binary system
DESCRIPTION
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 PresentationTRANSCRIPT
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.
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
• 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
• 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)
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
Decimal
Binary
÷ 2 * 2
(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
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
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
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
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
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 = ( … )2
0.250
0.125
0.50
2
2
20.50
0.250
0
1
0***
integer
1
.001
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
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
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
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.
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
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
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
1 + 0 = 10 + 1 = 10 + 0 = 01 + 1 = 0 carry 1
Addition
Example 19
11000 + 1100 100100
1
Example 20
11000 + 1 1100 110100
1
Example 21
1011000 + 111100 10010100
111
Example 22
100100.000 + 1111101.001 10100001.001
111
Example 23
1 100101.000 + 1111101.001 11100010.001
111
1 - 1 = 0 1 - 0 = 10 - 0 = 00 - 1 = 1 borrow 1
Subtraction
Ex 24
11000 - 1100 1100
borrow 1
Ex 25
1111101.000 - 100100.001 1011000.111
borrow 1
Ex 26
11110 - 1101 10001
borrow 1
Ex 27
11111.000 - 110.001 11000.111
borrow 1
1 * 0 = 00 * 1 = 00 * 0 = 01 * 1 = 1
Multiplication
Example 28
11000 * 110 00000
11000 11000 10010000
Example 29
11111 * 11
11111 11111
1011101
1101.1 * 110.01 11011
00000 00000
1101111011
1010000.011
Example 30
11111 * 11101 11111
00000 11111
1111111111
1110000011
Example 31
11.1 * 1.11 111
111 111101.001
Example 32
Exercises
1- Convert Decimal to binary
(25)10 = ( … )2
(80)10 = ( … )2
(59.125)10 = ( … )2
(31.0625)10 = ( … )2
(125)10 = ( … )2
Exercises
2- Convert binary to Decimal
(…..)10 = ( 0.01)2
(…..)10 = ( 0.001)2
(…..)10 = ( 0.0001)2
(….)10 = ( 0.1)2
Exercises
3- Convert Decimal to binary
(0.125)10 = ( … )2
(0.0625)10 = ( … )2
(0.03125)10 = ( … )2
(0.5)10 = ( … )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