```Notes on Basic Math by Serge Lang

Contents

Part I Algebra

Chapter 1 Numbers
The integers
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
Real numbers: positivity
Powers and roots
Inequalities

Interlude On Logic and Mathematical Expressions
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
Sine and cosine
The graphs
The tangent
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"

(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

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)

((*. /
(-. 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

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
note from now on: radians only
prove (x%o.1)=(mbdBV p0 1 p1 p2)%(mbdB p0 1)
*./(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

(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
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