Basic Mathematics - Serge Lang

Cheatsheet Content

Part I: Algebra 1 Numbers §1. The Integers Positive Integers: $1, 2, 3, \dots$ Zero: $0$ Negative Integers: $-1, -2, -3, \dots$ Integers: Positive integers, negative integers, and zero. Natural Numbers: Positive integers and zero. Representation on a line: $-4 \quad -3 \quad -2 \quad -1 \quad 0 \quad 1 \quad 2 \quad 3 \quad 4$ Addition with 0: $N1. a + 0 = a$ for any integer $a$. Additive Inverse: $N2. a + (-a) = 0$ and $-a + a = 0$. Also called "minus $a$". Notation: $a - b$ means $a + (-b)$. §2. Rules for Addition Commutativity: $a + b = b + a$. Associativity: $(a + b) + c = a + (b + c)$. Uniqueness of Additive Inverse: $N3. \text{If } a + b = 0, \text{ then } b = -a \text{ and } a = -b$. Double Negative: $N4. a = -(-a)$. Negative of a Sum: $N5. -(a + b) = -a - b$. Sum of positive integers is positive. Sum of negative integers is negative. $a+b = - (n+m)$ if $a=-n, b=-m$ for $n,m > 0$. Cancellation Law: If $a+b = a+c$, then $b=c$. §3. Rules for Multiplication Commutativity: $ab = ba$. Associativity: $(ab)c = a(bc)$. Multiplication by 1 and 0: $N6. 1a = a$ and $0a = 0$. Distributivity: $a(b + c) = ab + ac$ and $(b + c)a = ba + ca$. Multiplication by -1: $N7. (-1)a = -a$. Negative of a Product: $N8. -(ab) = (-a)b$. $N9. -(ab) = a(-b)$. Product of Two Negatives: $N10. (-a)(-b) = ab$. Powers: $a^n = a \cdot a \cdot \dots \cdot a$ ($n$ times). Power Rules: $N11. a^{m+n} = a^m a^n$. $N12. (a^m)^n = a^{mn}$. $(ab)^n = a^n b^n$. Important Formulas: $(a + b)^2 = a^2 + 2ab + b^2$. $(a - b)^2 = a^2 - 2ab + b^2$. $(a + b)(a - b) = a^2 - b^2$. §4. Even and Odd Integers; Divisibility Odd Integers: $1, 3, 5, \dots$ (form $2m-1$ or $2n+1$ for $n \ge 0$). Even Integers: $2, 4, 6, \dots$ (form $2n$). Theorem 1 (Parity of Sums): Even + Even = Even. Even + Odd = Odd. Odd + Even = Odd. Odd + Odd = Even. Theorem 2 (Parity of Squares): If $a$ is even, $a^2$ is even. If $a$ is odd, $a^2$ is odd. Corollary: If $a^2$ is even, $a$ is even. If $a^2$ is odd, $a$ is odd. Divisibility: $d$ divides $n$ (or $n$ is divisible by $d$) if $n = dk$ for some integer $k$. §5. Rational Numbers Definition: A quotient $m/n$ where $m, n$ are integers and $n \ne 0$. Cross-multiplying Rule: $m/n = r/s$ if and only if $ms = nr$ (for $n, s \ne 0$). Integer as Rational: $m = m/1$. Cancellation Rule: $am/an = m/n$ (for $a, n \ne 0$). Negative Rational: $-m/n = m/(-n) = -(m/n)$. Lowest Form: $r/s$ where $r, s$ are positive integers and their only common divisor is 1. Addition: $(m/n) + (r/s) = (ms + nr)/ns$. Multiplication: $(m/n) \cdot (r/s) = mr/ns$. Theorem 4: There is no positive rational number whose square is 2. (Proof by contradiction: assumes $m/n$ in lowest form, shows both $m, n$ must be even). §6. Multiplicative Inverses Definition: For $a \ne 0$, $a^{-1}$ is the rational number such that $a^{-1}a = aa^{-1} = 1$. If $a = m/n$, then $a^{-1} = n/m$. Uniqueness: If $ab = 1$, then $b = a^{-1}$. Zero Product Property: If $ab = 0$, then $a = 0$ or $b = 0$. Quotient Notation: $a/b$ means $ab^{-1}$. Cancellation Law for Multiplication: If $a \ne 0$ and $ab = ac$, then $b = c$. Cancellation for Fractions: $ab/ac = b/c$ (for $a, c \ne 0$). 2 Linear Equations §1. Equations in Two Unknowns System of equations: e.g., $2x + y = 1$, $3x - 2y = 4$. Elimination Method: Multiply equations by suitable numbers to make coefficients of one variable equal, then add/subtract to eliminate that variable. Example: Multiply first by 2: $4x + 2y = 2$. Add to second: $(4x+2y) + (3x-2y) = 2+4 \implies 7x=6 \implies x=6/7$. Substitute $x$ into first equation: $2(6/7) + y = 1 \implies 12/7 + y = 1 \implies y = 1 - 12/7 = -5/7$. No Solutions: Lines are parallel and distinct (e.g., $2x-y=5, 2x-y=7$). §2. Equations in Three Unknowns Method: Eliminate one variable from two pairs of equations, resulting in a system of two equations in two unknowns. Solve that system, then back-substitute. Example: $3x + 2y + 4z = 1$ $x + y + 2z = 2$ $x - 3y + z = -1$ Eliminate $x$ from 2nd and 3rd: Add (2nd - 3rd) * 0 (already done) $\implies 4y+z = 3$. Eliminate $x$ from 1st and 2nd: Multiply 2nd by 3: $3x+3y+6z = 6$. Subtract from 1st: $-y-2z = -5$. New system in $y, z$: $4y+z=3$, $-y-2z=-5$. Solve this to find $y, z$, then substitute back into one of original equations to find $x$. 3 Real Numbers §1. Addition and Multiplication Properties of Addition: Commutative, associative, $0+a=a$, $a+(-a)=0$. Properties of Multiplication: Commutative, associative, $1a=a$, $0a=0$. Distributivity: $a(b+c)=ab+ac$. Formulas: $(a+b)^2 = a^2+2ab+b^2$, $(a-b)^2 = a^2-2ab+b^2$, $(a+b)(a-b) = a^2-b^2$. Existence of Multiplicative Inverse: For $a \ne 0$, there exists $a^{-1}$ such that $a^{-1}a=aa^{-1}=1$. ($1/a$). §2. Real Numbers: Positivity Notation: $a > 0$ means $a$ is positive. POS 1: If $a, b > 0$, then $ab > 0$ and $a+b > 0$. POS 2: For any real $a$, exactly one of $a>0, a=0, -a>0$ is true. Consequences of POS 1 & 2: If $a>0, b If $a 0$. If $a>0$, then $1/a>0$. If $a Square Roots: For $a>0$, there exists a unique positive number $b$ such that $b^2=a$. Denoted $\sqrt{a}$. The solutions to $x^2=a$ are $x = \pm \sqrt{a}$. Absolute Value: $|x| = \sqrt{x^2}$. Properties: $|-3|=3$, $|a|=a$ if $a \ge 0$. Solving Equations with Absolute Value: $|x+5|=2 \implies x+5=2$ or $x+5=-2$. Rationalizing Denominators/Numerators: Use $(x+y)(x-y)=x^2-y^2$ to eliminate square roots. §3. Powers and Roots Powers: $a^n = a \cdot \dots \cdot a$. $n$-th Root: For $a>0$, $n>0$ integer, $a^{1/n}$ (or $\sqrt[n]{a}$) is the unique positive $r$ such that $r^n=a$. Theorem 1: $(ab)^{1/n} = a^{1/n}b^{1/n}$. Fractional Powers: For $a>0$, $x$ rational. POW 1: $a^{x+y} = a^x a^y$. POW 2: $(a^x)^y = a^{xy}$. POW 3: $(ab)^x = a^x b^x$. POW 4: If $a>1$, $x $a^0=1$. $a^{-x} = 1/a^x$. $a^{m/n} = (a^m)^{1/n} = (a^{1/n})^m$. §4. Inequalities Notation: $a>b \iff a-b>0$. $a 0$. $a \ge b$ means $a$ is greater than or equal to $b$. IN 1 (Transitivity): If $a>b$ and $b>c$, then $a>c$. IN 2 (Multiplication by positive): If $a>b$ and $c>0$, then $ac>bc$. IN 3 (Multiplication by negative): If $a>b$ and $c Adding/Subtracting a number: If $a Intervals: Open: $(a,b) = \{x \mid a Closed: $[a,b] = \{x \mid a \le x \le b\}$. Half-open/closed: $[a,b)$, $(a,b]$. Infinite: $(a, \infty) = \{x \mid x>a\}$, $(-\infty, b] = \{x \mid x \le b\}$, etc. 4 Quadratic Equations General Form: $ax^2 + bx + c = 0$ (where $a \ne 0$). Completing the Square: Method to solve quadratic equations by transforming $x^2 + \frac{b}{a}x = -\frac{c}{a}$ into $(x + \frac{b}{2a})^2 = \frac{b^2-4ac}{4a^2}$. Quadratic Formula: Solutions for $ax^2 + bx + c = 0$ are $$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$$ Discriminant: $\Delta = b^2-4ac$. If $\Delta > 0$: Two distinct real solutions. If $\Delta = 0$: One real solution (repeated root) $x = -b/2a$. If $\Delta Part II: Intuitive Geometry 5 Distance and Angles §1. Distance Distance between points $P, Q$: $d(P, Q)$. DIST 1: $d(P, Q) \ge 0$, and $d(P, Q) = 0 \iff P = Q$. DIST 2: $d(P, Q) = d(Q, P)$. DIST 3 (Triangle Inequality): $d(P, M) \le d(P, Q) + d(Q, M)$. Line Segment: Portion of line between $P, Q$, denoted $PQ$. Length is $d(P, Q)$. SEG 1: $d(P, M) = d(P, Q) + d(Q, M) \iff Q$ lies on segment $PM$. SEG 2: For $P, M$ and $0 \le c \le d(P, M)$, there's unique $Q$ on $PM$ with $d(P, Q) = c$. Circle: Set of points $Q$ with $d(P, Q) = r$ (center $P$, radius $r$). Disc: Set of points $Q$ with $d(P, Q) \le r$. Lines: Unique line $L_{PQ}$ through $P, Q$. Non-parallel lines meet at one point. Unique parallel/perpendicular line through a point. §2. Angles Ray: Half-line starting from $P$. $P$ is the vertex. $R_{PQ}$ is ray from $P$ through $Q$. Angle: Region formed by two rays $R_{PQ}, R_{PM}$ from same vertex $P$. Ordered Angles: $\angle QPM$ is counterclockwise arc from $R_{PQ}$ to $R_{PM}$. Zero Angle: Rays coincide, arc is a point. Straight Angle: Rays are opposite on a line ($180^\circ$). Right Angle: Half a straight angle ($90^\circ$). Degree Measure: Full angle is $360^\circ$. Measure $x^\circ$ means $\text{Area}(Sector)/\text{Area}(Disc) = x/360$. Radian Measure: (Introduced in Chapter 11) Full angle is $2\pi$ radians. Measure $x$ radians means $\text{Area}(Sector)/\text{Area}(Disc) = x/(2\pi)$. §3. The Pythagoras Theorem Triangle: $\triangle PQM$ is the set of segments $PQ, QM, MP$. Right Triangle: One angle is a right angle. Sides forming right angle are legs. Third side is hypotenuse. RT (Axiom): If two right triangles have corresponding legs of equal length, then: a) Corresponding angles have equal measure. b) Their areas are equal. c) Hypotenuses have equal length. PD (Axiom - Parallel Distance): Distance between parallel lines is constant. Rectangle: Four points $P, Q, N, M$ where $PQ \parallel NM$, $QN \parallel MP$, and adjacent sides are perpendicular. Opposite sides have equal length. Area of Rectangle: $ab$ for sides $a, b$. Theorem 1: In a right triangle, $A, B$ non-right angles, $m(A) + m(B) = 90^\circ$. Theorem 2: Area of a right triangle with legs $a, b$ is $ab/2$. Pythagoras Theorem: For a right triangle with legs $a, b$ and hypotenuse $c$, $a^2 + b^2 = c^2$. Corollary: $M$ lies on the perpendicular bisector of $PQ$ if and only if $d(P, M) = d(Q, M)$. 6 Isometries §1. Some Standard Mappings of the Plane Mapping (Map): Association $P \mapsto P'$ from point to point. $F(P)$ is the value/image of $P$. Identity: $I(P) = P$. Reflection through a line $L$: $R_L(P)$ is $P'$ such that $L$ is perpendicular bisector of $PP'$. Reflection through a point $O$: $R_O(P)$ is $P'$ such that $O$ is midpoint of $PP'$. Dilation (Stretching) by $r$ (relative to $O$): $F_r(P)$ is on ray $R_{OP}$ with $d(O, F_r(P)) = r \cdot d(O, P)$. Written as $rP$. Rotation by angle $A$ (relative to $O$): $G_A(P)$ is $P'$ on circle $d(O,P)$ such that $\angle POP'$ has measure $A$ (counterclockwise). Translation: $T_{OM}(P)$ is $P'$ such that vector $PP'$ is same as $OM$. §2. Isometries Isometry: A mapping $F$ that preserves distances: $d(P, Q) = d(F(P), F(Q))$ for all $P, Q$. Reflections, rotations, translations are isometries. Dilation by $r \ne 1$ is not. Theorem 1: Image of a line segment $PQ$ under an isometry $F$ is the line segment $F(P)F(Q)$. Corollary: Image of a straight line under an isometry is a straight line. Fixed Point: $P$ such that $F(P)=P$. Theorem 2: If $F$ is an isometry and $P, Q$ are distinct fixed points, then every point on line $L_{PQ}$ is a fixed point. Theorem 3: If $F$ is an isometry and $P, Q, M$ are three distinct non-collinear fixed points, then $F$ is the identity. §3. Composition of Isometries Composition: $(F \circ G)(P) = F(G(P))$. $F \circ G$ is an isometry. Associativity: $(F \circ G) \circ H = F \circ (G \circ H)$. Identity: $F \circ I = I \circ F = F$. Notation: $F^2 = F \circ F$, $F^n = F \circ \dots \circ F$ ($n$ times). Composition of Rotations: $G_x \circ G_y = G_{x+y}$ (relative to same point). Composition of Translations: $T_A \circ T_B = T_{A+B}$. §4. Inverse of Isometries Inverse Isometry $G$: $F \circ G = G \circ F = I$. Denoted $F^{-1}$. Uniqueness: Inverse is unique if it exists. Properties: $(F \circ G)^{-1} = G^{-1} \circ F^{-1}$. $F^{-k} = (F^{-1})^k$. $F^m \circ F^n = F^{m+n}$ for integers $m, n$. Corollary of Theorem 3: If $F, G$ are isometries with $F(P)=G(P)$, $F(Q)=G(Q)$, $F(M)=G(M)$ for distinct non-collinear $P, Q, M$, and $F^{-1}$ exists, then $F=G$. §5. Characterization of Isometries Theorem 4: If $F$ is an isometry leaving distinct $P, Q$ fixed, then $F$ is either the identity or reflection through $L_{PQ}$. Theorem 5: If $F$ is an isometry leaving one point $O$ fixed, then $F$ is either a rotation or a rotation composed with a reflection through a line. Theorem 6: Any isometry is either a translation, or a composite of a translation and a rotation, or a composite of a translation, a rotation, and a reflection through a line. §6. Congruences Congruence: Set $S$ is congruent to $S'$ if there exists an isometry $F$ such that $F(S)=S'$. Theorem 7: Two circles of the same radius are congruent. Theorem 8: Any two segments of the same length are congruent. Theorem 9 (SAS): Right triangles $\triangle PQM, \triangle P'Q'M'$ with $d(P,Q)=d(P',Q')$ and $d(Q,M)=d(Q',M')$ are congruent. Theorem 10 (SSS): Triangles $\triangle PQM, \triangle P'Q'M'$ with $d(P,Q)=d(P',Q')$, $d(P,M)=d(P',M')$, $d(Q,M)=d(Q',M')$ are congruent. Isometries preserve area and angle measure. 7 Area and Applications §1. Area of a Disc of Radius r Area of a square: $a^2$. Area of a rectangle: $ab$. Effect of Dilation on Area: If $S$ is a region, $rS$ is dilation by $r$. $\text{Area}(rS) = r^2 \text{Area}(S)$. Area of Disc: $\text{Area}(D_r) = \pi r^2$, where $\pi = \text{Area}(D_1)$. Approximation by squares: Sum of areas of small squares inside region. Generalization to 3D: $\text{Volume}(rS) = r^3 \text{Volume}(S)$. §2. Circumference of a Circle of Radius r Circumference of Circle: $C_r = 2\pi r$. Approximation by inscribed polygons: $A_n = \frac{1}{2} P_n h_n$. As $n \to \infty$, $A_n \to \pi r^2$, $P_n \to C_r$, $h_n \to r$. So $\pi r^2 = \frac{1}{2} C_r r \implies C_r = 2\pi r$. Effect of Dilation on Length: $d(rP, rQ) = r \cdot d(P, Q)$. $\text{Length}(rS) = r \cdot \text{Length}(S)$. Part III: Coordinate Geometry 8 Coordinates and Geometry §1. Coordinate Systems Coordinate Axes: Horizontal (x-axis) and vertical (y-axis) lines intersecting at origin $O=(0,0)$. Point Coordinates: $P=(x,y)$. $x$ is x-coordinate, $y$ is y-coordinate. Quadrants: I ($x>0, y>0$), II ($x 0$), III ($x 0, y 3-Space: Point is $(x,y,z)$. Denoted $\mathbb{R}^3$. §2. Distance Between Points On a Line: $d(x_1, x_2) = \sqrt{(x_1-x_2)^2}$. In the Plane: For $P_1=(x_1, y_1)$ and $P_2=(x_2, y_2)$, distance is $d(P_1, P_2) = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$. Definition of Plane: $\mathbb{R}^2$ is the set of all pairs $(x,y)$ of real numbers. In 3-Space: For $P_1=(x_1, y_1, z_1)$ and $P_2=(x_2, y_2, z_2)$, $d(P_1, P_2) = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$. §3. Equation of a Circle Definition: Set of points $(x,y)$ whose distance from center $(a,b)$ is $r$. Equation: $(x-a)^2 + (y-b)^2 = r^2$. Center at Origin: $x^2+y^2=r^2$. Completing the square: Use to find center and radius from general quadratic equations. Sphere in 3-Space: $(x-a)^2 + (y-b)^2 + (z-c)^2 = r^2$. §4. Rational Points on a Circle Pythagorean Triples: Integers $(a,b,c)$ such that $a^2+b^2=c^2$. Rational Points on Unit Circle: $(x,y)$ such that $x^2+y^2=1$ and $x,y$ are rational. Parametrization: If $x \ne -1$, then $t = y/(x+1)$ is rational. $$x = \frac{1-t^2}{1+t^2}, \quad y = \frac{2t}{1+t^2}$$ for rational $t$ gives all rational points on the unit circle except $(-1,0)$. 9 Operations on Points §1. Dilations and Reflections Point Notation: $A=(a_1, a_2)$. Scalar Multiplication (Dilation by $c$): $cA = (ca_1, ca_2)$. If $c>0$, it's dilation by $c$ relative to $O$. If $c Reflection through Origin: $R_O(A) = -A = (-a_1, -a_2)$. Theorem 1: For $r>0$, $d(rA, rB) = r \cdot d(A, B)$. Theorem 2: For any number $c$, $d(cA, cB) = |c| \cdot d(A, B)$. §2. Addition, Subtraction, and the Parallelogram Law Vector Addition: $A+B = (a_1+b_1, a_2+b_2)$. Properties: Commutative, associative, $A+O=A$, $A+(-A)=O$. Vector Subtraction: $A-B = A+(-B) = (a_1-b_1, a_2-b_2)$. Geometric Interpretation (Parallelogram Law): $O, A, B, A+B$ form a parallelogram. Translation by $A$: $T_A(P) = P+A$. Norm of a Point: $|A| = d(A,O) = \sqrt{a_1^2+a_2^2}$. $d(A,B) = |A-B|$. $|cA| = |c||A|$. Theorem 3: Circle of radius $r$ and center $A$ is the translation by $A$ of the circle of radius $r$ and center $O$. Properties of Scalar Multiplication: $b(cA)=(bc)A$, $(b+c)A=bA+cA$, $c(A+B)=cA+cB$, $1A=A$, $0A=O$. 10 Segments, Rays, and Lines §1. Segments Line Segment $PQ$: Set of points $P+tA$ for $0 \le t \le 1$, where $A=Q-P$. Also written as $(1-t)P+tQ$. Directed Segment (Located Vector): Ordered pair of points $PQ$. $P$ is beginning, $Q$ is end. §2. Rays Ray (vertex $P$, direction $A \ne O$): Set of points $P+tA$ for $t \ge 0$. Ray (vertex $P$, through $Q \ne P$): Set of points $P+t(Q-P)$ for $t \ge 0$. Same Direction: $A \ne O, B \ne O$ have same direction if $B=cA$ for some $c>0$. Parallel Located Vectors: $PQ$ and $MN$ are parallel if $Q-P = c(N-M)$ for $c \ne 0$. Application: Vector addition in physics (forces, velocities). §3. Lines Line (through $P$, parallel to $A \ne O$): Set of points $P+tA$ for all real numbers $t$. Parametric Representation: $(x(t), y(t)) = (p_1+ta_1, p_2+ta_2)$. $t$ is the parameter. Line through $P, Q$: $P+t(Q-P)$. Intersection of Lines: Solve simultaneous equations for $t$ and $s$. Intersection of Line and Circle: Substitute parametric equations of line into circle equation, solve for $t$. §4. Ordinary Equation for a Line Form: $y = ax+b$. $a$ is the slope, $b$ is the y-intercept. Slope: For two points $(x_1, y_1), (x_2, y_2)$ on the line, $a = (y_2-y_1)/(x_2-x_1)$. Equation from Point and Slope: $y-y_1 = a(x-x_1)$. Vertical Lines: Equation $x=c$. Slope is undefined. General Form: $Ax+By=C$. Finding Intersections: Substitute $y=ax+b$ into other equation (e.g., circle). 11 Trigonometry §1. Radian Measure Definition: Angle $A$ has $x$ radians if $\text{Area}(Sector)/\text{Area}(Disc) = x/(2\pi)$. Conversions: $360^\circ = 2\pi \text{ rad}$, $180^\circ = \pi \text{ rad}$. $1 \text{ rad} = 180/\pi \text{ deg}$. Arc Length: For a circle of radius $r$, arc length for angle $\theta$ (in radians) is $r\theta$. §2. Sine and Cosine Definition: For an angle $A$ with ray passing through $(a,b)$ (not origin), let $r=\sqrt{a^2+b^2}$. $\sin A = b/r$. $\cos A = a/r$. Unit Circle: If $r=1$, $(\cos A, \sin A)$ are coordinates. Signs by Quadrant: I: $\sin>0, \cos>0$. II: $\sin>0, \cos III: $\sin IV: $\sin 0$. Right Triangle Definition: $\sin A = \text{opposite}/\text{hypotenuse}$, $\cos A = \text{adjacent}/\text{hypotenuse}$. Periodicity: $\sin(x+2\pi) = \sin x$, $\cos(x+2\pi) = \cos x$. Basic Identity: $\sin^2 x + \cos^2 x = 1$. Co-function Identities: $\cos x = \sin(x+\pi/2)$, $\sin x = \cos(x-\pi/2)$. Even/Odd Functions: $\sin(-x) = -\sin x$ (odd), $\cos(-x) = \cos x$ (even). Polar Coordinates: $(x,y) = (r \cos \theta, r \sin \theta)$. $(r, \theta)$ are polar coordinates. §3. The Graphs Graph of $\sin x$: Wave-like, oscillates between -1 and 1, period $2\pi$. Graph of $\cos x$: Wave-like, oscillates between -1 and 1, period $2\pi$. Similar to $\sin x$ shifted by $\pi/2$. §4. The Tangent Definition: $\tan x = \sin x / \cos x$. Defined when $\cos x \ne 0$ (i.e., $x \ne \pi/2 + n\pi$). Right Triangle Definition: $\tan A = \text{opposite}/\text{adjacent}$. Graph of $\tan x$: Vertical asymptotes at $\pi/2 + n\pi$, period $\pi$. Applications: Measuring heights/distances. Other Functions: $\cot x = 1/\tan x$, $\sec x = 1/\cos x$, $\csc x = 1/\sin x$. Identities: $1 + \tan^2 x = \sec^2 x$. §5. Addition Formulas Theorem 3: $\sin(A+B) = \sin A \cos B + \cos A \sin B$. $\cos(A+B) = \cos A \cos B - \sin A \sin B$. Corollary: $\sin(A-B) = \sin A \cos B - \cos A \sin B$. $\cos(A-B) = \cos A \cos B + \sin A \sin B$. Double Angle Formulas: $\sin(2x) = 2 \sin x \cos x$. $\cos(2x) = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x$. Half Angle Formulas: $\cos^2 x = (1+\cos 2x)/2$. $\sin^2 x = (1-\cos 2x)/2$. §6. Rotations Rotation Matrix: For a point $P=(x,y)$ rotated by angle $\phi$ to $P'=(x',y')$: $$x' = (\cos \phi)x - (\sin \phi)y$$ $$y' = (\sin \phi)x + (\cos \phi)y$$ This can be written as a matrix multiplication: $$\begin{pmatrix} \cos \phi & -\sin \phi \\ \sin \phi & \cos \phi \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x' \\ y' \end{pmatrix}$$ 12 Some Analytic Geometry §1. The Straight Line Again Equation: $y = ax+b$. $a$ is slope, $b$ is y-intercept. Slope: $a = (y_2-y_1)/(x_2-x_1)$ for two points $(x_1,y_1), (x_2,y_2)$. Equation from point $(x_1,y_1)$ and slope $a$: $y-y_1 = a(x-x_1)$. Vertical Lines: $x=c$. Intersection of Line and Circle: Substitute $y=ax+b$ into $(x-h)^2+(y-k)^2=r^2$ to get a quadratic equation in $x$. §2. The Parabola Standard Form: $y=x^2$. Symmetric about y-axis. Translated Form: $(y-b) = c(x-a)^2$. Vertex at $(a,b)$. Horizontal Parabola: $x=y^2$. Symmetric about x-axis. Translated Horizontal Parabola: $(x-a) = c(y-b)^2$. Vertex at $(a,b)$. Completing the Square: Use to rewrite general quadratic equations into standard forms. §3. The Ellipse Mixed Dilation: $(x,y) \mapsto (ax, by)$ stretches coordinates. Equation (centered at origin): $x^2/a^2 + y^2/b^2 = 1$. Equation (centered at $(h,k)$): $(x-h)^2/a^2 + (y-k)^2/b^2 = 1$. Extremities: For $x^2/a^2 + y^2/b^2 = 1$, x-intercepts are $(\pm a, 0)$, y-intercepts are $(0, \pm b)$. §4. The Hyperbola Standard Form: $xy=c$. Asymptotes are x and y axes. Translated Form: $(x-h)(y-k)=c$. Asymptotes are $x=h, y=k$. Other Form: $y^2-x^2=c$ (or $x^2-y^2=c$). §5. Rotation of Hyperbolas Theorem 1: The hyperbola $xy=1$ (denoted $H$) rotated by $\pi/4$ (counterclockwise) becomes $v^2-u^2=2$. Rotation formulas: $u = (x-y)/\sqrt{2}$, $v = (x+y)/\sqrt{2}$. $v^2-u^2 = ((x+y)^2 - (x-y)^2)/2 = (4xy)/2 = 2xy$. So $xy=1 \implies v^2-u^2=2$. Hyperbola $xy=r^2$ rotated by $\pi/4$ becomes $v^2-u^2=2r^2$. Hyperbola $x^2-y^2=c$ is a rotated hyperbola $xy=c'$. Part IV: Miscellaneous 13 Functions §1. Definition of a Function Function: An association $f: S \to \mathbb{R}$ that maps each element $x$ in domain $S$ to a number $f(x)$. Examples: $f(x)=x^2$, $f(x)=\sin x$. Constant Function: $f(x)=c$. Image: $f(S)$ is the set of all values $f(x)$ for $x \in S$. Function Arithmetic: $(f+g)(x) = f(x)+g(x)$. $(fg)(x) = f(x)g(x)$. $(cf)(x) = c \cdot f(x)$. Properties: Commutativity, associativity, distributivity as for numbers. Zero function $f(x)=0$. Additive inverse $-f(x)$. Even Function: $f(-x)=f(x)$. Odd Function: $f(-x)=-f(x)$. §2. Polynomial Functions Definition: $f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$. Degree: If $a_n \ne 0$, degree is $n$. Root: $c$ is a root if $f(c)=0$. Theorem 1 (Factor Theorem): If $c$ is a root of polynomial $f(x)$, then $f(x) = (x-c)g(x)$ for some polynomial $g(x)$ of degree $n-1$. Theorem 2: A non-zero polynomial of degree $n$ has at most $n$ roots. Corollary: Coefficients of a polynomial are uniquely determined. Zero Polynomial: All coefficients are 0. Euclidean Algorithm for Polynomials: For non-zero polynomials $f(x), g(x)$, there exist polynomials $q(x), r(x)$ such that $f(x) = q(x)g(x) + r(x)$ with $\deg r Rational Function: A quotient of polynomials $f(x)/g(x)$. §3. Graphs of Functions Graph: Set of points $(x, f(x))$. Examples: Parabola ($y=x^2$), sine wave ($y=\sin x$). Piecewise Functions: e.g., $f(x)=[x]$ (greatest integer function), creates a staircase graph. §4. Exponential Function Definition: For $a>0$, $f(x)=a^x$. EXP 1: $a^{x+y} = a^x a^y$. EXP 2: $(a^x)^y = a^{xy}$. EXP 3: $(ab)^x = a^x b^x$. EXP 4: If $a>1$, $x $a^0=1$, $a^{-x}=1/a^x$. Graph: If $a>1$, increases steeply for $x>0$, approaches 0 for $x Exponential Growth/Decay: $P(t)=Ca^{Kt}$ or $P(t)=Ce^{Kt}$. $C$ is initial value ($P(0)$). §5. Logarithms Definition: For $a>1$, $x = \log_a y \iff y=a^x$. Properties: LOG 1: $\log_a(xy) = \log_a x + \log_a y$. LOG 2: $\log_a 1 = 0$. LOG 3: If $x $\log_a (x^k) = k \log_a x$. Solving Exponential Equations: Take logarithm of both sides. 14 Mappings §1. Definition Mapping: An association $f: S \to S'$ that maps each element $x \in S$ to an element $f(x) \in S'$. Functions are mappings where $S'=\mathbb{R}$. Isometries are mappings where $S=S'=\mathbb{R}^2$. Examples: Parametric curves (e.g., $t \mapsto (x(t), y(t))$), polar coordinate map. Image of a Subset: $f(T) = \{f(t) \mid t \in T\}$ for $T \subseteq S$. §2. Formalism of Mappings Identity Mapping: $I_S(x) = x$ for all $x \in S$. Composite Mapping: $(g \circ f)(x) = g(f(x))$. Associativity: $(h \circ g) \circ f = h \circ (g \circ f)$. Inverse Mapping: $g: S' \to S$ is inverse of $f: S \to S'$ if $g \circ f = I_S$ and $f \circ g = I_{S'}$. Denoted $f^{-1}$. Uniqueness of Inverse: Inverse is unique if it exists. Iteration: $f^k(x) = f(f(\dots f(x)\dots))$ ($k$ times). $f^0=I$. Power Rule: $f^{m+n} = f^m \circ f^n$. (Holds for negative exponents if $f^{-1}$ exists). Cancellation Law: If $h \circ f = h \circ g$ and $h^{-1}$ exists, then $f=g$. §3. Permutations Permutation of $J_n$: A mapping $\sigma: J_n \to J_n$ (where $J_n = \{1, \dots, n\}$) such that $\sigma(i) \ne \sigma(j)$ if $i \ne j$. Represented as $\begin{pmatrix} 1 & 2 & \dots & n \\ \sigma(1) & \sigma(2) & \dots & \sigma(n) \end{pmatrix}$. Product ($\sigma \circ \sigma'$): Apply $\sigma'$ first, then $\sigma$. Not necessarily commutative. Inverse Permutation: $\sigma^{-1}$ such that $\sigma \circ \sigma^{-1} = \sigma^{-1} \circ \sigma = I$. Transposition: Permutation that swaps two elements and leaves others fixed. Denoted $[a,b]$. $t^2=I$. Theorem 1: Every permutation can be expressed as a product of transpositions. Orbit: For an element $i$, the set $\{i, \sigma(i), \sigma^2(i), \dots, \sigma^{k-1}(i)\}$ where $k$ is the smallest positive integer such that $\sigma^k(i)=i$. Cycle: Notation $[i, \sigma(i), \dots, \sigma^{k-1}(i)]$. $k$ is the length. Theorem 3: Orbits of a permutation are either identical or disjoint. Orbit Decomposition: $\sigma = \gamma_1 \gamma_2 \dots \gamma_m$, where $\gamma_j$ are disjoint cycles. Theorem 2 (Parity of Permutations): In any expression of a permutation as a product of transpositions, the number of transpositions is either always even or always odd. Even Permutation: Product of even number of transpositions. Sign is $+1$. Odd Permutation: Product of odd number of transpositions. Sign is $-1$. Theorem 4: If $\sigma$ has $k$ orbits, its sign is $(-1)^{n-k}$. 15 Complex Numbers §1. The Complex Plane Definition: Numbers of the form $a+bi$, where $a,b \in \mathbb{R}$ and $i^2=-1$. Representation: $a+bi$ corresponds to point $(a,b)$ in the complex plane. Addition: $(x+iy)+(u+iv) = (x+u)+i(y+v)$. Multiplication: $(x+iy)(u+iv) = (xu-yv)+i(xv+yu)$. Complex Conjugate: For $z=a+bi$, $\bar{z}=a-bi$. $z\bar{z} = a^2+b^2$. $\overline{zw} = \bar{z}\bar{w}$, $\overline{z+w} = \bar{z}+\bar{w}$, $\overline{\bar{z}}=z$. Real Part: $\text{Re}(z) = (z+\bar{z})/2 = x$. Imaginary Part: $\text{Im}(z) = (z-\bar{z})/(2i) = y$. Absolute Value (Modulus): $|z| = \sqrt{x^2+y^2}$. $|z|^2=z\bar{z}$. $|zw|=|z||w|$. $|z+w| \le |z|+|w|$ (Triangle Inequality). Multiplicative Inverse: For $z \ne 0$, $z^{-1} = \bar{z}/|z|^2$. §2. Polar Form Polar Form: $z = r(\cos \theta + i \sin \theta)$, where $r=|z|$ and $\theta$ is the angle. Euler's Formula: $e^{i\theta} = \cos \theta + i \sin \theta$. So $z = re^{i\theta}$. Theorem 3: $e^{i\theta}e^{i\phi} = e^{i(\theta+\phi)}$. To multiply complex numbers, multiply their moduli and add their arguments (angles). $z w = (re^{i\theta})(se^{i\phi}) = (rs)e^{i(\theta+\phi)}$. De Moivre's Theorem: $(re^{i\theta})^n = r^n e^{in\theta}$. 16 Induction and Summations §1. Induction Principle of Mathematical Induction: To prove assertion $A(n)$ for all integers $n \ge 1$: IND 1: Prove $A(1)$ is true. IND 2: Assume $A(k)$ is true for all $1 \le k \le n$, then prove $A(n+1)$ is true. Summation Notation ($\Sigma$): $\sum_{k=1}^n f(k) = f(1)+f(2)+\dots+f(n)$. Binomial Coefficients: $\binom{n}{k} = \frac{n!}{k!(n-k)!}$. $(x+y)^n = \sum_{k=0}^n \binom{n}{k} x^k y^{n-k}$. §2. Summations Volume of Cylinder: $\pi r^2 h$. Approximation of Volume (Archimedes' Method): Sum of volumes of thin cylinders. Volume of Cone (radius $r$, height $h$): $\frac{1}{3}\pi r^2 h$. Mixed Dilation: For $F_{a,b,c}(x,y,z)=(ax,by,cz)$, $\text{Volume}(F_{a,b,c}(S)) = abc \cdot \text{Volume}(S)$. Volume of a Ball (radius $r$): $\frac{4}{3}\pi r^3$. §3. Geometric Series Finite Sum: $\sum_{k=0}^n c^k = 1+c+c^2+\dots+c^n = \frac{1-c^{n+1}}{1-c}$ (for $c \ne 1$). Infinite Sum: If $|c| Caution: Infinity ($\infty$) is a concept, not a number. 17 Determinants §1. Matrices Matrix: Rectangular array of numbers. E.g., $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$. Components: $a_{ij}$ is element in row $i$, column $j$. Row Vectors: Horizontal vectors. Column Vectors: Vertical vectors. Transpose ($^tA$): Rows become columns, columns become rows. $^tA = (a_{ji})$. Diagonal Components: $a_{ii}$. §2. Determinants of Order 2 Definition: For $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, $\det(A) = ad-bc$. Denoted $|A|$ or $D(A)$. Theorem 1 (Solving 2x2 Linear System): For $ax+by=u, cx+dy=v$, if $ad-bc \ne 0$, unique solution: $$x = \frac{ud-vb}{ad-bc}, \quad y = \frac{va-uc}{ad-bc}$$ Theorem 2: $\det(A) = \det(^tA)$. §3. Properties of 2x2 Determinants Notation: $D(A^1, A^2)$ where $A^1, A^2$ are column vectors. D1 (Linearity/Distributivity): $D(B'+B'', C) = D(B', C) + D(B'', C)$, and $D(B, C'+C'') = D(B, C') + D(B, C'')$. D2 (Scalar Multiplication): $D(xB, C) = x D(B, C) = D(B, xC)$. D3 (Unit Matrix): $D(E^1, E^2) = 1$ for $E^1=\begin{pmatrix} 1 \\ 0 \end{pmatrix}$, $E^2=\begin{pmatrix} 0 \\ 1 \end{pmatrix}$. D4 (Identical Columns): $D(B, B) = 0$. D5 (Adding Multiple of Column): $D(B+xC, C) = D(B, C)$. D6 (Column Interchange): $D(B, C) = -D(C, B)$. Cramer's Rule for 2x2: $$x = \frac{\det \begin{pmatrix} u & b \\ v & d \end{pmatrix}}{\det \begin{pmatrix} a & b \\ c & d \end{pmatrix}}, \quad y = \frac{\det \begin{pmatrix} a & u \\ c & v \end{pmatrix}}{\det \begin{pmatrix} a & b \\ c & d \end{pmatrix}}$$ §4. Determinants of Order 3 Definition (Expansion by first row): For $A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}$, $$\det(A) = a_{11} \det(A_{11}) - a_{12} \det(A_{12}) + a_{13} \det(A_{13})$$ where $A_{ij}$ is the $2 \times 2$ matrix after deleting row $i$ and column $j$. Sign Pattern: $\begin{pmatrix} + & - & + \\ - & + & - \\ + & - & + \end{pmatrix}$. Can expand along any row or column using the sign pattern. Theorem 3: $\det(A) = \det(^tA)$. §5. Properties of 3x3 Determinants Properties D1-D6 (linearity, scalar multiplication, unit determinant, identical columns, adding multiple of column, column interchange) hold for $3 \times 3$ determinants, analogous to $2 \times 2$. These properties can be used to simplify determinant calculation (e.g., creating zeros). §6. Cramer’s Rule System of 3x3 Linear Equations: $a_{11}x_1 + a_{12}x_2 + a_{13}x_3 = b_1$ $a_{21}x_1 + a_{22}x_2 + a_{23}x_3 = b_2$ $a_{31}x_1 + a_{32}x_2 + a_{33}x_3 = b_3$ In column vector notation: $x_1 A^1 + x_2 A^2 + x_3 A^3 = B$. Theorem 4 (Cramer's Rule): If $\det(A) \ne 0$, then $$x_1 = \frac{\det(B, A^2, A^3)}{\det(A^1, A^2, A^3)}, \quad x_2 = \frac{\det(A^1, B, A^3)}{\det(A^1, A^2, A^3)}, \quad x_3 = \frac{\det(A^1, A^2, B)}{\det(A^1, A^2, A^3)}$$ Theorem 5: If $\det(A) \ne 0$, the system has a unique solution.