Binary, Decimal and Hexadecimal Numbers Svetlin Nakov Telerik Corporation

1. Numerals Systems  Binary and Decimal Numbers  Hexadecimal Numbers  Conversion between Numeral Systems 3. Representation of Numbers  Positive and Negative Integer Numbers  Floating-Point Numbers 4. Text Representation 2

Conversion between Numeral Systems

 Decimal numbers (base 10 )  Represented using 10 numerals: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9  Each position represents a power of 10 :  401= 4* * *10 0 =  130= 1* * *10 0 =  9786= 9* * * *10 0 = = 9* * *10 + 6*1 4

 Binary numbers are represented by sequence of bits (smallest unit of information – 0 or 1 )  Bits are easy to represent in electronics

 Binary numbers (base 2 )  Represented by 2 numerals: 0 and 1  Each position represents a power of 2 :  101 b = 1* * *2 0 = 100 b + 1 b = = = 5  110 b = 1* * *2 0 = 100 b + 10 b = = = 6  b = 1* * * * * *2 0 = = = = 53 6

 Multiply each numeral by its exponent:  1001 b = 1* *2 0 = 1*8+ 1*1= = 9 = 9  0111 b = 0* * * *2 0 = = 100 b + 10 b + 1 b = = = 7  b = 1* * * * *2 1 = = b b b + 10 b = = = = 54 7

 Divide by 2 and append the reminders in reversed order: 500/2 = 250 (0) 250/2 = 125 (0) 125/2 = 62 (1) 62/2 = 31 (0) 500 d = b 62/2 = 31 (0) 500 d = b 31/2 = 15 (1) 31/2 = 15 (1) 15/2 = 7 (1) 15/2 = 7 (1) 7/2 = 3 (1) 7/2 = 3 (1) 3/2 = 1 (1) 3/2 = 1 (1) 1/2 = 0 (1) 1/2 = 0 (1) 8

 Hexadecimal numbers (base 16 )  Represented using 16 numerals: 0, 1, 2,... 9, A, B, C, D, E and F  Usually prefixed with 0x 0  0x0 8  0x8 1  0x1 9  0x9 2  0x210  0xA 3  0x311  0xB 4  0x412  0xC 5  0x513  0xD 6  0x614  0xE 7  0x715  0xF 9

 Each position represents a power of 16 :  9786 hex = 9* * * *16 0 = = 9* *256+ 8*16+ 6*1= =  0xABCDEF hex = 10* * * * * *16 0 = 13* * *16 0 = =

 Multiply each digit by its exponent  1F4 hex = 1* * *16 0 = = 1* *16+ 4*1 = = 500 d  FF hex = 15* *16 0 = = = = 255 d 11

 Divide by 16 and append the reminders in reversed order 500/16 = 31 (4) 31/16 = 1 (F) 500 d = 1F4 hex 1/16 = 0 (1) 12

 The conversion from binary to hexadecimal (and back) is straightforward: each hex digit corresponds to a sequence of 4 binary digits: 0x0 = 00000x8 = x1 = 00010x9 = x2 = 00100xA = x3 = 00110xB = x4 = 01000xC = x5 = 01010xD = x6 = 01100xE = x7 = 01110xF =

Positive and Negative Integers and Floating-Point Numbers

 A short is represented by 16 bits  100= = =  An int is represented by 32 bits  = = =  A char is represented by 16 bits  ‘0’= 48 = = =

 A number's sign is determined by the Most Significant Bit (MSB)  Only in signed integers: sbyte, short, int, long  Leading 0 means positive number  Leading 1 means negative number  Example: (8 bit numbers) 0XXXXXXX b > 0 e.g b = b = 0 1XXXXXXX b < 0 e.g b =

 The largest positive 8-bit sbyte number is: 127 ( ) = b  The smallest negative 8-bit number is: -128 ( -2 7 ) = b  The largest positive 32 -bit int number is: ( ) = 01111…11111 b  The smallest negative 32 -bit number is: ( ) = 10000…00000 b 17

+127= = = = = = = = = =  Positive 8-bit numbers have the format 0XXXXXXX  Their value is the decimal of their last 7 bits ( XXXXXXX)  Negative 8-bit numbers have the format 1YYYYYYY  Their value is 128 ( 2 7 ) minus ( - ) the decimal of YYYYYYY  b = 2 7 – b = = =

 Floating-point numbers representation (according to the IEEE 754 standard*):  Example: 19 2 k n SP 0...P k-1 M 0 M 1 M n-1 SignExponentMantissa Mantissa =1, Exponent= 4Sign=-1 Bits [22…0]Bits [30…23] Bit31 * See

How Computers Represent Text Data?   A text encoding is a system that uses binary numbers ( 1 and 0 ) to represent characters   Letters, numerals, etc.   In the ASCII encoding each character consists of 8 bits (one byte) of data   ASCII is used in nearly all personal computers   In the Unicode encoding each character consists of 16 bits (two bytes) of data   Can represent many alphabets 21

Character Codes – ASCII Table Excerpt from the ASCII table Binary Code Decimal Code Character A B C D # ~ 22

 Strings are sequences of characters  Null-terminated (like in C)  Represented by array  Characters in the strings can be:  8 bit (ASCII / windows / …)  16 bit (UTF- 16 ) ……………………\0 4 bytes length……………… 23

Questions?

1. Write a program to convert decimal numbers to their binary representation. 2. Write a program to convert binary numbers to their decimal representation. 3. Write a program to convert decimal numbers to their hexadecimal representation. 4. Write a program to convert hexadecimal numbers to their decimal representation. 5. Write a program to convert hexadecimal numbers to binary numbers (directly). 6. Write a program to convert binary numbers to hexadecimal numbers (directly). 25

7. Write a program to convert from any numeral system of given base s to any other numeral system of base d ( 2 ≤ s, d ≤ 16 ). 8. Write a program that shows the binary representation of given 16 -bit signed integer number (the C# type short ). 9. * Write a program that shows the internal binary representation of given 32 -bit signed floating-point number in IEEE 754 format (the C# type float ). Example: -27,25  sign = 1, exponent = , mantissa =