3 number systems

1CMPE12c Gabriel Hugh Elkaim
Number Systems

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CMPE12c Gabriel Hugh Elkaim
A Brief History of Numbers
From Gonick, Cartoon Guide to Computer Science

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Prehistoric Ledgers

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Elaborate Finger Counting

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Ancient Number Systems

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Positional Number Systems

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Number Systems
• Prehistory
Unary, or marks:
/////// = 7
/////// + ////// = /////////////
• Grouping lead to Roman Numerals:
VII + V = VVII = XII
• Better, Arabic Numerals:
7 + 5 = 12 = 1 x 10 + 2

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Positional Number System
• Base 10 is a special case of positional
number system
• PNS First used over 4000 years ago in
Mesopotamia (Iraq)
– Base 60
– 0...59 (written as 60 different symbols)
– 5,4560 = 5 x 60 + 45 = 34510
• Positional Number Systems are great for
algebra
• Why?

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Arabic Numerals
• 345 is really
– 3 x 102
+ 4 x 101
+ 5 x 100
– 3 x 100 + 4 x 10 + 5 x 1
– 3 is the most significant symbol (carries the most
weight)
– 5 is the least significant symbol (carries the least
weight)
• Digits (or symbols) allowed: 0-9
• Base (or radix): 10

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Try multiplication in (non-positional) Roman numerals!
XXXIII (33 in decimal)
XII (12 in decimal)
---------
XXXIII
XXXIII
CCCXXX
-----------
CCCXXXXXXXXXIIIIII
-----------
CCCLXXXXVI
-----------
CCCXCVI = 396 in decimal
The
Mesopotamians
wouldn’t have
had this
problem!!
*
+

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• There are many ways to “represent” a number
• Representation does not affect computation result
LIX + XXXIII = LXXXXII (Roman)
59 + 33 = 92 (Decimal)
• Representation affects difficulty of computing results
• Computers need a representation that works with fast
electronic circuits
• Positional numbers work great with 2-state devices

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What ’10’ Means

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Number Base Systems

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Binary Numbers

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The Powers of 2

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Equivalent Numbers

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Converting Binary to Decimal

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•Base (radix): 2
•Digits (symbols) allowed: 0, 1
•Binary Digits, or bits
•10012 is really
1 x 23
+ 0 x 22
+ 0 X 21
+ 1 X 20
910
•110002 is really
1 x 24
+ 1 x 23
+ 0 x 22
+ 0 x 21
+ 0 x 20
2410
Binary Number System

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Computers multiply Arabic numerals by
converting to binary, multiplying and
converting back (much as us with Roman
numerals)
Binary Number System
So if the computer is all binary how does
it multiply 5 by 324 when I type it in the
calculator program?

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Octal Number System
• Base (radix): 8
• Digits (symbols): 0 – 7
• 3458 is really
– 3 x 82
+ 4 x 81
+ 5 x 80
– 192 + 32 + 5
– 22910
• 10018 is really
– 1 x 83
+ 0 x 82
+ 0 x 81
+ 1 x 80
– 512 + 1
– 51310
• In C, octal numbers are represented with a
leading 0 (0345 or 01001).

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Hexadecimal Number System
• Base (radix): 16
• Digits (symbols) allowed: 0 – 9, a – f
Hex Decimal
a 10
b 11
c 12
d 13
e 14
f 15

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A316 is really:
A x 161
+ 3 x 160
160 + 3
16310
3E816 is really:
3 x 162
+ E x 161
+ 8 x 160
3 x 256 + 14 x 16 + 8 x 1
768 + 224 + 8
100010
Some Examples of converting hex numbers
to decimal

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10C16 is really:
1 x 162
+ 0 x 161
+ C x 160
1 x 256 + 12 x 16
256 + 192
44810
In C, hex numbers are represented with a leading
“0x” (for example “0xa3” or “0x10c”).

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For any positional number system
•Base (radix): b
•Digits (symbols): 0 … b – 1
•Sn-1Sn-2….S2S1S0
Use summation to transform any base to
decimal
Value = Σ (Sibi
)
n-1
i=0

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More PNS fun
• 21203 =
• 4035 =
• 2717 =
• 3569 =
• 11102 =
• 2A612 =
• BEEF16 =
=6910
=10310
=4110
=29410
=1410
=41410
=4887910

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Decimal → Binary Conversion
• Divide decimal value by 2 until the value is 0
• Know your powers of two and subtract
… 256 128 64 32 16 8 4 2 1
• Example: 42
• What is the biggest power of two that fits?
• What is the remainder?
• What fits?
• What is the remainder?
• What fits?
• What is the binary representation?

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0
2
1
2
2
2
5
2
10
2
21
2
43
2
86
2
172
2
345
345
10101
1001
→→→→→→
→→→→⇔
b10101100134510 ⇔

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• 12810=
• 31010=
• 2610=

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Binary → Octal Conversion
• Group into 3’s starting at least significant symbol
• Add leading 0’s if needed (why not trailing?)
• Write 1 octal digit for each group
• Examples:
100 010 111 (binary)
4 2 7 (octal)
10 101 110 (binary)
2 5 6 (octal)

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Octal → Binary Conversion
It is simple, just write down the 3-bit binary code for
each octal digit
Octal Binary
0 000
1 001
2 010
3 011
4 100
5 101
6 110
7 111

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Binary → Hex Conversion
• Group into 4’s starting at least significant symbol
| Adding leading 0’s if needed
• Write 1 hex digit for each group
• Examples:
1001 1110 0111 0000
9 e 7 0
0001 1111 1010 0011
1 f a 3

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Hex → Binary Conversion
Again, simply write down the 4 bit binary code for
each hex digit
Example:
3 9 c 8
0011 1001 1100 1000

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Conversion Table
Decimal Hexadecimal Octal Binary
0 0 0 0000
1 1 1 0001
2 2 2 0010
3 3 3 0011
4 4 4 0100
5 5 5 0101
6 6 6 0110
7 7 7 0111
8 8 10 1000
9 9 11 1001
10 A 12 1010
11 B 13 1011
12 C 14 1100
13 D 15 1101
14 E 16 1110
15 F 17 1111

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Hex → Octal
•Do it in 2 steps, hex → binary → octal
Decimal → Hex
•Do it in 2 steps, decimal → binary → hex
So why use hex and octal and not just binary
and decimal?

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Negative Integers
• Most humans precede number with “-”
(e.g., -2000)
• Accountants, however, use parentheses:
(2000) or color 2000
• Sign-magnitude format
• Example: -1000 in hex?
100010 = 3 x 162
+ e x 161
+ 8 x 160
-3E816

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Mesopotamians used positional fractions
Sqrt(2) = 1.24,51,1060
= 1 x 600
+ 24 x 60-1
+ 51 x 60-2
+
10 x 60-3
= 1.41422210
Most accurate approximation until the
Renaissance

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fn-1
fn-2
… f2
f1
f0
f-1
f-2
f-3
… fm-1
Radix point
Generalized Representation
For a number “f” with ‘n’ digits to the left and ‘m’ to the right
of the decimal place
Position is the power

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Fractional Representation
• What is 3E.8F16?
• How about 10.1012?
= 3 x 161
+ E x 160
+ 8 x 16-1
+ F x 16-2
= 48 + 14 + 8/16 + 15/256
= 1 x 21
+ 0 x 20
+ 1 x 2-1
+ 0 x 2-2
+ 1 x 2-3
= 2 + 0 + 1/2 + 1/8

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More PNS Fractional Fun
• 21.0123 =
• 4.1335 =
• 22.617 =
• A.3A12 =

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Converting Decimal → Binary fractions
• Consider left and right of the decimal point
separately.
• The stuff to the left can be converted to binary
as before.
• Use the following table/algorithm to convert the
fraction

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Fraction Fraction x 2 Digit left of decimal point
0.8 1.6 1  most significant (f-1)
0.6 1.2 1
0.2 0.4 0
0.4 0.8 0
0.8 (it must repeat
from here!!)
• Different bases have different repeating fractions.
• 0.810 = 0.110011001100…2 = 0.11002
• Numbers can repeat in one base and not in another.
For 0.810 to binary

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What is 2.210 in:
•Binary
•Hex

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3 number systems

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3 number systems