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Base Number Systems: Part 2 Honors Precalculus
Mr. Velazquez
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Reminder: General Statement
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Any natural number 𝑘 can be expressed as the sum of a series of powers of any natural number base 𝑏.
𝑘 = 𝑎𝑛𝑏
𝑛 + 𝑎𝑛−1𝑏𝑛−1 +⋯+ 𝑎2𝑏
2 + 𝑎1𝑏 + 𝑎0
Where 𝑎𝑛, 𝑎𝑛−1, … , 𝑎2, 𝑎1 and 𝑎0 are all equal or greater than zero and are the numeric digits of 𝑘 in base 𝑏.
Binary:
101102 = 1 24 + 0 23 + 1 22 + 1 21 + 0 20 = 2210
1100112 = 1(25) + 1(24) + 0(23) + 0(22) + 1(21) + 1(20) = 5110
Ternary:
22013 = 2 33 + 2 32 + 0 31 + 1 30 = 7310
121023 = 1(34) + 2(33) + 1(32) + 0(31) + 2(30) = 146
Bases Larger than 10
In this second part, we will examine base systems larger than 10, starting with the most popular: the base-12 duodecimal or dozenal system
Schoolhouse Rock, “Little Twelve Toes”
https://www.youtube.com/watch?v=_uJsoZheTR4
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Base 12: Dozenal System • Imagine that instead of two five-fingered hands, humans were instead born with
two six-fingered hands. • In this world, our number system might be in base 12. This would require using a
total of twelve symbols—the symbols from 0 to 9, plus one for 10 (called “dek”) and one for 11 (called “el”). Usually these are X and ℇ, respectively, but sometimes A and B are used. The number twelve is called “do” or “dozen.”
• Fun Fact: There is a society of people called the Dozenal Society who would like to see base 12 replace base 10 permanently
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ℇ
1
2
3 4 5
6 7
8 9 X
10
“dek”
“el”
“do”
Base 12: Dozenal System
1 2 3 4 5 6 7 8 9 X ℇ 10 1 2 3 4 5 6 7 8 9 10 11 12
11 12 13 14 15 16 17 18 19 1X 1ℇ 20 13 14 15 16 17 18 19 20 21 22 23 24
21 22 23 24 25 26 27 28 29 2X 2ℇ 30 25 26 27 28 29 30 31 32 33 34 35 36
31 32 33 34 35 36 37 38 39 3X 3ℇ 40 37 38 39 40 41 42 43 44 45 46 47 48 5
Base 12: Converting to Dozenal • Remember to select the highest
power of 12 less than the number you’re trying to convert, and also consider how many times you can subtract it (i.e. division).
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120 = 1 121 = 12 122 = 144 123 = 1,728 124 = 20,736 125 = 248,832 126 = 2,985,984
The largest power of 12 less than 131 is 121
13110
121 120
131 − 10 121 = 11 − 11 100 = 0
x10 x11
So we use the symbols for 10 and 11: 13110 = Xℇ12
Base 12: Converting to Dozenal
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120 = 1 124 = 20,736
121 = 12 125 = 248,832
122 = 144 126 = 2,985,984
123 = 1,728 127 = 35,831,808
56110 185710
Base 12: Converting to Dozenal
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120 = 1 124 = 20,736
121 = 12 125 = 248,832
122 = 144 126 = 2,985,984
123 = 1,728 127 = 35,831,808
40210 172710
Try it yourself! Convert the following decimal numbers into dozenal.
Base 12: Converting from Dozenal
9
120 = 1 124 = 20,736
121 = 12 125 = 248,832
122 = 144 126 = 2,985,984
123 = 1,728 127 = 35,831,808
X 3 7 2 1 ℇ 9 120 121 122 120 121 122 123
10 122 + 3 121 + 7(120) 2 123 + 1 122 + 11 121 + 9(120)
12 12
= 148310 = 374110
Base 12: Converting from Dozenal
10
120 = 1 124 = 20,736
121 = 12 125 = 248,832
122 = 144 126 = 2,985,984
123 = 1,728 127 = 35,831,808
142012 3XX512
Base 12: Converting from Dozenal
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120 = 1 124 = 20,736
121 = 12 125 = 248,832
122 = 144 126 = 2,985,984
123 = 1,728 127 = 35,831,808
ℇ22412 3X70112
Try it yourself! Convert the following dozenal numbers into decimal.
Base 12: Multiplication Table
12 This number has kind of a disgusting name in dozenal. Any idea what it is?
Base 16: Hexadecimal System • The base-16 hexadecimal system
may seem like overkill, but it has very practical purposes, mostly in programming
• We need 6 extra symbols here, so we use the letters from A to F to replace the numbers 10 through 15.
• Since 16 is a power of 2, we can now convert long, tedious binary numbers into shorter hexadecimal numbers quickly.
• 16 = 24, so one single digit in hexadecimal can be used to replace 4 binary digits (or “bits”)
• Because of this, we do not need to know the powers of 16 to convert from hexadecimal to binary.
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DECIMAL BINARY HEXADECIMAL
0 0000 0
1 0001 1
2 0010 2
3 0011 3
4 0100 4
5 0101 5
6 0110 6
7 0111 7
8 1000 8
9 1001 9
10 1010 A
11 1011 B
12 1100 C
13 1101 D
14 1110 E
15 1111 F
16 10000 10
Base 16: Converting from Binary
10110010101011100100
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B 2 A E 8
Which means 𝟏𝟎𝟏𝟏𝟎𝟎𝟏𝟎𝟏𝟎𝟏𝟎𝟏𝟏𝟏𝟎𝟎𝟏𝟎𝟎𝟐 = 𝐁𝟐𝐀𝐄𝟖𝟏𝟔
11101111011000000111 E F 6 0 7
Which means 𝟏𝟏𝟏𝟎𝟏𝟏𝟏𝟏𝟎𝟏𝟏𝟎𝟎𝟎𝟎𝟎𝟎𝟏𝟏𝟏𝟐 = 𝐄𝐅𝟔𝟎𝟕𝟏𝟔
Base 16: Converting from Binary Use the table to convert the following long binary numbers into shorter hexadecimal numbers.
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1101001001012 =
11010011101011112 =
11001010110010102 =
Going Further: Base 36 • Extending our logic, if we jumped up to base 36, we would need
26 extra symbols to represent the numbers 10 through 35, so we would utilize the entire alphabet.
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Base10 Base36 Base10 Base36 Base10 Base36 Base10 Base36
0 0 10 A 20 K 30 U
1 1 11 B 21 L 31 V
2 2 12 C 22 M 32 W
3 3 13 D 23 N 33 X
4 4 14 E 24 O 34 Y
5 5 15 F 25 P 35 Z
6 6 16 G 26 Q 36 10
7 7 17 H 27 R 37 11
8 8 18 I 28 S 38 12
9 9 19 J 29 T 39 13
Base 36: Converting from Decimal Using the base-36 system, certain numbers can be written using only letters, and we can convert strings of letters into a number by assigning each letter to a power of 36. For instance:
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88,597,882,558,33110 = VELAZQUEZ36
67610 = IS36
23,727,593,89410 = AWESOME36
= 𝟑𝟏 368 + 𝟏𝟒 367 + 𝟐𝟏 366 + 𝟏𝟎 365 + 𝟑𝟓 364 + 𝟐𝟔 363 + 𝟑𝟎(362) + 𝟏𝟒(361) + 𝟑𝟓(360)
= 𝟏𝟎 366 + 𝟑𝟐 365 + 𝟏𝟒 364 + 𝟐𝟖 363 + 𝟐𝟒(362) + 𝟐𝟐(361) + 𝟏𝟒(360)
= 𝟏𝟖(361) + 𝟐𝟖(360)
V E L A Z Q U E Z
I S
A W E S O M E
Your Assignment… (Due 11/24)
CHOOSE ONE (or both, if you want extra credit)
• Create a multiplication table using any base from 2 through 36 (NOT base 10, for obvious reasons). Must be at least 12 rows by 12 columns. All numbers in the table must be expressed in the chosen base. Color and/or decorate the table to make it look presentable, and make sure to check your values. Poster board or white printer paper only (NO LINED PAPER).
• Create a base-36 name tag. Fold a sheet of white paper in half (hot dog fold), and on the outside write and decorate your name expressed as a number in base-36, and its equivalent in base-10. Inside the fold, show the work proving that this number does indeed give the letters of your name.
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88,597,882,558,33110 = VELAZQUEZ36
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