Base-N Number Systems Overview

Base-N Number System

• Base N represents a number system with N unique digits.
• Digits range from 0 to N - 1.
• Positional number system assigns a value to each digit based on its position.
• The digit at the rightmost position is the Least Significant Digit (LSD) and the digit at the leftmost position is the Most Significant Digit (MSD).

Decimal Number System

• Uses base 10 with ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
• Also a positional number system.

Binary Number System

• Uses base 2 with two digits: 0, 1.
• Binary digits are called bits.
• The rightmost bit is the Least Significant Bit (LSB) and the leftmost bit is the Most Significant Bit (MSB).

Definitions

• Nybble: 4 bits
• Byte: 8 bits
• Short word: 2 bytes (16 bits)
• Double word: 4 bytes (32 bits)
• Long word: 8 bytes (64 bits)
• 1k (kilo): 1,024
• 1M (mega): 1,048,576
• 1G (giga): 1,073,741,824

Hexadecimal Number System

• Uses base 16 with sixteen digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F.
• Also a positional number system.

Binary Addition

• The single bit addition table guides addition with a carry if the sum is 2 or more.

Hex Addition

• 4-bit addition is used with carry when the sum exceeds F.

Hex Digit Addition Table

• Summarizes all possible hex digit additions, including carryovers for sums exceeding F.

1’s Complements

• Calculated by inverting each bit (0 becomes 1, 1 becomes 0).

2’s Complements

• Calculated by first taking the 1’s complement and then adding 1.

Complements

• The 2’s complement of the 2’s complement of a number results in the original number.

Two's Complement Representation for Signed Numbers

• Two’s complement notation is used to represent negative numbers.
• Two’s complement encoding is done by taking the 2’s complement of the positive value.