Lecture 1 Number Representation and Bit Models

• Modern computer hardware is based on binary digits, namely 0 and 1.
• Computer stores everything as numbers.
• The single symbols modern computers use are 0 and 1.
• A single 0 or 1 is a bit, which means binary digit.
• A group of 8 contiguous bits, treated as a unit is a Byte.
• A group of 4 bits is a Nybble.
• Bits are grouped, and methods are used for interpretation.
• Each method of interpretation is a bit model.
• The bit model for non-negative integers is unsigned binary.
• Examining the bit-models of C data types provides insight into memory usage.
• Numbers are written using a positional representation system.
• The decimal number 357 is represented as 3 x 10^2 + 5 x 10^1 + 7 x 10^0.

Numbers in Base r

• A number (dn-1dn-2...d1d0)r in base (or radix) r is expressed as dn-1 * r^(n-1) + dn-2 * r^(n-2) + ... + d1 * r^1 + d0 * r^0.
• (bn-1bn-2...b1b0)2 in base 2 or binary is expressed as bn-1 * 2^(n-1) + bn-2 * 2^(n-2) + ... + b1 * 2^1 + b0 * 2^0
• The unsigned binary number 1010102 equals 1 * 2^5 + 0 * 2^4 + 1 * 2^3 + 0 * 2^2 + 1 * 2^1 + 0 * 2^0 which is equal to 4210.

Conversion from Decimal to Binary

• Start with two columns: value and bit
• If the value is even, put a 0 in the bit column, and if odd, put a 1 in the bit column.
• Update the value column, by setting the new value to the old value divided by 2 and rounded down.
• The binary representation of the given value is the bit column read from bottom (MSB) to top (LSB).
• Given 123 represented as 12310, the bit column bottom->top gives 11110112
• Decimal numbers can be related to their bit patterns in binary.

Hexadecimal Representation

• Modern computers use 32 or 64-bit words and to overcome this, base 16 is used.
• Base 16 is hexadecimal (HEX) representation.
• Hexadecimal needs 16 symbols, identified as "higits" from 0 to 15.
• In HEX, symbols 0-9 are the same as in decimal.
• After that: 10 is A, 11 is B, 12 is C, 13 is D, 14 is E, and 15 is F.
• HEX values are identified with the prefix 0x, e.g., 0xF00D.

Converting Binary to Hex

• Group bits into sets of 4 starting from the right end (LSB).
• Pad the front with zeroes if necessary.
• Replace each group of 4 with the "higit".

Converting Hex to Binary

• Replace each "higit" with the group of 4 bits from the table.

Basic C Data Types

• Every computer as a word size.
• This is the largest number of contiguous bits that the ALU handles.
• It manifests as the nominal size of pointer data i.e. the number of bits required for a virtual address.
• A 32-bit word limits virtual address space to 4GB, while a 64-bit word allows 16 Exabytes.
• Given the signed data types [signed] char, short, int, long, int32_t, int64_t, char *, float and double
• The corresponding unsigned data types are unsigned char, unsigned short, unsigned int, unsigned long, unit_32, unit_64
• And respective byte allocations 1, 2, 4, 4/8, 4, 8, 8, 4, 8.
• Allocated sizes may vary but are typically as shown in the table.
• ISO C99 defines int32_t and int64_t which are guaranteed 4 & 8 bytes, respectively.
• C standard only guarantees minimum sizes and the actutal size is implementation dependant.
• intN_t is guaranteed for N = 8, 16, 32, 64.
• All pointers are 8 bytes (4 bytes for 32 bit systems).
• Compilers typically treat char as signed, though it is not guaranteed in the C standard unless the prefix signed is used.
• The C standard only guarantees lower bounds, except for intN_t types.
• Printf() is a general purpose output formatting function.
• Printf is not defined in C, and is part of the standard library of functions (stdio.h)
• The first argument is a string of chars to be printed, with each % indicating a substituted argument.
• For instance: printf("%2d\t%f\t%4.2f", dayNum, temperature, cost) DayNum is an integer at least 2 digits wide (%2d). Temperature is a floating point. Cost will print as floating point, at least 4 characters wide, 2 after the decimal point.
• Other format specifiers: u unsigned, x/X (int) hex, c single char, f/g/G floating point, p pointer (void *).

Multi-Byte Data and Addressing

• Programs use addresses to refer to data.
• Data is conceptually in a large array of bytes.
• An address is like an index into that array, and the pointer variable stores an address.
• Systems give each process private address spaces.
• Addresses specify byte locations, as such the address is of the first byte in the word.
• Successive word addresses differ by 8 bytes (for 64-bit) and

Endianness

• A variable x has a 4-byte value 0x01234567, and the address given by &x is 0x100.
• In big endian, the bytes are stored in memory as 01 23 45 67.
• In little endian, the bytes are stored in memory as 67 45 23 01.

Explanation of binary digits, bits, bytes, and nybbles in computer hardware. Discussion of bit models, positional representation systems, and base r numbers, with examples of decimal and binary representations.