🔢 Number System Lab & Complete Guide
Convert Bases, Flip Interactive Bits, Calculate Complements & Learn All Conversions
⚡ Interactive 8-Bit Simulator Click switches to flip bits
📘 Step-by-Step Conversion Breakdown
📋 Base Conversion Tables
Quick lookup reference tables mapping Octal and Hexadecimal digits to their equivalent binary bit representations.
Octal Conversion Table (3 Bits)
| Octal | Decimal | 3-Bit Binary |
|---|---|---|
| 0 | 0 | 000 |
| 1 | 1 | 001 |
| 2 | 2 | 010 |
| 3 | 3 | 011 |
| 4 | 4 | 100 |
| 5 | 5 | 101 |
| 6 | 6 | 110 |
| 7 | 7 | 111 |
Hexadecimal Conversion Table (4 Bits)
| Hex | Decimal | 4-Bit Binary |
|---|---|---|
| 0 | 0 | 0000 |
| 1 | 1 | 0001 |
| 2 | 2 | 0010 |
| 3 | 3 | 0011 |
| 4 | 4 | 0100 |
| 5 | 5 | 0101 |
| 6 | 6 | 0110 |
| 7 | 7 | 0111 |
| 8 | 8 | 1000 |
| 9 | 9 | 1001 |
| A | 10 | 1010 |
| B | 11 | 1011 |
| C | 12 | 1100 |
| D | 13 | 1101 |
| E | 14 | 1110 |
| F | 15 | 1111 |
1. Complete Guide to All 12 Inter-Base Conversions
Formatted like standard academic exam solutions using formal mathematical tables, division remainder steps, positional expansions, and side notes.
1. Decimal → BinaryProblem: Convert (254.2033)₁₀ to Binary
Part A: Integer Part (254)₁₀
| Division | Quotient | Remainder |
|---|---|---|
| 254 ÷ 2 | 127 | 0 (LSB) |
| 127 ÷ 2 | 63 | 1 |
| 63 ÷ 2 | 31 | 1 |
| 31 ÷ 2 | 15 | 1 |
| 15 ÷ 2 | 7 | 1 |
| 7 ÷ 2 | 3 | 1 |
| 3 ÷ 2 | 1 | 1 |
| 1 ÷ 2 | 0 | 1 (MSB) |
Part B: Fractional Part (0.2033)₁₀
| Multiplication | Product | Integer |
|---|---|---|
| 0.2033 × 2 | 0.4066 | 0 (MSB) |
| 0.4066 × 2 | 0.8132 | 0 |
| 0.8132 × 2 | 1.6264 | 1 |
| 0.6264 × 2 | 1.2528 | 1 (LSB) |
Problem: Convert (254.2033)₁₀ to Octal
Part A: Integer Part (254)₁₀
| Division | Quotient | Remainder |
|---|---|---|
| 254 ÷ 8 | 31 | 6 (LSB) |
| 31 ÷ 8 | 3 | 7 |
| 3 ÷ 8 | 0 | 3 (MSB) |
Part B: Fractional Part (0.2033)₁₀
| Multiplication | Product | Integer |
|---|---|---|
| 0.2033 × 8 | 1.6264 | 1 (MSB) |
| 0.6264 × 8 | 5.0112 | 5 (LSB) |
Problem: Convert (254.2033)₁₀ to Hexadecimal
Part A: Integer Part (254)₁₀
| Division | Quotient | Remainder (Hex) |
|---|---|---|
| 254 ÷ 16 | 15 | 14 ➜ E (LSB) |
| 15 ÷ 16 | 0 | 15 ➜ F (MSB) |
Part B: Fractional Part (0.2033)₁₀
| Multiplication | Product | Integer |
|---|---|---|
| 0.2033 × 16 | 3.2528 | 3 (MSB) |
| 0.2528 × 16 | 4.0448 | 4 (LSB) |
Problem: Convert (11111110.0011)₂ to Decimal
Part A: Integer Positional Expansion
= (1×2⁷) + (1×2⁶) + (1×2⁵) + (1×2⁴) + (1×2³) + (1×2²) + (1×2¹) + (0×2⁰)
= 128 + 64 + 32 + 16 + 8 + 4 + 2 + 0 = 254
Part B: Fractional Positional Expansion
= (0×2⁻¹) + (0×2⁻²) + (1×2⁻³) + (1×2⁻⁴)
= 0 + 0 + 0.125 + 0.0625 = 0.1875
Problem: Convert (11111110.0011)₂ to Octal
Grouping binary bits into 3-bit blocks relative to the radix point:
Integer Part (Group Leftward)
011 | 111 | 110
↓ ↓ ↓
3 7 6
Fractional Part (Group Rightward)
001 | 100
↓ ↓
1 4
Problem: Convert (11111110.0011)₂ to Hexadecimal
Grouping binary bits into 4-bit blocks relative to the radix point:
Integer Part (Group Leftward)
1111 | 1110
↓ ↓
15(F) 14(E)
Fractional Part (Group Rightward)
0011
↓
3
Problem: Convert (376.15)₈ to Decimal
Part A: Integer Expansion (Base 8)
= (3 × 8²) + (7 × 8¹) + (6 × 8⁰)
= (3 × 64) + (7 × 8) + (6 × 1) = 192 + 56 + 6 = 254
Part B: Fractional Expansion (Base 8)
= (1 × 8⁻¹) + (5 × 8⁻²)
= (1 / 8) + (5 / 64) = 0.125 + 0.078125 = 0.203125
Problem: Convert (376.15)₈ to Binary
Replace each octal digit with its 3-bit binary equivalent:
| Octal Digit | 3 | 7 | 6 | . | 1 | 5 |
|---|---|---|---|---|---|---|
| Binary (3 Bits) | 011 | 111 | 110 | . | 001 | 101 |
Problem: Convert (376.15)₈ to Hexadecimal
Step 1: Convert Octal to Binary
(376.15)₈ = 011 111 110 . 001 101₂
Step 2: Regroup Binary into 4-Bit Groups
Integer Part: [1111] [1110] ➜ 15(F) 14(E)
Fractional Part: [.0011] [0100] ➜ .3 4
Problem: Convert (FE.34)₁₆ to Decimal
Part A: Integer Expansion (Base 16)
Where F = 15, E = 14:
= (15 × 16¹) + (14 × 16⁰)
= 240 + 14 = 254
Part B: Fractional Expansion (Base 16)
= (3 × 16⁻¹) + (4 × 16⁻²)
= (3 / 16) + (4 / 256) = 0.1875 + 0.015625 = 0.203125
Problem: Convert (FE.34)₁₆ to Binary
Replace each hexadecimal digit with its 4-bit binary equivalent:
| Hex Digit | F | E | . | 3 | 4 |
|---|---|---|---|---|---|
| Binary (4 Bits) | 1111 | 1110 | . | 0011 | 0100 |
Problem: Convert (FE.34)₁₆ to Octal
Step 1: Convert Hexadecimal to Binary
(FE.34)₁₆ = 1111 1110 . 0011 0100₂
Step 2: Regroup Binary into 3-Bit Groups
Integer Part: [011] [111] [110] ➜ 3 7 6
Fractional Part: [.001] [101] [000] ➜ .1 5 0
2. 1's and 2's Complements Deep Dive
In digital circuitry, complements are used to convert subtraction operations into simple addition and represent negative signed integers.
A. 1's ComplementInvert every bit in the binary sequence: change all 0s to 1s and 1s to 0s.
Example: Find 1's Complement of 254 in an 8-bit register
Original 8-Bit Binary (+254) : 1 1 1 1 1 1 1 0
Invert Bits (1's Complement) : 0 0 0 0 0 0 0 1
Add 1 to the 1's complement result (2's Comp = 1's Comp + 1). This represents the negative equivalent of a binary integer.
Example: Find 2's Complement of +254 in a 16-bit register (to represent -254)
1. 16-bit Binary (+254) : 0000000011111110
2. 1's Complement : 1111111100000001
3. Add +1 for 2's Comp : 1111111100000001 + 1
Number systems can seem a bit strange at first. Most people are used to decimal numbers, the usual 0 to 9, but then binary, octal and hexadecimal show up, and suddenly the same value has several different ways of being written. A little confusing? Sure, especially when the numbers start looking nothing alike. The Number System Converter Learning Lab makes this topic more approachable by allowing learners to convert values between different number systems. Enter a number, select the system, and see how the value changes its form. The original value stays the same, even when the digits look completely different. That's the interesting part. For students exploring computer science, digital electronics or mathematics, this kind of practice can help make conversions easier to understand. Some examples feel straightforward, while others need a second look. No need to rush through every rule at once. Try different values, check the results, and let the patterns become clearer with practice.
What is the Number System Converter & Learning Lab?
The Number System Converter & Learning Lab is an educational tool that helps students learn and convert numbers between binary, decimal, octal, and hexadecimal number systems.
What number systems can I convert using this tool?
The tool focuses on four important number systems: binary (base 2), octal (base 8), decimal (base 10), and hexadecimal (base 16), which are widely used in mathematics and computer science.
How do I convert decimal numbers to binary?
To convert a decimal number to binary, repeatedly divide the number by 2 and record the remainders. Read the remainders in reverse order to obtain the binary representation.
How do I convert binary numbers to decimal?
To convert binary to decimal, multiply each digit by its corresponding power of 2, starting from the rightmost digit with power 0, and add the results.
What is the binary number system?
The binary number system is a base-2 system that uses only 0 and 1. Computers use binary values to represent and process digital information.
What is the difference between octal and hexadecimal?
Octal is a base-8 number system that uses digits from 0 to 7, while hexadecimal is a base-16 system that uses digits 0–9 and letters A–F. Both provide compact ways to represent binary values.
What is 2's complement in binary?
Two's complement is a method used to represent signed integers, including negative numbers, in binary. It is widely used in computer systems for representing numbers and performing arithmetic operations.
Why is number system conversion important in computer science?
Number system conversion helps students understand how computers represent numbers, store data, process information, and perform calculations using binary, octal, decimal, and hexadecimal values.
Who can use the Number System Converter & Learning Lab?
This learning tool is suitable for students, beginners, computer science learners, and anyone who wants to understand number systems and practise converting numbers between different bases.
How can I improve my number system conversion skills?
Practise converting numbers between different bases, compare the results, and review the steps involved in each conversion. Regular practice can improve accuracy and strengthen your understanding of number systems.
