Chapter 2
Binary Numbers and Number Systems
Number Categories – Ch. 2.1
Definition of
 number
 natural numbers
 negative numbers
 integers
 rational numbers
Positional notation – Ch. 2.2
base: b
any integer > 1
digits: 0, 1, ..., b−1
number
d n 1d n  2 d 2 d1d 0
its definition
d n 1  b n 1  d n  2  b n  2    d 2  b 2  d1  b1  d 0  b 0
Examples:
base
2
5
8
10
16
digits
0,1
0,1,2,3,4
0,1,2,3,4,5,6,7
0,1,2,3,4,5,6,7,8,9
0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F
CSE 1520 – Week 2.2
January 15, 2013
page 1
Binary, Octal and Hexadecimal
Decimal
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
Binary
Octal
0
1
10
11
100
101
110
111
1000
1001
1010
1011
1100
1101
1110
1111
10000
10001
10010
10011
10100
10101
10110
10111
Hexadecimal
0
1
2
3
4
5
6
7
10
11
12
13
14
15
16
17
20
21
22
23
24
25
26
27
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
F
10
11
12
13
14
15
16
17
etc.
Arithmetic in Other Bases
All the familiar rules and techniques of pencil-and-paper
decimal arithmetic carry over to any other base.
Addition
46
+ 27
Subtraction
101110
+ 11011
CSE 1520 – Week 2.2
5037
− 95
57
− 6
January 15, 2013
111001
−
110
page 2
Conversions between Decimal and Binary
Binary to Decimal
Technique
- use the definition of a number in a positional
number system with base 2
- evaluate the definition formula using decimal arithmetic
Example
543210 ← position = corresponding power of 2
||||||
101011 = 1  25 + 0  2 4 + 1  23 + 0  2 2 + 1  21 + 1  2 0 = 43 (decimal)
generalize to Octal to Decimal, Hexadecimal to Decimal
Decimal to Binary
Technique
- repeatedly divide by 2
- remainder is the next digit
- binary number is developed right to left
Example
173
86
43
21
10
5
2
1








2
2
2
2
2
2
2
2
86
43
21
10
5
2
1
0
1
0
1
1
0
1
0
1
1
01
101
1101
01101
101101
0101101
10101101
generalize to Decimal to Octal, Decimal to Hexadecimal
CSE 1520 – Week 2.2
January 15, 2013
page 3
Conversions between Binary and Octal/Hexadecimal
Binary to Octal
Technique
- group bits into threes, right to left
- convert each such group to an octal digit
Example
1011010111 = 1 011 010 111 = 1327 (octal)
Binary to Hexadecimal
Technique
- group bits into fours, right to left
- convert each such group to a hexadecimal digit
Example
1011001011 = 10 1100 1011 = 2CB (hexadecimal)
Octal to Binary
Technique
- convert each octal digit to a three-bit binary representation
Example
705 = 111 000 101 = 111000101 (binary)
CSE 1520 – Week 2.2
January 15, 2013
page 4
Hexadecimal to Binary
Technique
- convert each octal digit to a four-bit binary representation
Example
10AF = 0001 0000 1010 1111 = 1000010101111 (binary)
What about converting between Octal and Hexadecimal?
CSE 1520 – Week 2.2
January 15, 2013
page 5