p3 H p1 prove the distance between parallel lines is unique (*./(p0 p1 p2 H p3, p4)(p2 p3 p3 L p5)(p0 p1 i p5,p6)p2 p4 p4 L p6)<:(p3 d p5)=p4 d p6 define rectangle as four sides: opposites parallel and adjacents perpendicular (p0 p1 p2 p3 Z p4) if *. / (p0 p1 p2 H p3) , (p1 p2 p3 H p0) , (p0 p1 p1 L p2) , (p1 p2 p2 L p3) , (p2 p3 p3 L p0) , (p3 p0 p0 L p1) , p0 p1 p2 p3 p0 W p4 define (p0 p1 p2 p3 Z) as a rectangle with vertices p0 p1 p2 p3 prove the opposite sides of a rectangle have the same length note area of a rectangle means area of region bounded and containing a rectangle define (mZ p0 p1 p2 p3) as area of (p0 p1 p2 p3 Z) define a square as a rectangle all of whose sides have the same length prove the area of a square with side length a is a ^ 2 prove that (p0 p0 p1 p2 Z) uniquely determines (p3 p0 p1 p2 Z) prove the sum of the non-right angles in a right triangle is 90 deg (p0 p0 p1 p2 Z) implies 90 = (mV p1 p0 p2) + mV p1 p2 p0 prove the sum of the angles in a right triangle is 180 deg (p0 p0 p1 p2 Z) implies 180 = (mV p0 p1 p2) + (mV p1 p2 p0) + mV p2 p0 p1 prove the area of a right triangle with leg lengths a,b is -: a * b prove the Pythagorean theorem (p0 p1 p1 L p2) implies (*: p0 d p2) = + / *: (p0 d p1) , (p1 d p2) prove a triangle is right iff it satisfies the pythagorean theorem define the diagonals of (p0 p1 p2 p3 Z) as (p0 p2 W) and p1 p3 W prove the lengths of the diagonals of a rectangle (and square) are the same prove the length of the diagonal of a square with side length 1 is %: 2 prove a right triangle with legs of length 3,4 has hypotenuse of length 5 define perpendicular bisector as line perpendicular to segment through midpoint (p0 p1 t p2) if ((-: p0 d p1) = p0 d p3) implies +. / (p2 = p3) , p0 p3 p3 L p2 prove (p0 p1 t p2) iff (p0 d p2) = p1 d p2 prove the *: of the diagonal of a rectangular solid is + / *: of its sides prove the area of a triangle with base length b and height h is -: b * h prove the hypotenuse of a right triangle is greater than or equal to a leg prove (*. / (p0 p1 p2 L p3) , (p0 p1 i p3 , p4)) implies (p2 d p3) <: p2 d p4 prove opposite interior angles are the same prove corresponding angles are the same prove opposite angles are the same prove the perpendicular bisectors of the sides of a triangle meet at a point Isometries Some Standard Mappings Of The Plane define p0 is mapped to p1 as (p0 ; p1) note map is similar in meaning to association,function,verb,arrow define map of the plane as associating each point of the plane with another define the value of M0 at p0 or the image of p0 under M0 as (M0 p0) define M0 maps p0 onto p1 as p1 = M0 p2 define (M0 = M1) as (M0 p0) = M1 p0 for all p0 define the p0 constant map as (p0 Mp) p0 = p0 Mp p1 note (p0 [) is the constant map p0 = (p0 [ p1) define the identity map as ] p0 = ] p0 note ] is the identity map p0 = ] p0 define the reflection map about (p0 p1 i) as (p0 p1 Mt) p0 = p1 p2 Mt p3 if (p1 p2 i p4) iff p0 p3 t p4 define the reflection map about p0 as Mm (p0 = p1 Mm p2) if p0 p2 m p1 define the dilation about p0 of p1 to p2 as (p0 p1 p2 MH) (p0 = p1 p2 p3 MH p4) if (*. / (p3 p1 p1 L p5,p6)(p1 p2 o p5)(p1 p3 o p6))<:(p3 p5 p6 H p4)*.p0 p3 i p4 define dilation by r0 about p0 as (p0 r0 IH) (p0 = p1 r0 IH p2) if (p1 d p2)=r*p1 d p0 define the counterclockwise rotation about p1 by (p0 p1 p2 V) as (p0 p1 p2 Mo) (p0 = p1 p2 p3 Mo p4) if (*./(p2 p4 o p5)(p2 p1 i p5)(p2 p3 i p6)(p2 p6 p6 L p1)p5 p6 Ed p4 p7)<: (p2 p4 o p0)*.p2 p7 i p0 note the rotation map defined assumes acute angles define the counterclockwise rotation about p0 by r0 degrees as (p0 r0 Io) (p0 = p1 r0 Io p2) if *./(0<:r0)(r0<:360)r0=mV p2 p1 p0 note 0<:r0 implies (p0 r0 Io) is c.c. and r0<:0 implies (p0 r0 Io) is clockwise prove p0 180 Io = p0 Mm prove p0 180 Io = p0 _180 Io prove (p0 0 Io = ]) prove (p0 360 Io = ]) note rotation by 0 or 360 degrees is the identity transformation define (p0 r0 oV) as (p0 r1 oV) with *./(0<:r1),(r1<360),r0=r1+360*n for some n prove rotation by a negative angle is rotation by a positive angle define the arrow from p0 to p1 as a0 =: p0 ; p1 ((p0 S a0) *. p0 T a0) if a0 = p0 ; p1 define p0 is an object of a0 if p0 S a0 or p0 T a0 (p0 O a0) if (p0 S a0) +. p0 T a0 note, in general, a0;a1 is an arrow with objects a0,a1, source a0 and target a1 *. / ((a0 , a1) O a0 ; a1) , (a0 S a0 ; a1) , a1 T a0 ; a1 define p0 p1 W as the directed line segment associated with the arrow p0;p1 (p0 p1 W = p1 p0 W) iff p0 = p1 define translation by (p0 p1 W) as (p0 p1 MW) (p0 = p1 p2 MW p3) if ((p1 p3 p3 L p0) *. p1 p3 p2 H p0) +. *. / ((p1 p2 i p3)(-.p1 p2 i p4)(p1 p4 p2 H p5)p4 p5 p1 H p2)<:p0=p4 p5 MW p3 define p0 is a fixed point of M0 if p0 = M p0 prove that every point is a fixed point of ] prove that p0 is the only fixed point of p0 Mp prove p0 is the only fixed point of p0 Mm prove p0 is a fixed point of (p1 p2 Mt) iff (p1 p2 i p0) prove (-. 0 = mV p0 p1 p2) implies p1 is the only fixed point of p0 p1 p2 MV prove (-. 0 = r0) implies p0 is the only fixed point of p0 r0 IV prove that (-. p0 = p1) implies (p0 p1 MW) has no fixed points prove if -. 1 = r0 implies p0 is the only fixed point of p0 r0 IH prove every point is a fixed point of p0 1 IH Isometries define M0 is an isometry if it preserves distance: (d=d I0) (p0 d p1) = (I0 p0) d I0 p1 prove isometries map distinct points to distinct points (-. p0 = p1) implies -. (M0 p0) = M0 p1 define y is in the image of A under M0 if y = M0 x for some x in A assume point and line reflects, rotations, and translations are isometries prove isometries of points are points prove isometries of line segments are line segments prove isometries of lines are lines prove isometries of circles are circles prove isometries of discs are discs prove isometries of circular arcs are circular arcs prove if -. p0 = p1 fixed points of an isometry then so are points on p0 p1 i prove an isometry wit three fixed points is the identity prove (p0 1 IH) and (p0 _1 IH) are isometries (the only of the family IH) prove isometries of parallel lines are parallel prove isometries of perpendiculars are perpendicular note isometries in 3 space Composition of isometries define the composition of M0 with M1, M1 followed by M0, as (M1 M0) (p0 = (M0 M1) p1) if (p2 = M1 p1) implies p0 = M0 p2 prove if M0 is an isometry then M0 = (] M0) and M0 = (M0 ]) prove the composition of two (p0 180 Io) is ] prove the composition of isometries is an isometry prove the composition of rotations about a point is a rotation about that point p0 (r0 + r1) Io = (p0 r1 Io p0 r0 Io) prove that the composition of translations is a translation p0 p2 MW = (p1 p2 MW p0 p1 MW) prove the composition of dilations about a point is a dilation about that point p0 (r0 * r1) IH = (p0 r1 IH p0 r0 IH) prove the composition of isometries is associative (arrows in general) define (M0 ^: 2) as (M0 M0) define (M0 ^: 3) as (M0 M0 M0) define (M0 ^: 1 + n) as (M0 M0^:n) define (M0 ^: 0) as ] and (M0^:1) as M0 prove MI = (p0 Mm) ^: 2 prove MI = (p0 Mm) ^: 2 * n prove (p0 Mm) = (p0 Mm) ^: 1 + 2 * n prove (M0 ^: n0 + n1) = (M0 ^: n0 M0 ^: n1) prove if M0 is a reflection through a line then (M0 ^: 2) is MI note not all isometries commute Inverse Isometries define M0 as the inverse of M1 if (] = (M0 M1)) and (] = (M1 M0)) prove the inverse of a map is unique if it has one define (M0 ^: _1) as the inverse of M0 if it has one note (y = M0 x) iff (x = (M0 ^: _1)y) or ([ = (M0 ])) = (] = ((M0 ^: _1) [)) prove reflections are their own inverses prove identity is its own inverse prove ] = (p0 p1 MW p1 p0 MW) and ] = (p1 p0 MW p0 p1 MW) prove (p0 p1 MW) and (p1 p0 MW) are inverses of each other prove ] = (p0 r0 Io p0 -r0 Io) and ] = (p0 -r0 Io p0 r0 Io) prove (p0 r0 Io) and (p0 -r0 Io) are inverses of each other (p0 -r0 Io) = (p0 r0 Io) ^: _1 prove ((M0 M1) ^: _1) = (M1 ^: _1 M0 ^: _1) define M0 ^: _n0 as (M0 ^: _1) ^: n0 prove (M0 ^: n0 + n1) = (M0 ^: n0) M0 ^: n1 prove if M0,M1 are isometries *./(M0=M1)p0,p1,p2 then (M0=M1) if M0^:_1 exists prove every isometry actually does have an inverse prove reflections about perpendicular lines commute prove M0 , M1 , M2 isometries (M0 M1) = (M0 M2) implies M1 = M2 note symmetries of the square via isometries note symmetries of the triangle via isometries note symmetries of the hexagon via isometries note do these isometric symmetries characterize these shapes? Characterization Of Isometries prove -. p0 = p1 fixed points of isometry M0 implies +. / (MI = M0) , p0 p1 MT = M0 prove an isometry with only one fixed point is +. / Mo , Mo MT prove an isometry without a fixed point is +. / MW , (MW Mo) , ((MW Mo) Mm) Congruences define p00,p01,..,p0n is congruent to p10,p11,..,p1m if p00,..,p0n=M0 p11,..,p0m note if one set is the image of another under an isometry then they're congruent prove circles with the same radius are congruent prove discs with the same radius are congruent prove segments with the same length are congruent prove right triangles whose corresponding legs are congruent are congruent prove triangles whose corresponding sides are congruent are congruent prove squares whose sides are congruent are congruent prove rectangles whose corresponding sides are congruent are congruent assume the area of a region is equal to the area of its image under an isometry prove congruence is an equivalence relation prove any two lines are congruent prove the sides of a triangle with angle measures 60 deg have equal length define equilateral triangle if its sides are all the same length prove SAS characterization of congruence prove AAS characterization of congruence prove inscribed circle in a triangle angle bisectors Area And Applications Area Of A Disc Of Radius r note a unit length determines a unit area assume area of a square with side length a is a^2 assume area of a rectangle with side lengths a,b is a*b prove the area of the dilation by r of a square of area a is a*r^2 assume the area of the dilation by r of a region with area a is a*r^2 define o.1 as the length of of a circle with radius 1 prove the area of the dilation by r of a disc of radius 1 is o.-:r^2 note approximate regions with squares to find their area note upper/lower bounds as areas inside and outside of figure define ellipse as nonuniform scaling of a disc prove map circle to ellipse algebraically note scaling and volume in 3-space is similar Circumference Of A Circle Of Radius r assume ((o. 1) = mbdB p0 1) and (o. r) = mbdB p0 r note approximate by dividing disc into n sectors with angles 360%n note disc area to circle length prove the length of the dilation by r of a segment of length a is r*a assume the length of the dilation by r of an arbitrary curve of length a is r*a Coordinates And Geometry Coordinate Systems define an origin as the intersection of perpendicular lines (called axis) note the classical origin is the intersection of a horizontal and vertical line note pick unit length, cut axes into segments left/right up/down note cut plane into squares with unit side lengths note label each point of intersection with a pair of integers note intersection of perpendicular lines to axes through a point gives its coordinate define the coordinate of the origin as 0,0 note meaning of the positive/negative components as motions define x-coordinate is usually the first, y-coordinate is usually the second prove the axes divide the plane into four quadrants define the positive side of the second axis as counterclockwise the first note plot points assume/prove every point corresponds to a unique pair of numbers assume/prove every pair of numbers corresponds to a unique point note points in 3-space Distance Between Points points on the number line are labeled so that algebraic definitions work simply note the distance between points in the plane is found using the pythagorean theorem prove the distance between points p0 and p1 on a number line is %:(p0-p1)^2 (*./(p0=a0,b0),p1=a1,b1) implies (p0 d p1)=%:@+/@*:(a1-a0),(b1-b0) assume distance as d=:%:@+/@*:- satisfies the required geometric properties define the plane as all pairs of real numbers with distance %:@+/@*:- prove (0 = p0 d p1) iff p0 = p1 define dilation as * i.e. (r * x , y) = (r * x) , r * y prove (0 <: r) implies (d r * x , y) = r * d x , y prove ((r * [) d r * ]) = r * d prove distance works in 3-space Equation Of A Circle assume (p0 p1 o p2) iff (p0 d p1) = p0 d p2 assume p0 r0 bdB p1 if r0 = p0 d p1 define p0 r0 bdB as the circle centered at p0 with radius r0 prove ((p0=:r0,r1) r2 bdB p1=:r3,r4) iff (*:r0)=+/*:p0-p1 prove is the equation of a circle in r3,r4 with center r1,r2 and radius r0 is (*: r0) = + / *: (r1 , r2) - r3 , r4 prove (p0 r0 bdB p1) iff (*: r0) = + / *: p0 - p1 Rational Points On A Circle prove ((*:c)=+/*:a,b) iff (1=+/*:(a,b)%c) iff 1=+/*:(x=:a%c),(y=:b%c) when -.c=0 note to solve (*:c)=+/*:a," for integers a,b,c solve 1=+/*:x,y for rationals x,y define a rational point as one whose components are rational numbers prove (*./(t=:y%1+x),(1=+/*:x,y),-._1=x) <: *./x=((1- % 1+)*:t),y=(2* %(1+*:))t prove 1=+/*:x,y rational <: *./x=(1- % 1+)*:t),y=((2*)%(1+*:))t for rational t prove distinct rationals give distinct solutions (*./(0<:s),s<t) implies </((1-)%(1+))*:s,t Operations On Points Dilations And Reflections assume (r0*r1,r2)=(r0*r1),r0*r2 prove (p0= p1 r0 IH p2) iff (p0=p1+r0*p2-p1) or (p0=(r0*p2)+(1-r0)*p1) prove (p0= p1 Mm p2) iff (p0=p1-p2-p1) or (p0=(+:p1)-p2) prove ((r0*r1)d r0*r2)=(|r0)* r1 d r2 note the n-dimensional case Addition Subtraction And The Parallelogram Law assume ((a0,a1)+b0,b1)=(a0+b0),a1+b1 prove commutativity (p0+p1)=p1+p0 prove associativity: (p0+p1+p2)=(p0+p1)+p2 prove 0,0 is an additive identity: (p0=p0+0,0) and p0=(0,0)+p0 prove additive inverses exist: ((0,0)=p0+-p0) and (0,0)=(-p0)+p0 prove the points (0,0);p0;p1;p0+p1 are vertices of a parallelogram (0,0),p0,p1,:p0+p1 W is a parallelogram prove p0=(p0-p1)+p1 prove (0,0);p0;p1;p0-p1 are vertices of a parallelogram prove (p0=p1 p2 MW p3) iff (p0=p1+(p2-p1)+p3-p1) or p0=p3+p2-p1 define norm p0 as (0,0) d p0 norm =:(0,0) d prove (p0 d p1)=norm p0-p1 prove (p0 d p1)=norm p1-p0 prove M0 is an isometry iff (norm p0-p1)=norm (M0 p0)-M0 p1 prove (p0 r0 bdB p1) iff (p1=(0,0) p0 MW p2) for some p2 with r0=norm p1 p2 prove every circle is the translation of a circle about the origin (p0 r0 bdB p1) iff (p1=(0,0) p0 MW p2) for some p2 with (0,0) r0 bdB p2 prove associativity: (r0*r1*p0)=(r0*r1)*p0 prove distributivity: (r0*p0+p1)=(r0*p0)+r0*p0 prove identity: p0=1*p0 prove annihilator: (0,0)=0*p0 prove translation is an isometry (p0 d p1)=(p2 p3 MW p0) d p2 p3 MW p1 prove a reflection through the origin followed by a translation is a point-reflection (p0 p1 MW (0,0) Mm)= p2 Mm for some p2 prove a dilation through the origin followed by a translation is a point-dilation (p0 p1 MW (0,0) r0 MH)= p2 r1 MH for some p2 and r1 prove the reflection of a circle through a point is a circle for some p4,p5 (*./(p0=p1 Mm p2),p3 p4 o p2) iff (p4 p5 o p0) prove the dilation of a circle through a point is a circle prove ((]=(M0 p0 p1 MW) and ]=p0 p1 MW M0) iff (M0 p2)=p0+(p0-p1)+p2-p0 prove the inverse of a translation is a translation prove ((]=M0 p0 r0 IH) and ]=p0 r IH M0) iff (M0 p1)=p0+(%r)*p1-p0 prove the inverse of a dilation is a dilation prove (p0 = p1 p2 MW p0) iff (p0=p0+p2-p1) iff ((0,0)=p2-p1) iff p1=p2 prove translation doesn't have fixed points unless it is the identity prove the fixed points of a transformation via its coordinate definition prove (*./(p0=a0,a1),(e0=1,0),e1=0,1) implies p0=(a0*e0)+a1*e1 prove p0,(p0+r*e0),(p0+r*e1),:(p0+(r*e0)+r*e1) W is a rectangle Segments, Rays, And Lines Segments prove (p0 p1 W p2) iff *./(p2=p0+(p1-p0)*t),(0<:t),t<:1 prove the point halfway between p0 and p0+p1 is p0+-:p1 prove every segment is a translation of a segment from the origin prove every segment is a translation of a dilation of a unit segment from the origin prove (p0 p1 W p2) iff *./(p2=((1-t)*p0)+t*p1),(0<:t),t<:1 assume (p0 p1 W) is a directed segment ordered by ((1-t)*p0)+t*p1 with 0<:t and t<:1 note p0 p1 W is also called a located vector define the source of p0 p1 W as p0 define the target of p0 p1 W as p1 note p0 p1 W is also said to be located at p0 prove (p0 p1 MW = p1 p0 MW) iff p0=p1 note a point can be represented as an arrow whose source and target are equal Rays define the ray with vertex p0 in the direction of (0,0) p1 W as p0 (p0 + p1) R prove p0 p1 R p2 iff *. / (p2 = p0 + t * p1 - p0) , (R. *. 0 <:) t for some t prove p0 p1 R = p0 (p1 - p0) R prove (R. *. 0 <)t implies p0 p1 R = p0 (t * p1) R define p0 p1 R has the same direction as p2 p3 R if *. / ((R. *. 0 <:) t) , (p1 - p0) = t * p3 - p2 note this induces a sensed parallel axiom note multidimensional forms Lines define p0 p1 W is parallel to p2 p3 W if *. / (R. t) , (p1 - p0) = r * p3 - p2 prove parallelism in this way is an equivalence relation define p0 parallel to p1 if *. / (-. 0 = p0 , p1) , (R. t) , p0 = t * p1 for some t prove a located vector belongs to a unique line p0 p1 W p2 implies p0 p1 i p2 prove (-.p0=0,0) implies ((0,0),:p0 i p1) iff p1=t*p0 for some t note the line passing through p0 parallel to (0,0) p1 W is all points p0+t*p1 for some t prove p0 p1 i p2 iff p2=p0+t*p1 for some t note p0+t*p1 is called a parametric representation of the line i p0 (p0+p1) note in N the parametric representation is actually p0 + p1 * note t is called a parameter in p0+t*p1 note the following argument in N p0 =: a0 , a1 p0 is the ordered pair a0,a1 p1 =: b0 , b1 p1 is the ordered pair b0,b1 p =: p0 + p1 * parametric description of the line through p0 parallel to p1 x =: 0 { p zeroth coordinate of p y =: 1 { p first coordinate of p p = (x , y) x = a0 + b0 * y = a1 + b1 * xaxis =: 0 , ~ p = xaxis x suppose p is equal to a point on the xaxis (x , y) = 0 , ~ x p = (x , y) and (x , 0) = xaxis x (x = x) *. 0 = y pairs are equal iff their components are x = x this is always true, so we don't get any new information 0 = y thus (p=xaxis x) iff (0=y) (0 = y) t does there exist t such that 1=((0=y)t) ? (0 = a1 + b1 *) t (0 t) = (a1 + b1 *) t 0 = a1 + b1 * t t =: b1 % ~ s 0 = a1 + b1 * b1 % ~ s 0 = (a1 +) ] s by algebra 1=]*(%]) or (-.0=[)<: ]=[ * ] % [ 0 = a1 + s s =: (- a1) + u 0 = a1 + (- a1) + u 0 = ] u 0 = u t = b1 % ~ (- a1) + 0 t = b1 % ~ (- a1) t = (- a1) % b1 t = - a1 % b1 p - a1 % b1 yields a point on the x-axis, it is unique (by other arguments) note mW O p0 can be used to represent the magnitude of a velocity (speed) note when do two parametrically described lines intersect? prove when a line crosses a circle for what x and y does (p=(x,y))*.(*:r)=(+/(*:x,y)) prove if *./-.O=A,B then A=:a0,a1 is parallel to B=:b0,b1 iff 0=(a0*b1)-a1*b0 prove if two lines are not parallel then they have exactly one point in common prove if P=:p,q and (*:r)>:+/*:P then P+A* intersects (*:r)=(+/(*:(0 1{))) twice prove if A=:a0,a1 and B=:b0,b1 then (x,y)=(A +)(B *) iff x=a0 + b0 * and y=a1 + b1 * Ordinary Equation For A Line prove (x , y) = ((a0 , a1) +) ((b0 , b1) *) then x = a0 + b0 * y = a1 + b1 * ] (b % ~) (b *) ((b % ~) ]) (b *) ((b % ~) (a - ~ a +)) (b *) (b % ~) ((a - ~ a +) (b *)) (b % ~) (a - ~ ((a +) (b *))) (b % ~) (a - ~) x NB. alternatively (and going along the classical route) (a0 , a1) + (b0 , b1) * t (a0 , a1) + (b0 * t) , (b1 * t) (x =: a0 + b0 * t) , (y =: a1 + b1 * t) t t * 1 t * (b0 % b0) (t * b0) % b0 (b0 * t) % b0 (0 + b0 * t) % b0 ((- a0) + a0 + b0 * t) % b0 ((- a0) + x) % b0 (x - a0) % b0 t = (x - a0) % b0 t = (y - a1) % b1 NB. By a similar argument. prove the ordinary tacit form has x,y on the right (x , y) = (A +) (B *) ] (B % ~) (B *) (B % ~ A - ~ A + B *) (B % ~ A - ~) (x , y) ] = (b0 % ~ a0 - ~) x ] = (b1 % ~ a1 - ~) y ((b0 % ~ a0 - ~) x) = ((b1 % ~ a1 - ~) y) y = (a1 + b1 * b0 % ~ a0 - ~) x Trigonometry Radian Measure define x=mV p0 p1 p2 if *./(0<:x),(x<:o.1),(x%o.1)=(mclBV p1 1 p0 p2)%(mclB p1 1) prove if x=mV p0 p1 p2 then (mclB p1 1)=o.1r2 implies x=mclBV p1 1 p0 p2 prove (deg x)=((o.1)%180)*(rad x) note from now on: radians only prove (x%o.1)=(mbdBV p0 1 p1 p2)%(mbdB p0 1) if x>:o.2 then "x rad" means "w rad" with *./(0<:w),(w<o.2),(x=w+n*o.2) if *./(0<z),(x=-z) then (rad x) means "w rad" with *./(0<:w),(w<o.2),(z=(n*o.2)-w) Sine And Cosine if *. / (O p2 K p3) , (-. p3 = O) , (p3 = (a , b)) then "sine V p3 O (1,0)" is b % r =: %: + / *: a , b "cosine V p3 O (1,0)" is a%r sine and cosine are independent of the point p3 (prove) geometrically this means that any two such triangles are similar if O 1 bdB p3=:a,b then (sine V p3 O (1,0))=b and (cosine V p3 O (1,0))=a for O 1 bdB p3=:(a,b) define (sine mV p3 O (1,0))=b and (cosine mV p3 O (1,0))=a the sign of sine and cosine depending on the quadrant its relevant angle occupies Q1:+,+ Q2:-,+ Q3:-,- Q4:+,- if (LA p0 p1 p2) then (sin V p1 p0 p2)=(d p1 p2)%(d p0 p1) if (LA p0 p1 p2) then (cos V p1 p0 p2)=(d p0 p2)%(d p0 p1) "sin x" is (sine rad x) "cos x" is (cosine rad x) from the definition of rad (for an arbitrary angle) (sin x)=sin x+n*o.2 (cos x) = cos x + n * o. 1 using plane geometry and the Pythagorean theorem: ======================= x sin x cos x ----------------------- o.1r6 1r2 (%:3)%2 o.1r41 %%:21 %%:2 o.1r3 (%:3)%2 1r2 o.1r2 1 0 o.1 0 _1 o.2 0 1 ======================= consider 1,1,%:2 and 1,(%:3),2 right triangles (and their angles) reflect o.1r6, o.1r3, o.1 over longest leg and compute if 1=$x then 1=+/*:(sin,cos)x since 1 (*: r) % *: r ((*: a) + *: b) % *: r ((*: a)% *: r) + (*: b) % *: r (*: a % r) + *: b % r + / *: ((a % r) , b % r) + / *: (sin x) , cos x + / *: (sin , cos) x (cos x) = sin x + o. 1r2 and (sin x) = cos x - o. 1r2 (sin - x) = - sin x and (cos x) = cos - x determine a distance using small angle measurements and a known length polar coordinates r = %: + / *: x , y V =: mV (x , y) O (1 , 0) (x % r) = cos V (y % r) = sin V The Graphs plot ] , sin The Tangent tan =: sin % cos tan only gives relevant information when -.0=cos if *. / (O p2 K p3) , (-. p3 = O) , (p3 = a , b) then (b % a) = tan mV p3 O p2 tangent of the angle made by a line crossing the x-axis is the lines slope plot ],tan we only plot tables of values cot=: % tan sec=: % cos cosec =: % sin 1 = - / *: (tan , sec) x 1 = - / *: (csc , cot) x Addition Formulas (sin x + y) = ((sin x) * cos y) + (cos x) * sin y (cos x + y) = ((cos x) * sin y) - (sin x) * sin y (sin x - y) = ((sin x) * cos y) - (cos x) * sin y (cos x - y) = ((cos x) * sin y) + (sin x) * sin y (sin +: x) = +: * / (sin , cos) x (cos +: x) = - / *: (cos , sin) x (*: cos x) = (1 + cos +: x) % 2 or (+: *: cos x) = 1 + cos +: x (*: sin x) = (1 - cos +: x) % 2 or (+: *: sin x) = 1 - cos +: x (* / sin (m , n) * x) = -: - / cos (m (- , +) n) * x (((sin m *) * (cos n *)) x) = -: + / sin (m (+ , -) n) * x (* / cos (m , n) * x) = -: - / cos (m (+ , -) n) * x Rotations Since (r , V + x) = O x oV r , V then x0 = r * cos V y0 = r * sin V x1 = r * cos V + x x1 = r * ((cos V) * cos x) - (sin V) * sin x y1 = r * sin V + x y1 = r * ((sin V) * cos x) + (sin V) * cos x x1 = ((cos V) * x0) - (sin V) * y0 y1 = ((sin V) * x0) + (cos V) * y0 the rotation matrix for x is 2 2 $ (cos , (- sin) , sin , cos) x dilation matrix compositions of actions as multiplications of matrices Some Analytic Geometry The Straight Line Again the plot of points for which c = F yields 1 is called the graph of F an arbitrary point on the graph of ]=a* has the form (1 , a) * a point on the graph of ] = (- ]) is of the form (1 , -1) * the graph of [ = (b + a *) is a straight line parallel to the graph of [ = a * ] y1 =: y - b so y1 = a * x with points of the form (x , a * x) and [ = (b + a *) are (] , (b + a *)) the slope of a line that is the graph of [ = (b + a * ]) is a *. / (y0 = b + a * x0) , y1 = b + a * x1 implies *. / ((y1 - y0) = a * x1 - x0) , a = (y1 - y0) % x1 - x0 (a = (y - y0) % x - x0) iff ((y - y0) % x - x0) = (y1 - y0) % x1 - x0 0 = c + (a * x) + b * y equation of a line The Parabola (y - b) = c * (x - a) ^ 2 is called a parabola The Ellipse ((a , b) *) shear dilation 1 = + / *: (u % a) , (v % b) is an ellipse The Hyperbola c = x * y is a hyperbola Rotation Of Hyperbolas c = - / *: y , x