5  7  2  4  Face value 
10^{3}  10^{2}  10^{1}  10^{0}  Positional Value (powers of 10) 
1000  100  10  1  Resolved positional value 
To figure out the value of 5724, we do the following:
5 x 10^{3} = 5 x 1000 = 

7 x 10^{2} = 7 x 100 = 
700

2 x 10^{1} = 2 x 10 = 
20

4 x 10^{0} = 4 x 1 = 
4

5724

Binary Numbering systems:
The binary numbering system works much the same way as the decimal numbering
system except that now we are in base 2 so we only have 2 digits (0, 1). The
value of the number is still determined by the face value times the positional
value, but since we are in base 2, the positional values are the powers of 2.
Since the face values can only be 0 or 1, this means that the 0 or 1 is multiplied
by the positional place in which it is found.
Example: binary number 1011011
1  0  1  1  0  1  1  Face value 
2^{6}  2^{5}  2^{4}  2^{3}  2^{2}  2^{1}  2^{0}  Positional value 
64  32  16  8  4  2  1  Resolved positional value 
The positional values are first shown in the powers of 2 and then as the resolved
number  in other words, 2 to the 6th is equal to 64.
Converting binary to decimal:
In the previous example to find the decimal equivalent for the number 1011011,
we do the following:
1 x 2^{6} = 1 x 64 = 
64

0 x 2^{5} = 0 x 32 = 
0

1 x 2^{4} = 1 x 16 = 
16

1 x 2^{3} = 1 x 8 = 
8

0 x 2^{2} = 0 x 4 = 
0

1 x 2^{1} = 1 x 2 = 
2

1 x 2^{0} = 1 x 1 = 
1

91

Converting decimal to binary:
Before doing this it is important that we review the decimal equivalent for
the frequently used powers of 2:
2^{0} = 1  2^{1} = 2  2^{2} = 4  2^{3} = 8  2^{4} = 16  2^{5} = 32 
2^{6} = 64  2^{7} = 128  2^{8} = 256  2^{9} = 512  2^{10} = 1024  etc. 
1

_____

_____

_____

_____

_____

_____

2^{6}

2^{5}

2^{4}

2^{3}

2^{2}

2^{1}

2^{0}

64

32

16

8

4

2

1

1

0

_____

_____

_____

_____

_____

2^{6}

2^{5}

2^{4}

2^{3}

2^{2}

2^{1}

2^{0}

64

32

16

8

4

2

1

1

0

1

_____

_____

_____

_____

2^{6}

2^{5}

2^{4}

2^{3}

2^{2}

2^{1}

2^{0}

64

32

16

8

4

2

1

1

0

1

1

_____

_____

_____

2^{6}

2^{5}

2^{4}

2^{3}

2^{2}

2^{1}

2^{0}

64

32

16

8

4

2

1

1

0

1

1

0

_____

_____

2^{6}

2^{5}

2^{4}

2^{3}

2^{2}

2^{1}

2^{0}

64

32

16

8

4

2

1

1

0

1

1

0

1

_____

2^{6}

2^{5}

2^{4}

2^{3}

2^{2}

2^{1}

2^{0}

64

32

16

8

4

2

1

1

0

1

1

0

1

1

2^{6}

2^{5}

2^{4}

2^{3}

2^{2}

2^{1}

2^{0}

64

32

16

8

4

2

1

Binary counting:
Now, we are going to learn to count in binary and relate counting in binary
to counting in decimal. 0 and 1 are the same values in binary and decimal but
then we come to add 1 to 1 and we discover that there is no 2 in binary. Essentially
we have run out of digits. We stop and think what we do in decimal when we run
out of digits and we get the pattern to use in binary. For example, in decimal
when we try to add 1 to 9, we run out of digits.
In decimal: 
9

In binary: 
1


+1

+1


10

10

10

11

100

101

110

111

1000

+ 1

+ 1

+ 1

+ 1

+ 1

+ 1

+ 1

11

100

101

110

111

1000

1001

Decimal

Binary

0

0

1

1

2

10

3

11

4

100

5

101

6

110

7

111

8

1000

9

1001

10

1010

11

1011

12

1100

13

1101

14

1110

15

1111

16

10000

A  3  5  9  face value 
16^{3}  16^{2}  16^{1}  16^{0}  positional value (powers of 16) 
4096  256  16  1  resolved positional value 
A x 16^{3} =  10 x 4096 = 
40960

(note A is equivalent to decimal 10) 
3 x 16^{2} =  3 x 256 = 
768


5 x 16^{1} =  5 x 16 = 
80


9 x 16^{0} =  9 x 1 = 
9




41817

16^{0} = 1  16^{1} = 16  16^{2} = 256  16^{3} = 4096  16^{4} = 65536 
A  _____  _____  _____ 
16^{3}  16^{2}  16^{1}  16^{0} 
4096  256  16  1 
A  3  _____  _____ 
16^{3}  16^{2}  16^{1}  16^{0} 
4096  256  16  1 
A  3  5  _____ 
16^{3}  16^{2}  16^{1}  16^{0} 
4096  256  16  1 
A  3  5  9 
16^{3}  16^{2}  16^{1}  16^{0} 
4096  256  16  1 
Binary

Decimal

Hexadecimal

0

0

0

1

1

1

10

2

2

11

3

3

100

4

4

101

5

5

110

6

6

111

7

7

1000

8

8

1001

9

9

1010

10

A

1011

11

B

1100

12

C

1101

13

D

1110

14

E

1111

15

F

10000

16

10

10001

17

11

10010

18

12

Binary

Hexadecimal

0000

0

0001

1

0010

2

0011

3

0100

4

0101

5

0110

6

0111

7

1000

8

1001

9

1010

A

1011

B

1100

C

1101

D

1110

E

1111

F

1100/  1111  1100 is C and 1111 is F 
C

F


therefore the hexadecimal equivalent of 11001111 is CF 
11010101110101 =  11/  0101/  0111/  0101 
3

5

7

5

5

E

4

9

0101

1110

0100

1001

_  _  _  _  _  _  _  _ 
z  o  n  e/  d  i  g  it 
code  range 
00  A  I 
01  J  R 
10  S  Z 
11  numbers 
digit:  __  __  __  __ 
2^{3}  2^{2}  2^{1}  2^{0}  
8  4  2  1 
0  1  1  0 
2^{3}  2^{2}  2^{1}  2^{0} 
8  4  2  1 
0  1  0  0 
2^{3}  2^{2}  2^{1}  2^{0} 
8  4  2  1 
1  0  0  1 
2^{3}  2^{2}  2^{1}  2^{0} 
8  4  2  1 
0  0  1  0 
2^{3}  2^{2}  2^{1}  2^{0} 
8  4  2  1 
Binary

Hexadecimal

0000

0

0001

1

0010

2

0011

3

0100

4

0101

5

0110

6

0111

7

1000

8

1001

9

1010

A

1011

B

1100

C

1101

D

1110

E

1111

F

1. 
F = 11000110

1100

0110

in EBCDIC 
C

6

hexadecimal translation of EBCDIC  
2. 
M = 11010100

1101

0100

in EBCDIC 
D

4

hexadecimal translation of EBCDIC  
3. 
9 = 11111001

1111

1001

in EBCDIC 
F

9

hexadecimal translation of EBCDIC  
4. 
S = 11100010

1110

0010

in EBCDIC 
E

2

hexadecimal translation of EBCDIC 
Characters

EBCDIC Zone

Hexadecimal representation

A I

1100

C

J  R

1101

D

S  Z

1110

E

numbers

1111

F

A =  1  1  0  0  0  0  0  1  Face value 
2^{7}  2^{6}  2^{5}  2^{4}  2^{3}  2^{2}  2^{1}  2^{0}  Powers of 2  
128  64  32  16  8  4  2  1  Positional value from appropriate power of 2 
Character  EBCDIC Equivalent  Character  EBCDIC Equivalent 
A  193  N  213 
B  194  O  214 
C  195  P  215 
D  196  Q  216 
E  197  R  217 
F  198  S  226 
G  199  T  227 
H  200  U  228 
I  201  V  229 
J  209  W  230 
K  210  X  231 
L  211  Y  232 
M  212  Z  233 
__  __  __  __ 
2^{3}  2^{2}  2^{1}  2^{0} 
8  4  2  1 
0  0  0  0 
2^{3}  2^{2}  2^{1}  2^{0} 
8  4  2  1 
1  0  0  1 
2^{3}  2^{2}  2^{1}  2^{0} 
8  4  2  1 
__  __  __  __  __  face value 
2^{4}  2^{3}  2^{2}  2^{1}  2^{0}  positional value in powers of 2 
16  8  4  2  1  positional value in decimal 
__  __  __  __  1  face value 
2^{4}  2^{3}  2^{2}  2^{1}  2^{0}  positional value in powers of 2 
16  8  4  2  1  positional value in decimal 
0  1  1  0  0  face value 
2^{4}  2^{3}  2^{2}  2^{1}  2^{0}  positional value in powers of 2 
16  8  4  2  1  positional value in decimal 
1  1  0  1  0  face value 
2^{4}  2^{3}  2^{2}  2^{1}  2^{0}  positional value in powers of 2 
16  8  4  2  1  positional value in decimal 
__  __  __  __  __  __  __  face value 
2^{6}  2^{5}  2^{4}  2^{3}  2^{2}  2^{1}  2^{0}  positional value in powers of 2 
64  32  16  8  4  2  1  positional value in decimal 
1  0  0  1  1  0  0  face value 
2^{6}  2^{5}  2^{4}  2^{3}  2^{2}  2^{1}  2^{0}  positional value in powers of 2 
64  32  16  8  4  2  1  positional value in decimal 
1  0  1  1  0  1  0  face value 
2^{6}  2^{5}  2^{4}  2^{3}  2^{2}  2^{1}  2^{0}  positional value in powers of 2 
64  32  16  8  4  2  1  positional value in decimal 
Character  ASCII Equivalent  Character  ASCII Equivalent 
A  65  N  78 
B  66  O  79 
C  67  P  80 
D  68  Q  81 
E  69  R  82 
F  70  S  83 
G  71  T  84 
H  72  U  85 
I  73  V  86 
J  74  W  87 
K  75  X  88 
L  76  Y  89 
M  77  Z  90 