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32 Cards in this Set

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Factor

x²-y²
(x + y)(x - y)
Factor

x²+y²
No real factorization

Using complex numbers, with i = sqrt(-1)

(x+yi)(x-yi)
Factor

x³+y³
(x+y)(x²-xy+y²)
Factor

x³-y³
(x-y)(x²+xy+y²)
Factor

x²+2xy+y²
(x+y)²
Pythagorean Theorem
For a right triangle with legs of length a and b, and hypotenuse of length c

a²+b²=c²
Perfect Squares [0,50]
0²=0
1²=1
2²=4
3²=9
4²=16
5²=25
6²=36
7²=49
Perfect Squares [50,250]
8²=64
9²=81
10²=100
11²=121
12²=144
13²=169
14²=196
15²=225
Powers of 2 [1,500]
2^0 = 1
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16
2^5 = 32
2^6 = 64
2^7 = 128
2^8 = 256
Perfect Cubes [1,1000]
1³=1
2³=8
3³=27
4³=64
5³=125
6³=216
7³=343
8³=512
9³=729
Factorials [1,360000]
0! = 1
1! = 1
2! = 2
3! = 6
4! = 24
5! = 120
6! = 720
7! = 5040
8! = 40320
Factor

x^4 - y^4
(x-y)(x+y)(x²+y²)
Factor

x^5 - y^5
(x - y)(x^4 + x³y + x²y² + xy³ + y^4)
Factor

x^5 + y^5
(x + y)(x^4 - x³y + x²y² - xy³ + y^4)
Factor

x^6 - y^6
(x-y)(x+y)(x²+xy+y²)(x²-xy+y²)
Factor

x^6 + y^6
(x² + y²) (x^4 - x²y² + y^4)
ax² + bx + c = 0

i) Discriminant

ii) Quadratic Formula
i) D=b²-4ac

ii) x = [ -b ± sqrt(D)] / 2a


D < 0 -> complex roots
ax²+bx+c=0

Properties of the Discriminant

D = b² - 4ac
i) D>0 and a perfect square
-> 2 rational roots

ii) D>0 and not a perfect square
-> 2 irrational roots

iii) D=0 -> 1 root, multiplicty 2

iv) D<0 -> 2 complex conj. roots
Primative Pythagorean Triples

with perimeter < 100,
a b c p

3 4 5 12
5 12 13 30
8 15 17 40
7 24 25 56
20 21 29 70
12 35 37 84
9 40 41 90
Heron's Formula

for the area A of an arbitrary triangle
with side lengths a, b, c, and
semiperimeter s
A² = s(s-a)(s-b)(s-c)

-or-

A = sqrt[s(s-a)(s-b)(s-c)]
Primes [2,197]
2 29 67 107 157
3 31 71 109 163
5 37 73 113 167
7 41 79 127 173
11 43 83 131 179
13 47 89 137 181
17 53 97 139 191
19 59 101 149 193
23 61 103 151 197
Digits of pi
3.14159 26535 89793 23846
26433 83279 50288 4197 ...

irrational and transcendental

unknown whether normal
Digits of e
2.71828 18284 59045 23536
02874 71352 66249 7757...

Known to be both irrational and transcendental; e is maximally transcendental, with irrationanality measure, mu(e)=2.

it is unknown whether e is normal
The Golden Ratio phi
1.61803 39887 49894 84820
45868 34365 63811 7720...

irrational and algebraic

phi = ½ [ 1 + sqrt(5) ]
The Euler-Mascheroni constant gamma
0.57721 56649 01532 86060
65120 90082 40243 1042...

unknown if irrational
Catalan's Constant K
K = 0.915965594177...

unknown whether irrational
Apéry's constant zeta(3)
zeta(3) = 1.2020569...

known to be irrational

not known to be transcendental or normal
Pascal's Triangle
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
Perfect Number
a # equal to the sum of its factors
6
28
496
8128
33550336
8589869056
137438691328
2305843008139952128
The Hardy-Ramanujan Number
The smallest "taxicab number",
a number that has two representations
as the sum of two cubes.

1729 = 1³+12³ = 9³+10³
Platonic Solids
Solid f v e

Tetrahedron 4 4 6
Cube 6 8 12
Octahedron 8 6 12
Dodecahedron 12 20 30
Icosahedron 20 12 30

Note f+v-e=2
The Kepler-Poinsot Solids
great dodecahedron
great icosahedron
great stellated dodecahedron
small stellated dodecahedron