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Math fundamentals 2026-09-04 Quick reference for the material in chapter 1 of Ivan Savov's No bullshit guide to math & physics ,
one block per section. Nothing here is explained; it is the chart you look at once you already know
the idea. See also notation constants, units and conversions
1.1 Solving equations
Isolate the unknown by undoing operations, applying the same change to both sides.
To undo Apply Watch out + c +\,c + c − c -\,c − c × c \times\,c × c ÷ c \div\,c ÷ c c ≠ 0 c \neq 0 c = 0 x 2 x^2 x 2 \sqrt{\;} two roots: x = ± x = \pm\sqrt{\;} x = ± x \sqrt{x} x square requires x ≥ 0 x \ge 0 x ≥ 0 x n x^n x n n \sqrt[n]{\;} n even n n n ± \pm ± n n n e x e^x e x ln \ln ln argument must be positive 10 x 10^x 1 0 x log 10 \log_{10} log 10 sin , cos , tan \sin, \cos, \tan sin , cos , tan sin − 1 , cos − 1 , tan − 1 \sin^{-1}, \cos^{-1}, \tan^{-1} sin − 1 , cos − 1 , tan − 1 restrict to the principal range − c -\,c − c + c +\,c + c ÷ c \div\,c ÷ c × c \times\,c × c c ≠ 0 c \neq 0 c = 0 − x -x − x − ( ) -(\;) − ( ) its own inverse 1 / x 1/x 1/ x 1 / ( ) 1/(\;) 1/ ( ) x ≠ 0 x \neq 0 x = 0 x 3 x^3 x 3 3 \sqrt[3]{\;} 3 one root, negatives allowed ∣ x ∣ \lvert x \rvert ∣ x ∣ ± ( ) \pm(\;) ± ( ) two cases: x = ± ( ) x = \pm(\;) x = ± ( ) ( ) ≥ 0 (\;) \ge 0 ( ) ≥ 0 b x b^x b x log b \log_b log b argument must be positive ln x \ln x ln x e ( ) e^{(\;)} e ( ) result is always positive log b x \log_b x log b x b ( ) b^{(\;)} b ( ) c x \dfrac{c}{x} x c c ( ) \dfrac{c}{(\;)} ( ) c x ≠ 0 x \neq 0 x = 0
Order of operations: brackets, exponents, multiplication and division, addition and subtraction.
Situation Move unknown on both sides collect x x x unknown inside brackets expand, or divide by the bracket's coefficient first unknown in a denominator multiply both sides by the denominator (≠ 0 \neq 0 = 0 two fractions equal cross-multiply: a b = c d ⇒ a d = b c \tfrac{a}{b} = \tfrac{c}{d} \Rightarrow ad = bc b a = d c ⇒ a d = b c product equals zero a b = 0 ⇒ a = 0 ab = 0 \Rightarrow a = 0 ab = 0 ⇒ a = 0 b = 0 b = 0 b = 0 x 2 x^2 x 2 x x x quadratic: rearrange to a x 2 + b x + c = 0 ax^2 + bx + c = 0 a x 2 + b x + c = 0 same unknown under a root and outside isolate the root, square, check for extraneous roots done substitute back into the original equation
1.2 Numbers
Set Symbol Contains Closed under natural N \mathbb{N} N 0 , 1 , 2 , 3 , … 0, 1, 2, 3, \ldots 0 , 1 , 2 , 3 , … + + + × \times × integers Z \mathbb{Z} Z … , − 2 , − 1 , 0 , 1 , 2 , … \ldots, -2, -1, 0, 1, 2, \ldots … , − 2 , − 1 , 0 , 1 , 2 , … + + + − - − × \times × rational Q \mathbb{Q} Q p / q p/q p / q p , q ∈ Z p, q \in \mathbb{Z} p , q ∈ Z q ≠ 0 q \neq 0 q = 0 + + + − - − × \times × ÷ \div ÷ real R \mathbb{R} R rationals and irrationals (2 , π , e \sqrt{2}, \pi, e 2 , π , e limits of sequences complex C \mathbb{C} C a + b i a + bi a + bi i 2 = − 1 i^2 = -1 i 2 = − 1 roots of every polynomial
N ⊂ Z ⊂ Q ⊂ R ⊂ C \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C} N ⊂ Z ⊂ Q ⊂ R ⊂ C Irrational: cannot be written as p / q p/q p / q
1.3 Number representations
Form Example Convert fraction 3 8 \tfrac{3}{8} 8 3 divide: 3 ÷ 8 = 0.375 3 \div 8 = 0.375 3 ÷ 8 = 0.375 decimal 0.375 0.375 0.375 375 1000 = 3 8 \tfrac{375}{1000} = \tfrac{3}{8} 1000 375 = 8 3 repeating 0. 3 ‾ 0.\overline{3} 0. 3 x = 0. 3 ‾ x = 0.\overline{3} x = 0. 3 10 x − x = 3 10x - x = 3 10 x − x = 3 x = 1 3 x = \tfrac{1}{3} x = 3 1 percent 37.5 % 37.5\% 37.5% ÷ 100 \div 100 ÷ 100 scientific 3.75 × 10 − 1 3.75 \times 10^{-1} 3.75 × 1 0 − 1 mantissa in [ 1 , 10 ) [1, 10) [ 1 , 10 ) mixed number 2 1 2 2\tfrac{1}{2} 2 2 1 = 5 2 = \tfrac{5}{2} = 2 5 prime factors 360 = 2 3 ⋅ 3 2 ⋅ 5 360 = 2^3 \cdot 3^2 \cdot 5 360 = 2 3 ⋅ 3 2 ⋅ 5 divide out primes in order
Fraction arithmetic:
a b ± c d = a d ± b c b d , a b ⋅ c d = a c b d , a b ÷ c d = a b ⋅ d c \frac{a}{b} \pm \frac{c}{d} = \frac{ad \pm bc}{bd},
\qquad
\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd},
\qquad
\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c} b a ± d c = b d a d ± b c , b a ⋅ d c = b d a c , b a ÷ d c = b a ⋅ c d 1.4 Variables
Letters Usual meaning x , y , z x, y, z x , y , z unknowns and coordinates a , b , c a, b, c a , b , c known constants, coefficients n , m , k , i , j n, m, k, i, j n , m , k , i , j integers, counters, indices t t t time θ , ϕ , α , β \theta, \phi, \alpha, \beta θ , ϕ , α , β angles f , g , h f, g, h f , g , h functions λ , μ \lambda, \mu λ , μ rates, wavelengths, means ϵ , δ \epsilon, \delta ϵ , δ small quantities
Constant: fixed value. Parameter: fixed within one problem, may change between problems.
Unknown: the thing to solve for.
1.5 Functions and their inverses
f − 1 ( f ( x ) ) = x f^{-1}(f(x)) = x f − 1 ( f ( x )) = x f ( f − 1 ( y ) ) = y f(f^{-1}(y)) = y f ( f − 1 ( y )) = y f ( x ) = y f(x) = y f ( x ) = y f − 1 f^{-1} f − 1
f ( x ) f(x) f ( x ) f − 1 ( y ) f^{-1}(y) f − 1 ( y ) Domain of f − 1 f^{-1} f − 1 x + c x + c x + c y − c y - c y − c all y y y c x cx c x y / c y/c y / c all y y y c ≠ 0 c \neq 0 c = 0 x 2 x^2 x 2 ± y \pm\sqrt{y} ± y y ≥ 0 y \ge 0 y ≥ 0 x 3 x^3 x 3 y 3 \sqrt[3]{y} 3 y all y y y x \sqrt{x} x y 2 y^2 y 2 y ≥ 0 y \ge 0 y ≥ 0 1 / x 1/x 1/ x 1 / y 1/y 1/ y y ≠ 0 y \neq 0 y = 0 e x e^x e x ln y \ln y ln y y > 0 y > 0 y > 0 b x b^x b x log b y \log_b y log b y y > 0 y > 0 y > 0 sin x \sin x sin x sin − 1 y \sin^{-1} y sin − 1 y − 1 ≤ y ≤ 1 -1 \le y \le 1 − 1 ≤ y ≤ 1 [ − π 2 , π 2 ] [-\tfrac{\pi}{2}, \tfrac{\pi}{2}] [ − 2 π , 2 π ] cos x \cos x cos x cos − 1 y \cos^{-1} y cos − 1 y − 1 ≤ y ≤ 1 -1 \le y \le 1 − 1 ≤ y ≤ 1 [ 0 , π ] [0, \pi] [ 0 , π ] tan x \tan x tan x tan − 1 y \tan^{-1} y tan − 1 y all y y y ( − π 2 , π 2 ) (-\tfrac{\pi}{2}, \tfrac{\pi}{2}) ( − 2 π , 2 π ) x − c x - c x − c y + c y + c y + c all y y y x / c x/c x / c c y cy cy all y y y c ≠ 0 c \neq 0 c = 0 m x + b mx + b m x + b ( y − b ) / m (y - b)/m ( y − b ) / m all y y y m ≠ 0 m \neq 0 m = 0 − x -x − x − y -y − y all y y y x n x^n x n n n n y n \sqrt[n]{y} n y all y y y x n x^n x n n n n ± y n \pm\sqrt[n]{y} ± n y y ≥ 0 y \ge 0 y ≥ 0 x n \sqrt[n]{x} n x y n y^n y n y ≥ 0 y \ge 0 y ≥ 0 n n n y y y n n n 1 / x 2 1/x^2 1/ x 2 ± 1 / y \pm 1/\sqrt{y} ± 1/ y y > 0 y > 0 y > 0 ∣ x ∣ \lvert x \rvert ∣ x ∣ ± y \pm y ± y y ≥ 0 y \ge 0 y ≥ 0 ln x \ln x ln x e y e^y e y all y y y log b x \log_b x log b x b y b^y b y all y y y 10 x 10^x 1 0 x log 10 y \log_{10} y log 10 y y > 0 y > 0 y > 0 sin − 1 x \sin^{-1} x sin − 1 x sin y \sin y sin y − π 2 ≤ y ≤ π 2 -\tfrac{\pi}{2} \le y \le \tfrac{\pi}{2} − 2 π ≤ y ≤ 2 π f ( g ( x ) ) f(g(x)) f ( g ( x )) g − 1 ( f − 1 ( y ) ) g^{-1}(f^{-1}(y)) g − 1 ( f − 1 ( y )) undo the outer function first
A function has an inverse only where it is one-to-one; that is why x 2 x^2 x 2 sin x \sin x sin x f − 1 f^{-1} f − 1 f f f y = x y = x y = x
1.6 Basic rules of algebra
Rule Statement commutative a + b = b + a a + b = b + a a + b = b + a a b = b a ab = ba ab = ba associative ( a + b ) + c = a + ( b + c ) (a + b) + c = a + (b + c) ( a + b ) + c = a + ( b + c ) ( a b ) c = a ( b c ) (ab)c = a(bc) ( ab ) c = a ( b c ) distributive a ( b + c ) = a b + a c a(b + c) = ab + ac a ( b + c ) = ab + a c identity a + 0 = a a + 0 = a a + 0 = a a ⋅ 1 = a a \cdot 1 = a a ⋅ 1 = a inverse a + ( − a ) = 0 a + (-a) = 0 a + ( − a ) = 0 a ⋅ 1 a = 1 a \cdot \tfrac{1}{a} = 1 a ⋅ a 1 = 1 zero product a b = 0 ⇒ a = 0 ab = 0 \Rightarrow a = 0 ab = 0 ⇒ a = 0 b = 0 b = 0 b = 0
Expanding and factoring:
( 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 ( a + b ) 3 = a 3 + 3 a 2 b + 3 a b 2 + b 3 a 3 − b 3 = ( a − b ) ( a 2 + a b + b 2 ) a 3 + b 3 = ( a + b ) ( a 2 − a b + b 2 ) ( x + p ) ( x + q ) = x 2 + ( p + q ) x + p q ( a x + b ) ( c x + d ) = a c x 2 + ( a d + b c ) x + b d \begin{aligned}
(a + b)^2 &= a^2 + 2ab + b^2 \\
(a - b)^2 &= a^2 - 2ab + b^2 \\
(a + b)(a - b) &= a^2 - b^2 \\
(a + b)^3 &= a^3 + 3a^2 b + 3ab^2 + b^3 \\
a^3 - b^3 &= (a - b)(a^2 + ab + b^2) \\
a^3 + b^3 &= (a + b)(a^2 - ab + b^2) \\
(x + p)(x + q) &= x^2 + (p + q)x + pq \\
(ax + b)(cx + d) &= acx^2 + (ad + bc)x + bd
\end{aligned} ( a + b ) 2 ( a − b ) 2 ( a + b ) ( a − b ) ( a + b ) 3 a 3 − b 3 a 3 + b 3 ( x + p ) ( x + q ) ( a x + b ) ( c x + d ) = a 2 + 2 ab + b 2 = a 2 − 2 ab + b 2 = a 2 − b 2 = a 3 + 3 a 2 b + 3 a b 2 + b 3 = ( a − b ) ( a 2 + ab + b 2 ) = ( a + b ) ( a 2 − ab + b 2 ) = x 2 + ( p + q ) x + pq = a c x 2 + ( a d + b c ) x + b d Names: in a b c + d e = 0 abc + de = 0 ab c + d e = 0 a b c abc ab c d e de d e a a a b b b c c c d d d e e e
1.7 Solving quadratic equations
a x 2 + b x + c = 0 ⟹ x = − b ± b 2 − 4 a c 2 a ax^2 + bx + c = 0
\quad\Longrightarrow\quad
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} a x 2 + b x + c = 0 ⟹ x = 2 a − b ± b 2 − 4 a c Discriminant Δ = b 2 − 4 a c \Delta = b^2 - 4ac Δ = b 2 − 4 a c Roots Δ > 0 \Delta > 0 Δ > 0 two distinct real roots Δ = 0 \Delta = 0 Δ = 0 one repeated real root x = − b / 2 a x = -b/2a x = − b /2 a Δ < 0 \Delta < 0 Δ < 0 two complex conjugate roots
Form Expression Reads off standard a x 2 + b x + c ax^2 + bx + c a x 2 + b x + c y y y c c c factored a ( x − r 1 ) ( x − r 2 ) a(x - r_1)(x - r_2) a ( x − r 1 ) ( x − r 2 ) roots r 1 , r 2 r_1, r_2 r 1 , r 2 vertex a ( x − h ) 2 + k a(x - h)^2 + k a ( x − h ) 2 + k vertex ( h , k ) (h, k) ( h , k ) h = − b / 2 a h = -b/2a h = − b /2 a
Completing the square: x 2 + b x = ( x + b 2 ) 2 − b 2 4 x^2 + bx = \left(x + \tfrac{b}{2}\right)^2 - \tfrac{b^2}{4} x 2 + b x = ( x + 2 b ) 2 − 4 b 2 r 1 + r 2 = − b a r_1 + r_2 = -\tfrac{b}{a} r 1 + r 2 = − a b r 1 r 2 = c a r_1 r_2 = \tfrac{c}{a} r 1 r 2 = a c
1.8 Exponents
Rule Form product a m a n = a m + n a^m a^n = a^{m+n} a m a n = a m + n quotient a m / a n = a m − n a^m / a^n = a^{m-n} a m / a n = a m − n power of a power ( a m ) n = a m n (a^m)^n = a^{mn} ( a m ) n = a mn power of a product ( a b ) n = a n b n (ab)^n = a^n b^n ( ab ) n = a n b n power of a quotient ( a / b ) n = a n / b n (a/b)^n = a^n / b^n ( a / b ) n = a n / b n zero a 0 = 1 a^0 = 1 a 0 = 1 a ≠ 0 a \neq 0 a = 0 negative a − n = 1 / a n a^{-n} = 1/a^n a − n = 1/ a n fractional a 1 / n = a n a^{1/n} = \sqrt[n]{a} a 1/ n = n a a m / n = a m n a^{m/n} = \sqrt[n]{a^m} a m / n = n a m roots a b = a b \sqrt{ab} = \sqrt{a}\sqrt{b} ab = a b a / b = a / b \sqrt{a/b} = \sqrt{a}/\sqrt{b} a / b = a / b one a 1 = a a^1 = a a 1 = a 1 n = 1 1^n = 1 1 n = 1 zero base 0 n = 0 0^n = 0 0 n = 0 n > 0 n > 0 n > 0 0 0 0^0 0 0 1 1 1 negative fractional a − m / n = 1 a m n a^{-m/n} = \dfrac{1}{\sqrt[n]{a^m}} a − m / n = n a m 1 negative quotient ( a / b ) − n = ( b / a ) n (a/b)^{-n} = (b/a)^n ( a / b ) − n = ( b / a ) n negative base ( − a ) n = a n (-a)^n = a^n ( − a ) n = a n n n n − a n -a^n − a n n n n root of a power a n n = ∣ a ∣ \sqrt[n]{a^n} = \lvert a \rvert n a n = ∣ a ∣ n n n = a = a = a n n n nested roots a n m = a m n \sqrt[m]{\sqrt[n]{a}} = \sqrt[mn]{a} m n a = mn a root of a root index a m n = ( a n ) m \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m n a m = ( n a ) m rationalising 1 a = a a \dfrac{1}{\sqrt{a}} = \dfrac{\sqrt{a}}{a} a 1 = a a 1 a + b = a − b a − b \dfrac{1}{\sqrt{a} + \sqrt{b}} = \dfrac{\sqrt{a} - \sqrt{b}}{a - b} a + b 1 = a − b a − b any base as e e e a x = e x ln a a^x = e^{x \ln a} a x = e x l n a same base equal a m = a n ⇒ m = n a^m = a^n \Rightarrow m = n a m = a n ⇒ m = n a > 0 a > 0 a > 0 a ≠ 1 a \neq 1 a = 1 same exponent equal a n = b n ⇒ a = b a^n = b^n \Rightarrow a = b a n = b n ⇒ a = b n n n a = ± b a = \pm b a = ± b n n n scientific notation d . d d d × 10 n d.ddd \times 10^n d . ddd × 1 0 n
Fractional powers with even n n n a ≥ 0 a \ge 0 a ≥ 0 \sqrt{\;}
Not rules: ( a + b ) n ≠ a n + b n (a + b)^n \neq a^n + b^n ( a + b ) n = a n + b n a + b ≠ a + b \sqrt{a + b} \neq \sqrt{a} + \sqrt{b} a + b = a + b a m a n ≠ a m n a^m a^n \neq a^{mn} a m a n = a mn a m b n ≠ ( a b ) m + n a^m b^n \neq (ab)^{m+n} a m b n = ( ab ) m + n ( a m ) n ≠ a m n (a^m)^n \neq a^{m^n} ( a m ) n = a m n a ( m n ) a^{(m^n)} a ( m n ) ( a m ) n = a m n (a^m)^n = a^{mn} ( a m ) n = a mn
1.9 Logarithms
log b x = y \log_b x = y log b x = y b y = x b^y = x b y = x b > 0 b > 0 b > 0 b ≠ 1 b \neq 1 b = 1 x > 0 x > 0 x > 0
Rule Form product log b ( x y ) = log b x + log b y \log_b(xy) = \log_b x + \log_b y log b ( x y ) = log b x + log b y quotient log b ( x / y ) = log b x − log b y \log_b(x/y) = \log_b x - \log_b y log b ( x / y ) = log b x − log b y power log b ( x n ) = n log b x \log_b(x^n) = n \log_b x log b ( x n ) = n log b x base log b b = 1 \log_b b = 1 log b b = 1 log b 1 = 0 \log_b 1 = 0 log b 1 = 0 inverse b log b x = x b^{\log_b x} = x b l o g b x = x log b ( b x ) = x \log_b(b^x) = x log b ( b x ) = x change of base log b x = log c x log c b = ln x ln b \log_b x = \dfrac{\log_c x}{\log_c b} = \dfrac{\ln x}{\ln b} log b x = log c b log c x = ln b ln x reciprocal log b ( 1 / x ) = − log b x \log_b(1/x) = -\log_b x log b ( 1/ x ) = − log b x root log b x n = 1 n log b x \log_b \sqrt[n]{x} = \dfrac{1}{n}\log_b x log b n x = n 1 log b x swap log b a = 1 log a b \log_b a = \dfrac{1}{\log_a b} log b a = log a b 1 power of base log b n x = 1 n log b x \log_{b^n} x = \dfrac{1}{n}\log_b x log b n x = n 1 log b x log 1 / b x = − log b x \log_{1/b} x = -\log_b x log 1/ b x = − log b x exponent swap x log b y = y log b x x^{\log_b y} = y^{\log_b x} x l o g b y = y l o g b x equality log b x = log b y ⟺ x = y \log_b x = \log_b y \iff x = y log b x = log b y ⟺ x = y sign log b x < 0 \log_b x < 0 log b x < 0 0 < x < 1 0 < x < 1 0 < x < 1 > 0 > 0 > 0 x > 1 x > 1 x > 1 b > 1 b > 1 b > 1 solving a x = c a^x = c a x = c x = log a c = ln c ln a x = \log_a c = \dfrac{\ln c}{\ln a} x = log a c = ln a ln c solving log b x = c \log_b x = c log b x = c x = b c x = b^c x = b c any base a x = e x ln a a^x = e^{x \ln a} a x = e x l n a ln ( a x ) = x ln a \ln(a^x) = x \ln a ln ( a x ) = x ln a
Not rules: log b ( x + y ) ≠ log b x + log b y \log_b(x + y) \neq \log_b x + \log_b y log b ( x + y ) = log b x + log b y log b ( x y ) ≠ log b x ⋅ log b y \log_b(xy) \neq \log_b x \cdot \log_b y log b ( x y ) = log b x ⋅ log b y ( log b x ) n ≠ n log b x (\log_b x)^n \neq n \log_b x ( log b x ) n = n log b x log b x log b y ≠ log b ( x / y ) \dfrac{\log_b x}{\log_b y} \neq \log_b(x/y) log b y log b x = log b ( x / y )
Name Notation Base natural ln x \ln x ln x e ≈ 2.71828 e \approx 2.71828 e ≈ 2.71828 common log x \log x log x 10 10 10 binary log 2 x \log_2 x log 2 x 2 2 2
1.10 The Cartesian plane
Quantity Formula point P = ( x , y ) P = (x, y) P = ( x , y ) distance d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 midpoint ( x 1 + x 2 2 , y 1 + y 2 2 ) \left(\tfrac{x_1 + x_2}{2}, \tfrac{y_1 + y_2}{2}\right) ( 2 x 1 + x 2 , 2 y 1 + y 2 ) slope m = y 2 − y 1 x 2 − x 1 = Δ y Δ x m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{\Delta y}{\Delta x} m = x 2 − x 1 y 2 − y 1 = Δ x Δ y quadrants I ( + , + ) (+,+) ( + , + ) ( − , + ) (-,+) ( − , + ) ( − , − ) (-,-) ( − , − ) ( + , − ) (+,-) ( + , − )
Parallel lines share a slope; perpendicular lines have slopes with m 1 m 2 = − 1 m_1 m_2 = -1 m 1 m 2 = − 1
1.11 Functions
f : A → B f : A \to B f : A → B x x x A A A f ( x ) f(x) f ( x ) B B B
Term Meaning domain inputs where f f f image / range outputs { f ( x ) : x ∈ A } \{ f(x) : x \in A \} { f ( x ) : x ∈ A } composition ( g ∘ f ) ( x ) = g ( f ( x ) ) (g \circ f)(x) = g(f(x)) ( g ∘ f ) ( x ) = g ( f ( x )) one-to-one f ( x 1 ) = f ( x 2 ) ⇒ x 1 = x 2 f(x_1) = f(x_2) \Rightarrow x_1 = x_2 f ( x 1 ) = f ( x 2 ) ⇒ x 1 = x 2 even f ( − x ) = f ( x ) f(-x) = f(x) f ( − x ) = f ( x ) y y y odd f ( − x ) = − f ( x ) f(-x) = -f(x) f ( − x ) = − f ( x ) periodic f ( x + T ) = f ( x ) f(x + T) = f(x) f ( x + T ) = f ( x ) zero / root x x x f ( x ) = 0 f(x) = 0 f ( x ) = 0 x x x
Restriction on the domain Reason x ≠ 0 x \neq 0 x = 0 1 / x 1/x 1/ x division by zero x ≥ 0 x \ge 0 x ≥ 0 x \sqrt{x} x real square root x > 0 x > 0 x > 0 ln x \ln x ln x log of a positive number x ≠ π 2 + n π x \neq \tfrac{\pi}{2} + n\pi x = 2 π + nπ tan x \tan x tan x cos x = 0 \cos x = 0 cos x = 0
1.12 Functions reference
Function f ( x ) f(x) f ( x ) Domain Image Notes line m x + b mx + b m x + b R \mathbb{R} R R \mathbb{R} R slope m m m b b b − b / m -b/m − b / m square x 2 x^2 x 2 R \mathbb{R} R [ 0 , ∞ ) [0, \infty) [ 0 , ∞ ) even; vertex at origin square root x \sqrt{x} x [ 0 , ∞ ) [0, \infty) [ 0 , ∞ ) [ 0 , ∞ ) [0, \infty) [ 0 , ∞ ) inverse of x 2 x^2 x 2 x ≥ 0 x \ge 0 x ≥ 0 absolute value ∣ x ∣ \lvert x \rvert ∣ x ∣ R \mathbb{R} R [ 0 , ∞ ) [0, \infty) [ 0 , ∞ ) = x 2 = \sqrt{x^2} = x 2 polynomial a n x n + ⋯ + a 0 a_n x^n + \cdots + a_0 a n x n + ⋯ + a 0 R \mathbb{R} R depends on degree at most n n n n n n C \mathbb{C} C sine sin x \sin x sin x R \mathbb{R} R [ − 1 , 1 ] [-1, 1] [ − 1 , 1 ] odd; period 2 π 2\pi 2 π n π n\pi nπ cosine cos x \cos x cos x R \mathbb{R} R [ − 1 , 1 ] [-1, 1] [ − 1 , 1 ] even; period 2 π 2\pi 2 π = sin ( x + π 2 ) = \sin(x + \tfrac{\pi}{2}) = sin ( x + 2 π ) tangent tan x \tan x tan x x ≠ π 2 + n π x \neq \tfrac{\pi}{2} + n\pi x = 2 π + nπ R \mathbb{R} R odd; period π \pi π = sin x / cos x = \sin x / \cos x = sin x / cos x exponential e x e^x e x R \mathbb{R} R ( 0 , ∞ ) (0, \infty) ( 0 , ∞ ) e 0 = 1 e^0 = 1 e 0 = 1 natural log ln x \ln x ln x ( 0 , ∞ ) (0, \infty) ( 0 , ∞ ) R \mathbb{R} R ln 1 = 0 \ln 1 = 0 ln 1 = 0 ln e = 1 \ln e = 1 ln e = 1 e x e^x e x
Shapes, drawn over each function's usual window:
-3 -2 -1 1 2 3 -2 -1 1 2 3 line x/2 + 1 -3 -2 -1 1 2 3 -1 1 3 5 7 9 square x² 2 4 6 8 -1 1 2 3 square root √x -3 -2 -1 1 2 3 -1 1 2 3 absolute value |x| -2 -1 1 2 -4 -3 -2 -1 1 2 3 4 polynomial x³ − 3x -3 -2 -1 1 2 -1 1 3 5 7 exponential eˣ 1 2 3 4 5 6 7 8 -3 -2 -1 1 2 3 natural log ln x -2π -π π 2π -1 1 sine sin x -2π -π π 2π -1 1 cosine cos x -π π -4 -3 -2 -1 1 2 3 4 tangent tan x 1 2 3 4 1 2 3 4 inverses reflect in y = x ── x²── √x── y = xNew function Effect on the graph of f f f f ( x ) + c f(x) + c f ( x ) + c shift up by c c c f ( x − c ) f(x - c) f ( x − c ) shift right by c c c a f ( x ) a f(x) a f ( x ) stretch vertically by a a a a > 1 a > 1 a > 1 0 < a < 1 0 < a < 1 0 < a < 1 f ( x / a ) f(x / a) f ( x / a ) stretch horizontally by a a a − f ( x ) -f(x) − f ( x ) reflect in the x x x f ( − x ) f(-x) f ( − x ) reflect in the y y y f − 1 ( x ) f^{-1}(x) f − 1 ( x ) reflect in the line y = x y = x y = x
General sinusoid: A sin ( ω x + ϕ ) + C A \sin(\omega x + \phi) + C A sin ( ω x + ϕ ) + C A A A 2 π / ω 2\pi/\omega 2 π / ω − ϕ / ω -\phi/\omega − ϕ / ω C C C
Each plot shows f f f
-3 -2 -1 1 2 3 -1 1 3 5 7 9 shifts ── f(x) = x²── f(x) + 2── f(x − 2)-2π -π π 2π -2 -1 1 2 stretches ── f(x) = sin x── 2 f(x)── f(2x)-3 -2 -1 1 2 3 -8 -6 -4 -2 2 4 6 8 reflections ── f(x) = eˣ── −f(x)── f(−x)-2π -π π 2π -2 -1 1 2 3 4 sinusoid ── sin x── 2 sin(2(x − 0.5)) + 11.14 Geometry
Shape Area / surface Perimeter / volume triangle 1 2 b h \tfrac{1}{2} b h 2 1 bh a + b + c a + b + c a + b + c square s 2 s^2 s 2 4 s 4s 4 s rectangle ℓ w \ell w ℓ w 2 ℓ + 2 w 2\ell + 2w 2 ℓ + 2 w parallelogram b h b h bh trapezoid 1 2 ( a + b ) h \tfrac{1}{2}(a + b) h 2 1 ( a + b ) h circle π r 2 \pi r^2 π r 2 2 π r 2\pi r 2 π r sphere 4 π r 2 4\pi r^2 4 π r 2 4 3 π r 3 \tfrac{4}{3}\pi r^3 3 4 π r 3 cylinder 2 π r h + 2 π r 2 2\pi r h + 2\pi r^2 2 π r h + 2 π r 2 π r 2 h \pi r^2 h π r 2 h cone π r ℓ + π r 2 \pi r \ell + \pi r^2 π r ℓ + π r 2 1 3 π r 2 h \tfrac{1}{3}\pi r^2 h 3 1 π r 2 h box 2 ( ℓ w + w h + ℓ h ) 2(\ell w + w h + \ell h) 2 ( ℓ w + w h + ℓ h ) ℓ w h \ell w h ℓ w h
Pythagoras: a 2 + b 2 = c 2 a^2 + b^2 = c^2 a 2 + b 2 = c 2 c c c A = s ( s − a ) ( s − b ) ( s − c ) A = \sqrt{s(s-a)(s-b)(s-c)} A = s ( s − a ) ( s − b ) ( s − c ) s = a + b + c 2 s = \tfrac{a+b+c}{2} s = 2 a + b + c π \pi π n n n ( n − 2 ) π (n - 2)\pi ( n − 2 ) π
1.15 Trigonometry
For a right triangle with angle θ \theta θ o o o a a a h h h
sin θ = o h , cos θ = a h , tan θ = o a = sin θ cos θ \sin\theta = \frac{o}{h}, \qquad \cos\theta = \frac{a}{h}, \qquad \tan\theta = \frac{o}{a} = \frac{\sin\theta}{\cos\theta} sin θ = h o , cos θ = h a , tan θ = a o = cos θ sin θ θ \theta θ rad sin \sin sin cos \cos cos tan \tan tan 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 30 30 30 π / 6 \pi/6 π /6 1 2 \tfrac{1}{2} 2 1 3 2 \tfrac{\sqrt{3}}{2} 2 3 1 3 \tfrac{1}{\sqrt{3}} 3 1 45 45 45 π / 4 \pi/4 π /4 2 2 \tfrac{\sqrt{2}}{2} 2 2 2 2 \tfrac{\sqrt{2}}{2} 2 2 1 1 1 60 60 60 π / 3 \pi/3 π /3 3 2 \tfrac{\sqrt{3}}{2} 2 3 1 2 \tfrac{1}{2} 2 1 3 \sqrt{3} 3 90 90 90 π / 2 \pi/2 π /2 1 1 1 0 0 0 undefined 180 180 180 π \pi π 0 0 0 − 1 -1 − 1 0 0 0 270 270 270 3 π / 2 3\pi/2 3 π /2 − 1 -1 − 1 0 0 0 undefined
Reciprocals: csc = 1 / sin \csc = 1/\sin csc = 1/ sin sec = 1 / cos \sec = 1/\cos sec = 1/ cos cot = 1 / tan \cot = 1/\tan cot = 1/ tan θ \theta θ ( cos θ , sin θ ) (\cos\theta, \sin\theta) ( cos θ , sin θ ) π / 180 \pi/180 π /180 2 π 2\pi 2 π
Any triangle with sides a , b , c a, b, c a , b , c A , B , C A, B, C A , B , C
sin A a = sin B b = sin C c , c 2 = a 2 + b 2 − 2 a b cos C \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c},
\qquad
c^2 = a^2 + b^2 - 2ab\cos C a sin A = b sin B = c sin C , c 2 = a 2 + b 2 − 2 ab cos C 1.16 Trigonometric identities
sin 2 θ + cos 2 θ = 1 1 + tan 2 θ = sec 2 θ sin ( a ± b ) = sin a cos b ± cos a sin b cos ( a ± b ) = cos a cos b ∓ sin a sin b tan ( a ± b ) = tan a ± tan b 1 ∓ tan a tan b sin 2 θ = 2 sin θ cos θ cos 2 θ = cos 2 θ − sin 2 θ = 2 cos 2 θ − 1 = 1 − 2 sin 2 θ sin 2 θ = 1 2 ( 1 − cos 2 θ ) cos 2 θ = 1 2 ( 1 + cos 2 θ ) sin a sin b = 1 2 [ cos ( a − b ) − cos ( a + b ) ] cos a cos b = 1 2 [ cos ( a − b ) + cos ( a + b ) ] sin a cos b = 1 2 [ sin ( a + b ) + sin ( a − b ) ] \begin{aligned}
\sin^2\theta + \cos^2\theta &= 1 \\
1 + \tan^2\theta &= \sec^2\theta \\
\sin(a \pm b) &= \sin a \cos b \pm \cos a \sin b \\
\cos(a \pm b) &= \cos a \cos b \mp \sin a \sin b \\
\tan(a \pm b) &= \frac{\tan a \pm \tan b}{1 \mp \tan a \tan b} \\
\sin 2\theta &= 2 \sin\theta \cos\theta \\
\cos 2\theta &= \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta \\
\sin^2\theta &= \tfrac{1}{2}(1 - \cos 2\theta) \\
\cos^2\theta &= \tfrac{1}{2}(1 + \cos 2\theta) \\
\sin a \sin b &= \tfrac{1}{2}[\cos(a - b) - \cos(a + b)] \\
\cos a \cos b &= \tfrac{1}{2}[\cos(a - b) + \cos(a + b)] \\
\sin a \cos b &= \tfrac{1}{2}[\sin(a + b) + \sin(a - b)]
\end{aligned} sin 2 θ + cos 2 θ 1 + tan 2 θ sin ( a ± b ) cos ( a ± b ) tan ( a ± b ) sin 2 θ cos 2 θ sin 2 θ cos 2 θ sin a sin b cos a cos b sin a cos b = 1 = sec 2 θ = sin a cos b ± cos a sin b = cos a cos b ∓ sin a sin b = 1 ∓ tan a tan b tan a ± tan b = 2 sin θ cos θ = cos 2 θ − sin 2 θ = 2 cos 2 θ − 1 = 1 − 2 sin 2 θ = 2 1 ( 1 − cos 2 θ ) = 2 1 ( 1 + cos 2 θ ) = 2 1 [ cos ( a − b ) − cos ( a + b )] = 2 1 [ cos ( a − b ) + cos ( a + b )] = 2 1 [ sin ( a + b ) + sin ( a − b )] Symmetry Identity parity sin ( − θ ) = − sin θ \sin(-\theta) = -\sin\theta sin ( − θ ) = − sin θ cos ( − θ ) = cos θ \cos(-\theta) = \cos\theta cos ( − θ ) = cos θ cofunction sin ( π 2 − θ ) = cos θ \sin(\tfrac{\pi}{2} - \theta) = \cos\theta sin ( 2 π − θ ) = cos θ cos ( π 2 − θ ) = sin θ \cos(\tfrac{\pi}{2} - \theta) = \sin\theta cos ( 2 π − θ ) = sin θ half turn sin ( θ + π ) = − sin θ \sin(\theta + \pi) = -\sin\theta sin ( θ + π ) = − sin θ cos ( θ + π ) = − cos θ \cos(\theta + \pi) = -\cos\theta cos ( θ + π ) = − cos θ period sin ( θ + 2 π ) = sin θ \sin(\theta + 2\pi) = \sin\theta sin ( θ + 2 π ) = sin θ tan ( θ + π ) = tan θ \tan(\theta + \pi) = \tan\theta tan ( θ + π ) = tan θ
1.17 Circles and polar coordinates
Quantity Formula circle, center origin x 2 + y 2 = r 2 x^2 + y^2 = r^2 x 2 + y 2 = r 2 circle, center ( h , k ) (h,k) ( h , k ) ( x − h ) 2 + ( y − k ) 2 = r 2 (x - h)^2 + (y - k)^2 = r^2 ( x − h ) 2 + ( y − k ) 2 = r 2 parametric x = r cos θ x = r\cos\theta x = r cos θ y = r sin θ y = r\sin\theta y = r sin θ arc length s = r θ s = r\theta s = r θ θ \theta θ sector area A = 1 2 r 2 θ A = \tfrac{1}{2} r^2 \theta A = 2 1 r 2 θ chord length 2 r sin ( θ / 2 ) 2r\sin(\theta/2) 2 r sin ( θ /2 )
Polar ( r , θ ) (r, \theta) ( r , θ ) ( x , y ) (x, y) ( x , y )
x = r cos θ , y = r sin θ r = x 2 + y 2 , θ = atan2 ( y , x ) \begin{aligned}
x &= r\cos\theta, & y &= r\sin\theta \\
r &= \sqrt{x^2 + y^2}, & \theta &= \operatorname{atan2}(y, x)
\end{aligned} x r = r cos θ , = x 2 + y 2 , y θ = r sin θ = atan2 ( y , x ) 1.18 Ellipse
x 2 a 2 + y 2 b 2 = 1 ( a ≥ b ) \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \qquad (a \ge b) a 2 x 2 + b 2 y 2 = 1 ( a ≥ b ) Quantity Formula semi-axes a a a b b b foci ( ± c , 0 ) (\pm c, 0) ( ± c , 0 ) c = a 2 − b 2 c = \sqrt{a^2 - b^2} c = a 2 − b 2 eccentricity ε = c / a \varepsilon = c/a ε = c / a 0 ≤ ε < 1 0 \le \varepsilon < 1 0 ≤ ε < 1 area π a b \pi a b π ab defining property sum of distances to the foci is 2 a 2a 2 a parametric x = a cos t x = a\cos t x = a cos t y = b sin t y = b\sin t y = b sin t polar (focus at origin) r ( θ ) = a ( 1 − ε 2 ) 1 + ε cos θ r(\theta) = \dfrac{a(1 - \varepsilon^2)}{1 + \varepsilon\cos\theta} r ( θ ) = 1 + ε cos θ a ( 1 − ε 2 )
1.19 Parabola
Form Vertex Opens Focus / directrix y = a x 2 + b x + c y = ax^2 + bx + c y = a x 2 + b x + c x = − b / 2 a x = -b/2a x = − b /2 a up if a > 0 a > 0 a > 0 y = a ( x − h ) 2 + k y = a(x - h)^2 + k y = a ( x − h ) 2 + k ( h , k ) (h, k) ( h , k ) up if a > 0 a > 0 a > 0 focus ( h , k + 1 4 a ) (h, k + \tfrac{1}{4a}) ( h , k + 4 a 1 ) y = k − 1 4 a y = k - \tfrac{1}{4a} y = k − 4 a 1 x 2 = 4 p y x^2 = 4py x 2 = 4 p y ( 0 , 0 ) (0, 0) ( 0 , 0 ) up if p > 0 p > 0 p > 0 focus ( 0 , p ) (0, p) ( 0 , p ) y = − p y = -p y = − p y 2 = 4 p x y^2 = 4px y 2 = 4 p x ( 0 , 0 ) (0, 0) ( 0 , 0 ) right if p > 0 p > 0 p > 0 focus ( p , 0 ) (p, 0) ( p , 0 ) x = − p x = -p x = − p
Defining property: every point is equidistant from the focus and the directrix. Eccentricity ε = 1 \varepsilon = 1 ε = 1
1.20 Hyperbola
x 2 a 2 − y 2 b 2 = 1 \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 a 2 x 2 − b 2 y 2 = 1 Quantity Formula vertices ( ± a , 0 ) (\pm a, 0) ( ± a , 0 ) foci ( ± c , 0 ) (\pm c, 0) ( ± c , 0 ) c = a 2 + b 2 c = \sqrt{a^2 + b^2} c = a 2 + b 2 asymptotes y = ± b a x y = \pm \tfrac{b}{a} x y = ± a b x eccentricity ε = c / a > 1 \varepsilon = c/a > 1 ε = c / a > 1 defining property difference of distances to the foci is 2 a 2a 2 a vertical form y 2 a 2 − x 2 b 2 = 1 \tfrac{y^2}{a^2} - \tfrac{x^2}{b^2} = 1 a 2 y 2 − b 2 x 2 = 1 rectangular x y = k xy = k x y = k
Conics by eccentricity: circle 0 0 0 0 < ε < 1 0 < \varepsilon < 1 0 < ε < 1 1 1 1 > 1 > 1 > 1
1.21 Solving systems of linear equations
Method Steps substitution solve one equation for a variable; substitute into the other elimination scale equations so a variable cancels when they are added graphical the solution is where the lines cross determinant Cramer's rule, below
For a 1 x + b 1 y = c 1 a_1 x + b_1 y = c_1 a 1 x + b 1 y = c 1 a 2 x + b 2 y = c 2 a_2 x + b_2 y = c_2 a 2 x + b 2 y = c 2
D = a 1 b 2 − a 2 b 1 , x = c 1 b 2 − c 2 b 1 D , y = a 1 c 2 − a 2 c 1 D D = a_1 b_2 - a_2 b_1,
\qquad
x = \frac{c_1 b_2 - c_2 b_1}{D},
\qquad
y = \frac{a_1 c_2 - a_2 c_1}{D} D = a 1 b 2 − a 2 b 1 , x = D c 1 b 2 − c 2 b 1 , y = D a 1 c 2 − a 2 c 1 D D D Solutions D ≠ 0 D \neq 0 D = 0 exactly one D = 0 D = 0 D = 0 none (parallel lines) or infinitely many (same line)
1.22 Compound interest
Scheme Amount after t t t simple A = P ( 1 + r t ) A = P(1 + rt) A = P ( 1 + r t ) compounded yearly A = P ( 1 + r ) t A = P(1 + r)^t A = P ( 1 + r ) t compounded n n n A = P ( 1 + r n ) n t A = P\left(1 + \tfrac{r}{n}\right)^{nt} A = P ( 1 + n r ) n t continuous A = P e r t A = Pe^{rt} A = P e r t
Effective annual rate: ( 1 + r n ) n − 1 (1 + \tfrac{r}{n})^n - 1 ( 1 + n r ) n − 1 t = ln 2 ln ( 1 + r ) ≈ 72 100 r t = \dfrac{\ln 2}{\ln(1 + r)} \approx \dfrac{72}{100r} t = ln ( 1 + r ) ln 2 ≈ 100 r 72 P = A ( 1 + r ) − t P = A(1 + r)^{-t} P = A ( 1 + r ) − t
1.23 Set notation
Symbol Reads { a , b , c } \{a, b, c\} { a , b , c } the set containing a a a b b b c c c { x : P ( x ) } \{x : P(x)\} { x : P ( x )} the set of x x x P ( x ) P(x) P ( x ) x ∈ A x \in A x ∈ A x x x A A A x ∉ A x \notin A x ∈ / A x x x A A A A ⊆ B A \subseteq B A ⊆ B A A A B B B A ⊂ B A \subset B A ⊂ B proper subset A ∪ B A \cup B A ∪ B union: in A A A B B B A ∩ B A \cap B A ∩ B intersection: in both A ∖ B A \setminus B A ∖ B difference: in A A A B B B A ˉ \bar{A} A ˉ A c A^c A c complement ∅ \varnothing ∅ the empty set ∣ A ∣ \lvert A \rvert ∣ A ∣ number of elements A × B A \times B A × B Cartesian product: pairs ( a , b ) (a, b) ( a , b ) ∀ \forall ∀ ∃ \exists ∃ for all, there exists
Interval Set [ a , b ] [a, b] [ a , b ] { x : a ≤ x ≤ b } \{x : a \le x \le b\} { x : a ≤ x ≤ b } ( a , b ) (a, b) ( a , b ) { x : a < x < b } \{x : a < x < b\} { x : a < x < b } [ a , b ) [a, b) [ a , b ) { x : a ≤ x < b } \{x : a \le x < b\} { x : a ≤ x < b } [ a , ∞ ) [a, \infty) [ a , ∞ ) { x : x ≥ a } \{x : x \ge a\} { x : x ≥ a }
Laws: A ∪ B = B ∪ A A \cup B = B \cup A A ∪ B = B ∪ A A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ) A \cap (B \cup C) = (A \cap B) \cup (A \cap C) A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ) A ∪ B ‾ = A ˉ ∩ B ˉ \overline{A \cup B} = \bar{A} \cap \bar{B} A ∪ B = A ˉ ∩ B ˉ A ∩ B ‾ = A ˉ ∪ B ˉ \overline{A \cap B} = \bar{A} \cup \bar{B} A ∩ B = A ˉ ∪ B ˉ
1.24 Math problems
A checklist for working through the chapter's problem sets.
Step Do read name the unknown; write down what is given with units model pick the equation or identity that links given and unknown solve isolate the unknown symbolically before substituting numbers check units, sign, magnitude, special cases (0 0 0 ∞ \infty ∞ verify substitute the answer back into the original equation
References
Ivan Savov, No bullshit guide to math & physics (opens in a new tab) OpenStax, College Algebra 2e (opens in a new tab) OpenStax, Precalculus 2e (opens in a new tab) Paul's Online Math Notes: Algebra (opens in a new tab) Paul's Online Math Notes: Trig cheat sheet (PDF) (opens in a new tab) Wikipedia: List of trigonometric identities (opens in a new tab) NIST DLMF, chapter 4: Elementary functions (opens in a new tab) Khan Academy: Algebra 2 (opens in a new tab) Trigonometry (opens in a new tab) 