1. Binary Numbers
 Strings of binary digits (“bits”)
 One bit can store a number from 0 to 1
 n bits can store numbers from 0 to 2n
1

2. Binary – Powers of 2
 Positional representation
 Each digit represents a power of 2
So 101 binary is
1 • 22 + 0 • 21 + 1 • 20
or
1•4 + 0•2 + 1•1=5
2

3. Converting Binary to Decimal
 Easy, just multiply digit by power of 2
 Just like a decimal number is represented
 Example follows
3

4. Binary
Decimal Example
7
6
5
4
3
2
1
0
27
26
25
24
23
22
21
20
128
64
32
16
8
4
2
1
What is 10011100 in decimal?
1
0
0
1
1
1
0
0
128 + 0 + 0 + 16 + 8 + 4 + 0 + 0 = 156
4

5. Decimal to Binary
 A little more work than binary to decimal
 Some examples
 3 = 2 + 1 = 11 (that‟s 1•21 + 1•20)
 5 = 4 + 1 = 101 (that‟s 1•22 + 0•21 + 1•20)
5

6. Algorithm – Decimal to Binary
 Find largest power-of-two smaller than decimal
number
 Make the appropriate binary digit a „1‟
 Subtract the power of 2 from decimal
 Do the same thing again
6

7. Decimal
Binary Example
 Convert 28 decimal to binary
32 is too large, so use 16
Binary
10000
Decimal
Next is 8
Binary
11000
Decimal
Next is 4
Binary
11100
Decimal
28 – 16 = 12
12 – 8 = 4
4–4=0
7
6
5
4
3
2
1
0
27
26
25
24
23
22
21
20
128
64
32
16
8
4
2
1
7

8. Hexadecimal
 Strings of 0s and 1s too hard to write
 Use base-16 or hexadecimal – 4 bits
Dec
Bin
Hex
Dec
Bin
Hex
0
0000
0
8
1000
8
1
0001
1
9
1001
9
2
0010
2
10
1010
?
3
0011
3
11
1011
?
4
0100
4
12
1100
?
5
0101
5
13
1101
?
6
0110
6
14
1110
?
7
0111
7
15
1111
?
8

9. Hexadecimal
 Letters to represent 10-15
Dec
Bin
Hex
Dec
Bin
Hex
0
0000
0
8
1000
8
1
0001
1
9
1001
9
2
0010
2
10
1010
a
3
0011
3
11
1011
b
4
0100
4
12
1100
c
5
0101
5
13
1101
d
6
0110
6
14
1110
e
•Power of 2
7
0111
7
15
1111
f
•Size of byte
Why use
base 16?
9

10. Hex to Binary
Bin
Hex
0000
0
0001
1
0010
2
 Convention – write 0x before number
0011
3
 Hex to Binary – just convert digits
0100
4
0101
5
0110
6
0111
7
1000
8
1001
9
1010
a
1011
b
1100
c
1101
d
1110
e
1111
f
0x2ac
0010
1010 1100
0x2ac = 001010101100
No magic – remember hex digit = 4 bits
10

11. Binary to Hex
Bin
0000
0011
0100
0101
3
7
b
101001101111011 = 0x537b
5
0110
1011
4
6
0111
7
1000
8
9
a
1011
b
1100
c
1101
5
0111
3
1010
0011
2
1001
0101
1
0010
101001101111011
0
0001
 Just convert groups of 4 bits
Hex
d
1110
e
1111
f
11

12. Hex to Decimal
Dec
Hex
0
0
 Just multiply each hex digit by decimal value,
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
a
11
b
12
c
13
d
14
e
15
f
and add the results.
0x2ac
2 • 256

+ 10 • 16 + 12 • 1 = 684



163
162
161
160
4096
256
16
1
12

13. Decimal to Hex
1.
2.
3.
4.
Analogous to decimal
binary.
Find largest power-of-16 smaller than decimal
number
Divide by power-of-16. The integer result is hex
digit.
The remainder is new decimal number.
Do the same thing again
13

14. Decimal to Hex
Dec
Hex
0
0
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
a
11
b
12
c
13
d
160
14
e
1
15
f
684
0x2__
684/256 = 2
684%256 = 172
0x2a_
172%16 = 12 = c 0x2ac
172/16 = 10 = a

163
4096

162
256

161
16

14

15. Octal
 Octal is base 8
 Similar to hexadecimal
 Conversions
 Less convenient for use with 8-bit bytes
15

16. Arithmetic -- addition
 Binary similar to decimal arithmetic
1
No carries
0
1
1
0
0
0
1
1
0
0
1
0
1
1
0
+ 1
0
0
0
1
+ 1
0
1
1
1
1
1
1
0
1
1
1
1
0
Carries
1
0
1+1 is 2 (or 102), which results in a carry
16

17. Arithmetic -- subtraction
0
1
-
0
1
1
1
0
0
1
0
0
0
1
0
0
-
1
1
0
1
1
1
0
1
0
0
1
1
0
0
0
1
No borrows
1
0
1
1
Borrows
0 - 1 results in a borrow
17

18. Arithmetic -- multiplication
1
0
1
1
1
0
1
1
0
1
1
0
0
0
0
1
0
1
1
1
1
0
1
X
1
Successive additions of
multiplicand or zero,
multiplied by 2 (102).
Note that multiplication
by 102 just shifts bits
left.
1
18

19. Hexadecimal Arithmetic
 Similar
 If you‟re doing by hand, easiest to convert each set
of digits to decimal and back
19