Base Number Systems: Part 2 Honors Precalculus

Mr. Velazquez

1

Reminder: General Statement

2

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

3

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

4

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).

6

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

7

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

8

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

11

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.

13

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

14

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.

15

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.

16

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:

17

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.

18

88,597,882,558,33110 = VELAZQUEZ36