Download tmsca math formulas and memorization

MATH STUDY GUIDE
By Intellectuality123
1)
2)
3)
4)
5)
6)
7)
8)
Basic Memorization
Number Theory
Algebra
Geometry
Advanced Algebra
Precalculus
Competition Math
Miscellaneous
1. Basic Memorization
i.
x
1
Squares, Cubes, 4th powers
X2
X3
1
1
2
4
8
3
9
27
4
16
64
5
25
125
6
36
216
7
49
343
8
64
512
9
81
729
10
100
1000
11
121
1331
12
144
1728
13
169
2197
14
196
2744
15
225
3375
16
256
4096
17
289
4913
18
324
5832
18
19
20
21
22
23
24
25
26
27
28
29
30
324
361
400
441
484
529
576
625
676
729
784
841
900
5832
6859
8000
9261
10648
12167
13824
15625
17576
19683
21952
24389
27000
*To square any integer X...
(X+N)(X-N)+N2
Make X+N or X-N a base 10 number.
*To square any number x between 41 and 49...
25-(50-x) (50-x)2
First 2 digits
Last 2 digits
*To square any number x between 51 and 59...
25+(units digit of x) (units digit of x)2
First 2 digits
last 2 digits
2. NUMBER THEORY
Fundamental Theorem of Arithmetic
“Every integer can be written as the product of 2 or more primes; Every integer has its
own unique prime factorization.”
i. Divisibility Rules










2- # is even
3- sum of digits divisible by 3
4- last 2 digits divisible by 4
5- ends in 5 or 0
6- even and divisible by 3
7- no good trick
8- last 3 digits divisible by 8
9- sum of digits divisible by 9
10- ends in 0
11-sum of ones, hundreds, ten thousands, etc. digits - sum of tens, thousands,
hundred thousands, etc. digits= 0 or divisible by 11
ii. Primes
 There are an infinite number of prime numbers.
P1*P2*...*Pn +1 is prime.
 To check if a number is prime, check divisibility of all primes less than the square
root of that number.
iii. Factors
of
X1a1*X2a2*...*Xnan, where X is prime...



Number of Factors
(a1+1)(a2+2)...(an+1)
Sum of factors
(X01+X12+...+X1a1)(X02+X12+...+X2a2)...(X0n+X1n+...+Xnan)
Product of Factors
n(x/2), where n is the number, and x is the number of divisors.
iv. 142857, the first cyclic number
142857*2=285714
142857*3=571428
...
142857*9=999999
1 / 7  0.142857repeating
v. Types of numbers
n(n  1)
 Triangular
2
 Rectangular n(n+1)
n(3n  1)
 Pentagonal
2
n ( 4 n  2)
 Hexagonal
2
n ( 6 n  4)
 Octagonal
2
n(7 n  5)
 Nonagonal
2
n(m  2)n  (m  4)
 M-gonal
2
3. ALGEBRA
Fundamental Theorem of Algebra:
P( x)  an ( x  r1 )( x  r2 )...( x  rn )
i. Means

a1  a2  ...  an
n
Geometric mean n a1a2 ...an

Harmonic Mean

Arithmetic mean
n
(1 / a1 )  (1 / a2 )  ...  (1 / an )
ii. Sequences and series
 Arithmetic series x  ( x  d )  ( x  2d )  ...  [ x  (n  1)d ] =A
( x  x )n
A 1 n
2
x x
n  n 1 1
d
 Geometric Series a  ar  ar 2  ...  ar n  Z
1 r n 

Z  a
 1 r 
 Infinite Geometric Series a  ar  ar 2  ...  Z , where r<1
a
Z
1 r
iii. LINEAR AND QUADRATIC EQUATIONS
 Linear Equations
y  y2
Slope: m  1
x1  x2
Slope-Intercept Form y  mx  b
Point-Slope Form y1  y2  mx1  x2 
Standard Form Ax  By  C
Slope: -A/B

Distance Between 2 points ( x2  x1 ) 2  ( y2  y1 ) 2
Quadratic Equations
Standard Form ax 2  bx  c  y
Quadratic Formula
 b  b 2  4ac
x
2a
b  4ac  b 2

y
Vertex Form/Complete the Square  x   
2a 
4a

Vieta’s Formulas: Sum of Roots: c/a
Product of Roots: -b/a
2
iv. Expanding, Factoring,
 Expanding
a (b  c)  ac  bc
(a  b) 2  a 2  2ab  b 2
(a  b) 2  a 2  2ab  b 2
(a  b)(c  d )  ab  ad  bc  bd
(a  b) 3  a 3  3a 2b  3ab 2  b 3

(a  b) 3  a 3  3a 2b  3ab 2  b 3
Factoring
(Sorry, factoring kind of got deleted, I’ll move on)
a n  1  (a  1)(a n  a n1  ...  a1  a 0 )
 Binomial Theorem
x  an n Coa0 xn n C1a1xn1 n C2a2 xn2  ... n Cn1an1x1 n Cn an x0
m(m  1)( 2m  1)
6

Sum of first m squares

 m( m  1) 
Sum of first m cubes 

2



Sum of first m alternating squares 
2
m( m  1)
2
4. GEOMETRY
i. Area
 Area of any regular polygon, side length S, number of sides N, apothem A, or Radius
R
When you know the side length S
S 2N
Area 
4 tan(180 / N )
When you know the Apothem A
 180 
Area  A2 N tan 

 N 
When you know the Radius R
R 2 N sin 360 / N 
Area 
2
Also, Area=ap/2, where a is apothem, p is perimeter
 Area of various polygons
(Sorry, I haven’t memorized all this yet, I’ll update it as soon as possible!)
 Area of Conics
Circle Area  r 2
Ellipse Area  ab , where a is semimajor axis and b is semiminor axis
 Area of Triangles
-Heron’s Formula s(s  a)( s  b)( s  c)  Area
-a, b, are side lengths,  is angle in between
1
a  b  sin 
2
 Area of Quadrilateral (s  a)( s  b)( s  c)( s  d )
ii. Perimeter
For a polygon, add together all the side lengths
Circle Perimeter  2r
Ellipse perimeter approximations
a2  b2
2
2) Perimeter   3(a  b)  3(a  b)(a  3b)
1) Perimeter  2

h

( a  b)
( a  b) 2
2
3)
3h


Perimeter   (a  b)1 

10

4

3
h


Note: I will not include the basic volume formulas and surface area formulas, since they
are so common and won’t be forgotten.
*Surface area of cone ( I always forget this one ;))
r 2  rl , where l is slant height
iv. Diagonals and angles
 # of Distinct Diagonals from one vertex of a polygon
n-3
 # of total diagonals
n(n  3)
2
 Diagonal length in a square: s 2




Diagonal length in a rectangle: l 2  w2
d 2  2s 2 in a square
Basically, diagonal squared= 2*area
Inner Diagonal length in a cube s 3


Inner Diagonal length in a rectangular prism l 2  w2  h 2
Surface area of cube when known diagonal 2d 2

 d 
Volume of a cube when known diagonal 

 3
3
V. Polyhedra/Platonic Solids
Name
Shape of Faces
Tetrahedron
Hexahedron
Triangles
Squares
Number of
Faces
4
6
Number of
Vertices
5
8
Number of
Edges
6
12
vi. Trigonometry
Function
Sin
Cos
Tan
Csc
Sec
Cot
In terms of
Sin and Cos
sin
cos
sin/cos
1/sin
1/cos
cos/sin
Radians to Degrees: ×
Degrees to Radians: ×

180
180

Table of Trig Values (in degrees)
0°
30°
45°
0
1
0
undef.
1
undef.
½
(√3)/2
(√3)/3
2
(2√3)/3
√3
(√2)/2
(√2)/2
1
√2
√2
1
60°
90°
(√3)/2
½
√3
(2√3)/3
2
(√3)/3
1
0
undef.
1
undef.
0