Binary System

Binary System

Bits, Bytes, and Buzzwords 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, and Buzzwords MB: (megabyte); about 1 million bytes of information GB: (gigabyte); about 1 billion bytes of information TB: (terabyte); about 1 million megabytes of information

Decimal 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

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

Negative Binary Integers From 255 to 128 are treated as negative decimal Number From -1 to 128

Decimal integer -1

Binary integer

(255)

11111111

-10 (246)

11110110

-20 (236) -30 (226)

11101100 11100010

………..

……………

Positive Binary Integers Decimal 0 1 2 3 4 5 6 7 8 9 10 11

Binary 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011

Decimal

÷2

*2 Binary

Example 1 Convert Decimal to binary (35)10

= (… )2 (110001) 2

35

÷2

1

17

÷2

1

8

÷2

0

4

÷2

0

2

÷2

0

1

÷2

1

NO

Example 2 Convert Decimal to binary (45)10 = ( (101101) … )2 2 45

÷2

1

22

÷2

0

11

÷2

1

5

÷2

1

2

÷2

0

1

÷2

1

NO

Example 3 Convert Decimal to binary (55)10 = ( (110111) … )2 2 55

÷2

1

27

÷2

1

13

÷2

1

6

÷2

0

3

÷2

1

1

÷2

1

NO

Example 4 Convert Decimal to binary (72)10

= ((1001000) … )2 2

72

÷2

0

36

÷2

0

18

÷2

0

9

÷2

1

4

÷2

0

2

÷2

0

1

÷2

1

NO

Example 5 Convert Decimal to binary (85)10

= ((1010101) … )2 2

85

÷2

1

42

÷2

0

21

÷2

1

10

÷2

0

5

÷2

1

2

÷2

0

1

÷2

1

NO

Example 6 convert binary to decimal ( 100101)2 = (….)10

1*20+0*21+1*22+0*23+0*24+1*25 = 37

Example 7 convert binary to decimal ( 111101)2 = (….)10

1*20+0*21+1*22+1*23+1*24+1*25 = 61

Example 8 convert binary to decimal ( 111111)2 = (….)10

1*20+1*21+1*22+1*23+1*24+1*25 = 63

Example 9 convert binary to decimal ( 1100000)2 = (….)10

0*20+0*21+0*22+0*23+0*24+1*25+1*22 = 96

Example 10 convert binary to decimal ( 1101001)2 = (….)10

1*20+0*21+0*22+1*23+0*24+1*25+1*22 = 105

Example 11 Convert decimal fraction into binary

(0.125)10

= ( .001 … )2

0.125 0.250 0.50

* * *

integer

2

0.250

0

2

0.50

0

2

1

1

Example 12 Convert decimal fraction into binary

(0.0625)10

.0001 = (… )2

0.0625 0.1250 0.2500 0.5000

* * * *

integer

2

0.1250

0

2

0.2500

0

2

0.5000

0

2

1

1

Example 13 Convert decimal fraction into binary

(0.03125)10

=.00001 ( … )2

integer

0.03125

*

2

0.06250

0

0.06250

*

2

0.12500

0

0.12500

*

2

0.25000

0

0.25000

*

2

0.5000

0

0.5000

*

1

1

2

Example 14 convert fraction binary to decimal

( .0001)2 = (….) 0.0625 10

0*2-1 + 0*2-2 +0*2-3 + 1*2-4 = 1/16 =0.0625

Example 15 convert fraction binary to decimal

( .0011)2 = (….) 0.0625 10

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

1 1 = + = 0.1875 8 16

Example 16 Convert decimal into binary

(35.125)10

= (100011.001 … )2

35

2

1

17

2

1

8

2

0

4

2

0

2

2

0

1

2

1

NO

100011

integer

0.125 *2

0.250

0

0.250 *2

0.50

0

0.50 *2

1

1

.001

Example 17 Convert decimal into binary

(42.125)10

= (101010.001 … )2

42

2

0

21

2

1

10

2

0

5

2

1

2

2

0

1

2

1

NO

101010

integer

0.125 *2

0.250

0

0.250 *2

0.50

0

0.50 *2

1

1

.001

Example 18 Convert decimal into binary

(39.125)10

= (100111.001 … )2

39

2

1

19

2

1

9

2

1

4

2

0

2

2

0

1

2

1

NO

101010

integer

0.125 *2

0.250

0

0.250 *2

0.50

0

0.50 *2

1

1

.001

Addition

1+0=1 0+1=1 0+0=0 1+1=0

carry 1

Example 19

1

11000 + 1100 100100

Example 20

1

11000 + 1 1100 110100

Example 21

1

1

1

1011000 + 111100 10010100

Example 22

1 1 1

+

100100.000 1111101.001 10100001.001

Example 23

1 1 1

+

1 100101.000 1111101.001 11100010.001

Subtraction

1-1=0 1-0=1 0-0=0 0-1=1

borrow 1

Ex 24

borrow 1

-

11000 1100 1100

Ex 25

borrow 1

-

1111101.000 100100.001 1011000.111

Ex 26

borrow 1

-

11110 1101 10001

Ex 27

borrow 1

-

11111.000 110.001 11000.111

Multiplication

1*0=0 0*1=0 0*0=0 1*1=1

Example 28

11000 * 110 00000 11000 11000 1001000

Example 29

11111 * 11 11111 11111 1011101

Example 30

1101.1 * 110.01 11011 00000 00000 11011 11011 1010100.011

Example 31

11111 * 11101 11111 00000 11111 11111 11111 1110000011

Example 32

11.1 * 1.11 111 111 111 100.001

Exercises 1- Convert Decimal to binary (25)10

= ( … )2

(80)10

= ( … )2

(59.125)10

= ( … )2

(31.0625)10 (125)10

= ( … )2

= ( … )2

Exercises 2- Convert binary to Decimal

(…..)10

= ( 0.01)2

(…..)10

= ( 0.001)2

(…..)10 (….)10

= ( 0.0001)2 = ( 0.1)2

Exercises 3- Convert Decimal to binary

(0.125)10

= ( … )2

(0.0625)10

= ( … )2

(0.03125)10 (0.5)10

= ( … )2

= ( … )2

Exercises 4- binary addition 11011 + 111 1111001 + 111111 10001 + 100001 0.1101 + 0.101 0.010001+ 0.0101 + 0.010111

Exercises 5- binary multiplication 11011 * 111 11001 * 11101 1001 * 1001 0.101 * 0.01 0.0111 * 0.0101