Lecture 2
Binary Values and Number Systems
Positional Notation
The number 943 is an example of a number written in positional notation. The relative positions of the digits determine the values
they represent. In base-10 each successive digit (reading from right to left) is worth ten times the previous position.
94310 = 9(100) + 4(10) + 3
In modern computers numbers are represented in base-2. This means that 0 and 1 are the only digits permitted and each position in
a multi-digit number is worth two time the previous position.
1101012 = 1(32) + 1(16) + 0(8) + 1(4) + 0(2) + 1 = 5310
Binary, Octal, and Hexadecimal Numbers
Base 2 (binary), base 8 (octal), and base 16 (hexadecimal) are all powers of 2 bases so it is easy to convert between them.
decimal
Base 2 – digits 0,1
Base 8 – digits 0,1,2,3,4,5,6,7
Base 16 – digits 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F
A single binary digit is called a bit. Eight bits is called a byte, four bits is called a nibble.
base 2
10111010001010101111000010101011
we can convert from base 2
to base 8 by evaluating groups
of 3 bits at a time
10 111 010 001 010 101 111 000 010 101 011
we an convert from base 2
to base 16 by evaluating groups
of 4 bits at a time
1011 1010 0010 1010 1111 0000 1010 1011
2 7
B
2
A
1
2 5
2
A
7
F
0
0
2
5
A
3
B
base 8
base 16
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
binary
octal
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
0
1
2
3
4
5
6
7
10
11
12
13
14
15
16
17
hexadecimal
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
F
Converting Between Base 10 and Base 2
Let’s convert 142 to base 2
142/2 = 71
71/2 = 35
35/2 = 17
17/2 = 8
8/2 = 4
4/2 = 2
2/2 = 1
1/2 = 0
remainder 0
remainder 1
remainder 1
remainder 1
remainder 0
remainder 0
remainder 0
remainder 1
Let’s convert 101100101 to base 10
10001110
1
4
32
64
+ 256
357
Left to Right Evaluation
1=1
10 = 2x1 = 2
101 = 2x2 + 1 = 5
1011 = 2x5 + 1 = 11
10110 = 2x11 = 22
101100 = 2x22 = 44
1011001 = 2x44 + 1 = 89
10110010 = 2x89 = 178
101100101 = 2x178 + 1 = 357
Base 2 Arithmetic
optional
Addition
10001010
+ 00101110
10111000
Converting a Number to Two’s Complement*
00011101
11100010
+
1
11100011
Subtraction
this is the base 2 representation for 29
flip all the bits
add 1
this is the two’s complement representation for -29
ignore
overflow
01111011
+ 11100011
this is 123
this is -29
101011110
this is 94
When the binary value has a leading 1, the number is negative...to find the magnitude of this value we need to take the two’s complement
again. Lets add -29 to 12...
00001100
+ 11100011
11101111
most significant digit a 1 means values is negative
*Two’s complement is a special way to represent negative values for base-2 integers.
11101111
00010000
+
1
00010001
magnitude is 17
so 11100011 = -17