| Mathematics |
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| Author: |
William Pantoja |
| Illustrations: |
William Pantoja |
| Date: |
January 14, 2012 |
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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 |
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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 twenty-seven hundredths.
329.582
is pronounced three hundred twenty-nine and five hundred eighty-two thousandths.
$7,293.23
is pronounced seven thousand two hundred ninety-three and twenty-three 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 (base-10) |
Binary (base-2) |
Octal (base-8) |
Hexadecimal (base-16) |
| 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
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