Binary numbers represent values using strings of binary digits (bits) that are equal to powers of two. Converting between binary, decimal, and hexadecimal involves representing the value as a sum of place values and multiplying/dividing place values by powers of two or sixteen. Arithmetic in binary follows similar rules to decimal, with carries for addition and borrows for subtraction, and multiplication accomplished by successive additions and left shifts. Hexadecimal uses groups of four bits and letters to represent values for convenience.

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