Mathematics 

Author: 
William Pantoja 
Illustrations: 
William Pantoja 
Date: 
January 14, 2012 




Contents
1. Overview
2. Numbers
2.1. Decimal Numbers
2.1.1. Basic Numbers
2.1.2. The "Teens"
2.1.3. The "Tens"
2.1.4. Larger Numbers
2.1.5. Fractions
2.1.6. Roman Numerals
2.1.7. Other Bases
3. Algebra
3.1. Addition
3.2. Subtraction
3.3. Multiplication
3.4. Division
3.5. Exponentiation
1. Overview
2. Numbers
2.1. Decimal Numbers
2.1.1. Basic Numbers
Number 
Name 
Amount 
0 
zero 

1 
one 
• 
2 
two 
•• 
3 
three 
••• 
4 
four 
•••• 
5 
five 
••••• 
6 
six 
••••• • 
7 
seven 
••••• •• 
8 
eight 
••••• ••• 
9 
nine 
••••• •••• 
10 
ten 
••••• ••••• 
2.1.2. The "Teens"
Number 
Name 
Amount 
11 
eleven 
••••• ••••• • 
12 
twelve 
••••• ••••• •• 
13 
thirteen 
••••• ••••• ••• 
14 
fourteen 
••••• ••••• •••• 
15 
fifteen 
••••• ••••• ••••• 
16 
sixteen 
••••• ••••• ••••• • 
17 
seventeen 
••••• ••••• ••••• •• 
18 
eighteen 
••••• ••••• ••••• ••• 
19 
nineteen 
••••• ••••• ••••• •••• 
2.1.3. The "Tens"
Number 
Name 
Amount 
20 
twenty 
two tens 
30 
thirty 
three tens 
40 
fourty 
four tens 
50 
fifty 
five tens 
60 
sixty 
six tens 
70 
seventy 
seven tens 
80 
eighty 
eight tens 
90 
ninety 
nine tens 
2.1.4. Larger Numbers
A comma is used with larger numbers to group digits in sets of three. After one million some of these definitions differ depending on which country you live in. The definitions below are used in the United States and France.
Number 
Name 
Amount 
100 
one hundred 
ten tens 
1,000 
one thousand 
ten one hundreds 
10,000 
ten thousand 
ten one thousands 
100,000 
one hundred thousand 
one hundred one thousands 
1,000,000 
one million 
one thousand one thousands 
1,000,000,000 
one billion 
one thousand one millons 
1,000,000,000,000 
one trillion 
one thousand one billions 
1,000,000,000,000,000 
one quadrillion 
one thousand one trillions 
1,000,000,000,000,000,000 
one quintillion 
one thousand one quadrillions 
1,000,000,000,000,000,000,000 
one sextillion 
one thousand one quintillions 
1,000,000,000,000,000,000,000,000 
one septillion 
one thousand one sextillions 
1,000,000,000,000,000,000,000,000,000 
one octillion 
one thousand one septillions 
1 followed by 100 zeros 
one googol 

1 followed by a googol zeros 
one googolplex 

2.1.5. Fractions
Digits to the right of the decimal point represent the fractional value of a number. Each decimal place is one tenth the magnitude of the decimal place to its immediate left. When writing our pronouncing a decimal number greater than one, the decimal point is written and prounounced as "and".
Number 
Name 
Fraction 
0.1 
one tenth 
1/10 
0.01 
one hundredth 
1/100 
0.001 
one thousandth 
1/1000 
0.0001 
one ten thousandth 
1/10000 
0.00001 
one hundred thousandth 
1/100000 
Examples:
0.27
is pronounced twentyseven hundredths.
329.582
is pronounced three hundred twentynine and five hundred eightytwo thousandths.
$7,293.23
is pronounced seven thousand two hundred ninetythree and twentythree dollars.
2.1.6. Roman Numerals
• 
Roman numerals has no representation of zero. 
• 
The numbers are created starting from the largest digit on the left (skipping any zeros) to the smallest on the right. 
• 
If a numeral to the immediate left of a numeral is smaller, the numeral to the left is subtracted from the numeral on the right (e.g., IX = 10  1 = 9). 
Number 
Value 
I 
1 
II 
2 
III 
3 
IV 
4 
V 
5 
VI 
6 
VII 
7 
VIII 
8 
IX 
9 
X 
10 
XL 
40 
L 
50 
XC 
90 
C 
100 
CD 
400 
D 
500 
CM 
900 
M 
1,000 
MV 
4,000 
V 
5,000 
MX 
9,000 
X 
10,000 
XL 
40,000 
L 
50,000 
XC 
90,000 
C 
100,000 
CD 
400,000 
D 
500,000 
CM 
900,000 
M 
1,000,000 
Examples:
LXXXVII
is 87.
MMCDXCIII
is 2,493.
DCMMXXXXL
is 602,350.
2.1.7. Other Bases
Numbers may be represented in other bases than 10. A single digit in a number is a value between 0 and one less then the base inclusive (e.g., base 8 uses the digits 0 through 7). Some bases, such as base sixteen (or hexadecimal) use alpha characters to represent digits greater than nine. A number written in a base other than base 10 (decimal) will have its base written in subscript after the number (e.g., 2538 is a base eight number).
Decimal (base10) 
Binary (base2) 
Octal (base8) 
Hexadecimal (base16) 
0 
02 
08 
016 
1 
12 
18 
116 
2 
102 
28 
216 
3 
112 
38 
316 
4 
1002 
48 
416 
5 
1012 
58 
516 
6 
1102 
68 
616 
7 
1112 
78 
716 
8 
10002 
108 
816 
9 
10012 
118 
916 
10 
10102 
128 
A16 
11 
10112 
138 
B16 
12 
11002 
148 
C16 
13 
11012 
158 
D16 
14 
11102 
168 
E16 
15 
11112 
178 
F16 
16 
100002 
208 
1016 
17 
100012 
218 
1116 
18 
100102 
228 
1216 
19 
100112 
238 
1316 
20 
101002 
248 
1416 
21 
101012 
258 
1516 
22 
101102 
268 
1616 
23 
101112 
278 
1716 
24 
110002 
308 
1816 
3. Algebra
3.1. Addition
Identity
a + 0 = a
Inverse
a + (a) = 0
Associativity
(a + b) + c = a + (b + c)
Commutativity
a + b = b + a
3.2. Subtraction
Identity
a  0 = a
Inverse
a  a = 0
Associativity
(a  b)  c = a  (b  c)
Commutativity
Subtraction has no commutativity.
a  b ≠ b  a
3.3. Multiplication
Identity
a × 1 = a
Inverse
a ÷ a = 1
Associativity
(a × b) × c = a × (b × c)
Commutativity
a × b = b × a
3.4. Division
Identity
a ÷ 1 = a
Inverse
a × a = 1
Associativity
(a ÷ b) ÷ c = a ÷ (b ÷ c)
Commutativity
Division has no commutativity.
a ÷ b ≠ b ÷ a
3.5. Exponentiation
an = a × a × a × a × … × a (n number of times)
Identity
an × am = an + m
(an)m = an × m
(a × b)n = an × bn
Inverse
alogab = b
elna = a
Associativity
Exponentiation has no associativity.
abc ≠ (ab)c
Commutativity
Exponentiation has no commutativity.
ab ≠ ba
Properties
Exponentiation has these properties.
a0 = 1
Note that in the case of 00 the value may differ depending on the application. In general, 00 = 1.
a1/n = n√a
an/m = m√an
