3.0 Introduction to Number Systems

Tags: #◇Computer Science #◇CCICOMP #◇Problem Solving #◇M2

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◇Overview

A number system is a method of representing numerical values. The base (or radix) determines the number of symbols used to represent digits, and each digit is weighted based on its position relative to the radix point. While computers natively operate exclusively on binary (base 2) numbers, humans often use decimal (base 10), octal (base 8), and hexadecimal (base 16) to conveniently represent and manipulate computer data.

◇Important Formulas

Common Number Systems Binary (Base 2): Symbols $0, 1$. Denoted as $1011_2$ or $1011\text{B}$. Octal (Base 8): Symbols $0, 1, 2, 3, 4, 5, 6, 7$. Denoted as $327_8$ or $327\text{o}$. Decimal (Base 10): Symbols $0-9$. Denoted as $25_{10}$ or $25\text{d}$. Hexadecimal (Base 16): Symbols $0-9$, $\text{A (10)}, \text{B (11)}, \text{C (12)}, \text{D (13)}, \text{E (14)}, \text{F (15)}$. Denoted as $5\text{C}8_{16}$ or $5\text{C}8\text{h}$.

Base Conversion Rules Any Base to Decimal: Multiply each digit by the base raised to the power of its positional index (starting at $0$ for the rightmost digit before the radix point) and sum them up. Decimal to Any Base: Use continuous division by the target base (e.g., $2$, $8$, or $16$) for the integer part. The remainders, read from the bottom up (last remainder to first), form the new digits. Binary to Octal/Hexadecimal: Group the binary digits starting from the rightmost (least significant) bit. Use groups of $3$ for Octal and groups of $4$ for Hexadecimal. Pad with $0$s on the left if necessary. Convert each group to its corresponding octal/hex digit. Octal/Hexadecimal to Binary: Convert each octal digit into its corresponding $3$-bit binary equivalent, or each hexadecimal digit into its corresponding $4$-bit binary equivalent.

◇Worked Examples

1. Binary to Decimal Conversion

Convert $101010_2$ to Decimal.
$101010_2 = (1 \times 2^5) + (0 \times 2^4) + (1 \times 2^3) + (0 \times 2^2) + (1 \times 2^1) + (0 \times 2^0)$
$= 32 + 0 + 8 + 0 + 2 + 0 = 42_{10}$

2. Decimal to Binary Conversion

Convert $23_{10}$ to Binary. $23 \div 2 = 11$ R $1$ $11 \div 2 = 5$ R $1$ $5 \div 2 = 2$ R $1$ $2 \div 2 = 1$ R $0$ $1 \div 2 = 0$ R $1$
Reading remainders from bottom to top: $10111_2$.

3. Binary to Octal Conversion

Convert $1101010_2$ to Octal. Group by 3 from the right: $001 \mid 101 \mid 010$ (padding the leftmost group with two $0$s). Convert groups: $001_2 = 1_8$, $101_2 = 5_8$, $010_2 = 2_8$. Result: $152_8$.

4. Decimal to Hexadecimal Conversion

Convert $351_{10}$ to Hexadecimal. $351 \div 16 = 21$ R $15 (\text{F})$ $21 \div 16 = 1$ R $5$ $1 \div 16 = 0$ R $1$
Reading from bottom to top: $15\text{F}_{16}$.

◇References & Navigation

• Previous Topic: ◇Computers, Data, and Information
• Next Topic: ◇Data Representation - Integers
• Related Notes: ◇Positional Number System, ◇Number System Base Conversion