Date post: | 02-Apr-2015 |
Category: |
Documents |
Upload: | kelly-moris |
View: | 232 times |
Download: | 6 times |
04/11/23 Slide # 1
Binary, Octal and Hex NumbersCopyright Thaddeus Konar
Introduction to Binary, Octal and Hexadecimal Numbers
Thaddeus Konar
04/11/23 Slide # 2
Binary, Octal and Hex NumbersCopyright Thaddeus Konar
Decimal Integers
There is nothing ‘unique’ about number 10, but because we have 10 fingers, the decimal notation (from Latin decem and Greek Deka: 10) seems ‘natural’ to us.
If the world would be like Simpsons (and I am glad it is not) the natural notation would be octal (8 fingers)
457678543
965432783745673
01
8934098798347298763287632
09832198798237986498762380236409
04/11/23 Slide # 3
Binary, Octal and Hex NumbersCopyright Thaddeus Konar
Decimal Integers
Each digit (counting from the right) represents next power of ten, the rightmost digit represents 1s, next digit represents 10s, next 100s, and so on:
…,10000, 1000, 100, 10, 1which is the same as:
…,104 ,103, 102, 101, 100
(Please remember that any number X to zero (0) power equals 1!)
X0 = 1
04/11/23 Slide # 4
Binary, Octal and Hex NumbersCopyright Thaddeus Konar
Decimal Integers (cont)
7845 means:
(5*1)+(4*10)+(8*100)+(7*1000)
and this is same as:
(5*100)+(4*101)+(8*102)+(7*103)
04/11/23 Slide # 5
Binary, Octal and Hex NumbersCopyright Thaddeus Konar
Decimal Integers (Cont)
What does 58345 ‘really’ mean:
58345
5 * 1 4 * 103 * 100
=5=40
=3008 * 1000 =8000
5 * 10000 =50000
=58345
04/11/23 Slide # 6
Binary, Octal and Hex NumbersCopyright Thaddeus Konar
Decimal Integers (cont) Lets look at the properties of the decimal integers:
Base = 10 (1, 10, 100, …) (100, 101 , 102 , …)
Digits range: 0 -> 9 (0,1,2,3,4,5,6,7,8,9)
Number of values represented by a single digit: 10
Please note that number of digits equals Base, and range goes from zero to (Base –1).
Digits range: 0 -> (Base - 1)
Number of values represented by a single digit: Base
04/11/23 Slide # 7
Binary, Octal and Hex NumbersCopyright Thaddeus Konar
General NotationAny number is represented by combination of single digits Dx, where x is the position of the digit counting from the right. The value of Dx can be only the digits between (and including) 0 and (Base-1).
…D5D4D3D2D1D0
Using our example decimal number 7845
D0 =5, D1=4, D2=8, and D3 =7
04/11/23 Slide # 8
Binary, Octal and Hex NumbersCopyright Thaddeus Konar
General Notation (cont)
We can see that any number really means:
(D0*B0)+(D1*B1)+(D2*B2)+(D3*B3)+…(Dn*Bn)
In our example number 7845 (base 10) means:(5*100)+(4*101)+(8*102)+(7*103)=5+40+800+7000
04/11/23 Slide # 9
Binary, Octal and Hex NumbersCopyright Thaddeus Konar
Binary Integers 1Just as the each digit (position) in decimal integer is represented by the power of 10, binary integers are numbers where each digit is represented by the power of 2 (Base = 2).
Digits range: 0 -> (Base - 1)Number of values represented by a single digit: Base
Digits range: 0 -> 1 (0,1)Number of values represented by a single digit: 2
04/11/23 Slide # 10
Binary, Octal and Hex NumbersCopyright Thaddeus Konar
Binary Integers 2
In the decimal number the digits could be 0,1,2,3,4,5,6,7,8,9. (0 -> Base-1). As we can see the binary number digits could only be either 0 or 1 (0 ->Base-1).
The single decimal number can represent 10 values, and the single binary number can represent only 2 values.
04/11/23 Slide # 11
Binary, Octal and Hex NumbersCopyright Thaddeus Konar
Binary Integers 3
Binary integer will be a combination of 1s and 0s.
Please recall the formula (and remember that now Base=2):
(D0*B0)+(D1*B1)+(D2*B2)+(D3*B3)+…(Dn*Bn)
Lets look at the binary number 101110
D0=0, D1=1, D2=1, D3=1, D4=0, D5=1
B0=1, B1=2, B2=4, B3 =8, B4=16, B5=32, B6 =64…
04/11/23 Slide # 12
Binary, Octal and Hex NumbersCopyright Thaddeus Konar
Binary Integers 4
Here are the first few binary numbers:
…101110 means:
(0*20)+(1*21)+(1*22)+(1*23)+(0*24)+(1*25) = (0*1)+(1*2)+(1*4)+(1*8)+(0*16)+(1*32) = 4610
00002 = 010 01002 = 410 10002 = 810 11002 = 1210 100002 = 1610
00012 = 110 01012 = 510 10012 = 910 11012 = 1310 100012 = 1710
00102 = 210 01102 = 610 10102 = 1010 11102 = 1410 100102 = 1810
00112 = 310 01112 = 710 10112 = 1110 11112 = 1510 100112 = 1910
04/11/23 Slide # 13
Binary, Octal and Hex NumbersCopyright Thaddeus Konar
Binary Conversion Example
Convert binary number 10111 to decimal:
10111
1 * 1 1 * 21 * 4
=1=2
=40 * 8 =0
1 * 16 =16
=23
04/11/23 Slide # 14
Binary, Octal and Hex NumbersCopyright Thaddeus Konar
Convert binary number to decimal: 10111001
1 * 1
0 * 2
0 * 4
=1
=0
=0
1 * 8 =8
1 * 16 =16
=185
1 * 32 =32
0 * 64 =01 * 128 =128
04/11/23 Slide # 15
Binary, Octal and Hex NumbersCopyright Thaddeus Konar
Octal Integers 1Just as the each digit (position) in decimal integer is represented by the power of 10, in binary integer each digit represents power of 2, in octal numbers each digit is represented by the power of 8 (Base = 8).
Digits range: 0 -> (Base - 1)Number of values represented by a single digit: Base
Digits range: 0 -> 7 (0,1, 2, 3, 4, 5, 6, 7)Number of values represented by a single digit: 8
04/11/23 Slide # 16
Binary, Octal and Hex NumbersCopyright Thaddeus Konar
Octal Integers 2
In the decimal number the digits could be 0,1,2,3,4,5,6,7,8,9. (0 -> Base-1), in binary 0,1 (Base-1), so as you can suspect in octal numbers the digits would be 0,1,2,3,4,5,6,7 (0 -> Base-1).
The single decimal number can represent 10 values, the single binary number can represent only 2 values, and single octal number can represent 8 values.
04/11/23 Slide # 17
Binary, Octal and Hex NumbersCopyright Thaddeus Konar
Octal Integers 3
In octal integers Base = 8.
Please recall the formula:
(D0*B0)+(D1*B1)+(D2*B2)+(D3*B3)+…(Dn*Bn)
Lets look at the octal number 4153
D0=3, D1=5, D2=1, D3=4
B0=1, B1=8, B2=64, B3 =512, B4=4096…
04/11/23 Slide # 18
Binary, Octal and Hex NumbersCopyright Thaddeus Konar
Octal Integers 4
Here are the first few octal numbers:
…4153 means:
(3*80)+(5*81)+(1*82)+(4*83) = (3*1)+(5*8)+(1*64)+(4*512) = 215510
00008 = 010 00048 = 410 00108 = 810 00148 = 1210 00208 = 1610
00018 = 110 00058 = 510 00118 = 910 00158 = 1310 00218 = 1710
00028 = 210 00068 = 610 00128 = 1010 00168 = 1410 00228 = 1810
00038 = 310 00078 = 710 00138 = 1110 00178 = 1510 00238 = 1910
04/11/23 Slide # 19
Binary, Octal and Hex NumbersCopyright Thaddeus Konar
Hexadecimal (Hex) Integers 1Just as the each digit (position) in decimal integer is represented by the power of 10, in binary integer - power of 2, in octal numbers - power of 8, and in hex integers – power of 16 (Base = 16).
Digits range: 0 -> (Base - 1)Number of values represented by a single digit: Base
Digits range: 0 -> 15 (0,1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F)Number of values represented by a single digit: 16
04/11/23 Slide # 20
Binary, Octal and Hex NumbersCopyright Thaddeus Konar
Hex Integers 2
In the decimal number the digits could be 0,1,2,3,4,5,6,7,8,9. (0 -> Base-1), in binary 0,1 (Base-1), in octal 0,1,2,3,4,5,6,7. (0 -> Base-1). In hex numbers the digits would be 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F (0 -> Base-1). The letters A through F represent the decimal numbers 10 to 15.
The single decimal number can represent 10 values, the single binary number can represent only 2 values, the single octal number can represent 8 values, and the single hex number can represent 16 values.
04/11/23 Slide # 21
Binary, Octal and Hex NumbersCopyright Thaddeus Konar
Hex Integers 3
In Hex integers Base = 16.
Please recall the formula:
(D0*B0)+(D1*B1)+(D2*B2)+(D3*B3)+…(Dn*Bn)
Lets look at the octal number A59C
D0=C, D1=9, D2=5, D3=A
B0=1, B1=16, B2=256, B3 =4096, B4=65536…
04/11/23 Slide # 22
Binary, Octal and Hex NumbersCopyright Thaddeus Konar
Hex Integers 4
Here are the first few hex numbers:
…A59C means:
(12*160)+(9*161)+(5*162)+(A*163) = (12*1)+(9*16)+(5*256)+(10*4096) = 4239610
00016 = 010 00416 = 410 00816 = 810 00C16 = 1210 01016 = 1610
00116 = 110 00516 = 510 00916 = 910 00D16 = 1310 01116 = 1710
00216 = 210 00616 = 610 00A16 = 1010 00E16 = 1410 01216 = 1810
00316 = 310 00716 = 710 00B16 = 1110 00F16 = 1510 01316 = 1910
04/11/23 Slide # 23
Binary, Octal and Hex NumbersCopyright Thaddeus Konar
Relationship between Binary and Octal Numbers
10110102 = 9010
Converting to Octal:
Binary: 001 011 010 = 9010
Octal: 1 3 2 = 9010 each octal digit is 3 bits
04/11/23 Slide # 24
Binary, Octal and Hex NumbersCopyright Thaddeus Konar
Relationship between Binary and Hex Numbers
10110102 = 9010
Converting to Hex:
Binary: 0101 1010 = 9010
Hex: 5 A = 9010 each hex digit is 4 bits
04/11/23 Slide # 25
Binary, Octal and Hex NumbersCopyright Thaddeus Konar
Relationship between Binary, Octal and Hex Numbers 1
1111011011100101101010110102
Converting to Octal:
111 101 101 110 010 110 101 011 010 7 5 5 6 2 6 5 3 2
Converting to Hex:
0111 1011 0111 0010 1101 0101 1010 7 B 7 2 D 5 A
04/11/23 Slide # 26
Binary, Octal and Hex NumbersCopyright Thaddeus Konar
Relationship between Binary, Octal and Hex Numbers 2
Decimal Binary Octal (3 binary digits) Hex (4 binary digits)
0 00000 0 0
1 00001 1 1
2 00010 2 2
3 00011 3 3
4 00100 4 4
5 00101 5 5
6 00110 6 6
7 00111 7 7
8 01000 10 8
9 01001 11 9
10 01010 12 A
11 01011 13 B
12 01100 14 C
13 01101 15 D
14 01110 16 E
15 01111 17 F
16 10000 20 10
17 10001 21 11
18 10010 22 12
19 10011 23 (10 + 011) 13 (1 + 0011)
04/11/23 Slide # 27
Binary, Octal and Hex NumbersCopyright Thaddeus Konar
"nobody wants to be a 0 but everybody wants to be a 1"
Laurie Anderson - Home of the Brave