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Binary System

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Binary System. Common terms might describe file size or memory size: Bit : smallest unit of information Byte : a grouping of eight bits of information K : (kilobyte); 1024 bytes equals 1K of storage. Bits, Bytes. MB : (megabyte); about 1 million bytes of information - PowerPoint PPT Presentation

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Page 1: Binary System

Binary System

Page 2: Binary System

Bits, Bytes

Common terms might describe file size or memory size:

Bit: smallest unit of information

Byte: a grouping of eight bits of information

K: (kilobyte); 1024 bytes equals 1K of storage.

Page 3: Binary System

Bits, Bytes

MB: (megabyte); about 1 million bytes of information

GB: (gigabyte); about 1 billion bytes of information

TB: (terabyte); about 1 million megabytes of information

Page 4: Binary System

• Numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ...

• Use 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9  

• We know that 5304 = 5*103+3*102+0*101+4*100

• Base 10

Decimal System

Page 5: Binary System

• BINARY numbers: 0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001,

1010, 1011, .... • Written with only 2 digits: "0" and "1" • In the same way as for decimal, 1011 (binary)

= 1*23+0*22+1*21+1*20= 11 (decimal) • Base 2 • Converting from binary to decimal is simple,

just as for 1011 above.

Binary System

Page 6: Binary System

• How to know if 1011 is in binary or in decimal? Subscripts are used to show the base: 10112 (binary number), 101110

(decimal number)

Binary System

Page 7: Binary System

Decimal      Binary            0               0000             1               0001                            2               0010                            3               0011                           4               0100                            5               0101                            6               0110                            7               0111                            8              1000                           9              1001                          10             1010                          11             1011                    

Positive Binary Integers

Page 8: Binary System

Decimal

Binary

÷ 2 * 2

Page 9: Binary System

(35)10 = ( … )2

17

35

4

8

÷2

÷2

÷2

÷2

1

1

0

0

1

2

÷2

÷2

1

0

(110001)2

NO

Example 1 Convert Decimal to binary

Page 10: Binary System

Example 2 Convert Decimal to binary(45)10 = ( … )2

22

45

5

11

÷2

÷2

÷2

÷2

0

1

1

1

1

2

÷2

÷2

1

0

(101101)2

NO

Page 11: Binary System

Example 3 Convert Decimal to binary(55)10 = ( … )2

27

55

6

13

÷2

÷2

÷2

÷2

1

1

0

1

1

3

÷2

÷2

1

1

(110111)2

NO

Page 12: Binary System

Example 4 Convert Decimal to binary

(72)10 = ( … )2

36

72

9

18

÷2

÷2

÷2

÷2

0

0

1

0

2

4

÷2

÷2

0

0

(1001000)2

NO÷2 11

Page 13: Binary System

Example 5 Convert Decimal to binary

(85)10 = ( … )2

42

85

10

21

÷2

÷2

÷2

÷2

0

1

0

1

2

5

÷2

÷2

0

1

(1010101)2

NO÷2 11

Page 14: Binary System

Example 6 convert binary to decimal

( 100101)2 = (….)10

1*20+0*21+1*22+0*23+0*24+1*25

= 37

Page 15: Binary System

Example 7 convert binary to decimal

( 111101)2 = (….)10

1*20+0*21+1*22+1*23+1*24+1*25

= 61

Page 16: Binary System

Example 8 convert binary to decimal

( 111111)2 = (….)10

1*20+1*21+1*22+1*23+1*24+1*25

= 63

Page 17: Binary System

Example 9 convert binary to decimal

( 1100000)2 = (….)10

0*20+0*21+0*22+0*23+0*24+1*25+1*22

= 96

Page 18: Binary System

Example 10 convert binary to decimal

( 1101001)2 = (….)10

1*20+0*21+0*22+1*23+0*24+1*25+1*22

= 105

Page 19: Binary System

Example 11 Convert decimal fraction into binary

(0.125)10 = ( … )2

0.250

0.125

0.50

2

2

20.50

0.250

0

1

0***

integer

1

.001

Page 20: Binary System

Example 12 Convert decimal fraction into binary

(0.0625)10 = ( … )2

0.1250

0.0625

0.2500

2

2

20.2500

0.1250

0

0

0***

integer

0.5000

.0001

0.5000 * 2 1 1

Page 21: Binary System

Example 13 Convert decimal fraction into binary

(0.03125)10 = ( … )2

0.06250

0.03125

0.12500

2

2

2

0.12500

0.06250

0

0

0*

*

*

integer

0.25000

.00001

0.25000 * 2 0.5000 0

0.5000 * 2 1 1

Page 22: Binary System

Example 14 convert fraction binary to decimal

( .0001)2 = (….)10

0*2-1 + 0*2-2 +0*2-3+ 1*2-4

= 1/16 =0.0625

0.0625

Page 23: Binary System

Example 15 convert fraction binary to decimal

( .0011)2 = (….)10

0*2-1 + 0*2-2 +1*2-3+ 1*2-4

0.0625

1875016

1

8

1.

Page 24: Binary System

Example 16 Convert decimal into binary

(35.125)10 = ( … )2

17

35

4

8

2

2

2

2

1

1

0

0

1

2

2

2

1

0

NO

0.250 *2

0.125 *2

0.50 *2

0.50

0.250

1

integer

1

0

0

100011.001

.001

100011

Page 25: Binary System

Example 17 Convert decimal into binary

(42.125)10 = ( … )2

21

42

5

10

2

2

2

2

1

0

1

0

1

2

2

2

1

0

NO

0.250 *2

0.125 *2

0.50 *2

0.50

0.250

1

integer

1

0

0

101010.001

.001

101010

Page 26: Binary System

Example 18 Convert decimal into binary

(39.125)10 = ( … )2

19

39

4

9

2

2

2

2

1

1

0

1

1

2

2

2

1

0

NO

0.250 *2

0.125 *2

0.50 *2

0.50

0.250

1

integer

1

0

0

100111.001

.001

101010

Page 27: Binary System

1 + 0 = 10 + 1 = 10 + 0 = 01 + 1 = 0 carry 1

Addition

Page 28: Binary System

Example 19

11000 + 1100 100100

1

Page 29: Binary System

Example 20

11000 + 1 1100 110100

1

Page 30: Binary System

Example 21

1011000 + 111100 10010100

111

Page 31: Binary System

Example 22

100100.000 + 1111101.001 10100001.001

111

Page 32: Binary System

Example 23

1 100101.000 + 1111101.001 11100010.001

111

Page 33: Binary System

1 - 1 = 0 1 - 0 = 10 - 0 = 00 - 1 = 1 borrow 1

Subtraction

Page 34: Binary System

Ex 24

11000 - 1100 1100

borrow 1

Page 35: Binary System

Ex 25

1111101.000 - 100100.001 1011000.111

borrow 1

Page 36: Binary System

Ex 26

11110 - 1101 10001

borrow 1

Page 37: Binary System

Ex 27

11111.000 - 110.001 11000.111

borrow 1

Page 38: Binary System

1 * 0 = 00 * 1 = 00 * 0 = 01 * 1 = 1

Multiplication

Page 39: Binary System

Example 28

11000 * 110 00000

11000 11000 10010000

Page 40: Binary System

Example 29

11111 * 11

11111 11111

1011101

Page 41: Binary System

1101.1 * 110.01 11011

00000 00000

1101111011

1010000.011

Example 30

Page 42: Binary System

11111 * 11101 11111

00000 11111

1111111111

1110000011

Example 31

Page 43: Binary System

11.1 * 1.11 111

111 111101.001

Example 32

Page 44: Binary System

Exercises

1- Convert Decimal to binary

(25)10 = ( … )2

(80)10 = ( … )2

(59.125)10 = ( … )2

(31.0625)10 = ( … )2

(125)10 = ( … )2

Page 45: Binary System

Exercises

2- Convert binary to Decimal

(…..)10 = ( 0.01)2

(…..)10 = ( 0.001)2

(…..)10 = ( 0.0001)2

(….)10 = ( 0.1)2

Page 46: Binary System

Exercises

3- Convert Decimal to binary

(0.125)10 = ( … )2

(0.0625)10 = ( … )2

(0.03125)10 = ( … )2

(0.5)10 = ( … )2

Page 47: Binary System

Exercises

4- binary addition

11011 + 111

1111001 + 111111

10001 + 100001

0.1101 + 0.101

0.010001+ 0.0101 + 0.010111

Page 48: Binary System

Exercises

5- binary multiplication

11011 * 111

11001 * 11101

1001 * 1001

0.101 * 0.01

0.0111 * 0.0101