Number System Quiz

Study Notes

Number System Overview

Digital systems utilize positional number systems where digits represent different values based on their positions.
Each digit's value in a number is determined by its positional weight according to the base of the number system.

Decimal Numbers

The decimal system uses ten digits (0-9) where each digit’s position determines its value.
Whole number weights are positive powers of ten, increasing from right to left.
Fractional weights are negative powers of ten, decreasing from left to right.
Example: For 87, the representation is ( (8 \times 10^1) + (7 \times 10^0) ).
Example: For 725.45, it can be expressed as ( (7 \times 10^2) + (2 \times 10^1) + (5 \times 10^0) + (4 \times 10^{-1}) + (5 \times 10^{-2}) ).

Binary Numbers

The binary system is simpler than decimal, using only two digits: 0 and 1 (base two system).
Each digit's position indicates its value based on powers of two.
A 4-bit binary number can represent values from 0 to 15; ( 2^n - 1 ) gives maximum decimal value for n bits.
Example: With 4 bits, the maximum number is binary 1111, which equals decimal 15.

Decimal to Binary Conversion

Converting decimal to binary involves repeated division by 2.
Example: ( (42){10} ) converts to ( (101010){2} ).

Binary to Decimal Conversion

The decimal value of a binary number is the sum of the weights for bits that are 1.
Example: For binary ( (1110)_{2} ), the decimal equivalent is calculated by summing relevant weights.

Hexadecimal Numbers

The hexadecimal system comprises 16 digits (0-9 and A-F) and is used for compact binary representation.
Maximum 3-digit hexadecimal number is FFF (decimal 4095); maximum 4-digit is FFFF (decimal 65535).

Hexadecimal to Binary Conversion

Each hexadecimal digit converts to a 4-digit binary equivalent.
Example: Hexadecimal ( 9AF ) converts to binary ( 1001\ 1010\ 1111 ).

Binary to Hexadecimal Conversion

To convert binary to hexadecimal, group the binary number into 4 bits and replace each group with its corresponding hexadecimal symbol.
Example: ( 1100101001010111 ) converts to hexadecimal CA57.

Hexadecimal to Decimal Conversion

Convert hexadecimal to binary first, then convert binary to decimal for the result.

Decimal to Hexadecimal Conversion

Decimal to hexadecimal conversion involves repeated division by 16, forming the hexadecimal number from remainders. The first remainder is the least significant digit.

Description

Test your knowledge of the positional number system and its fundamentals. This quiz covers the significance of digits, their values based on position, and the concept of bases in different number systems. Perfect for students looking to strengthen their understanding of numerical systems.