Binary Calculator
Perform binary arithmetic operations and number system conversions with step-by-step solutions.
Perform binary add operation
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A Comprehensive Guide to Binary Arithmetic and Number Systems
Binary numbers are the foundation of digital computing, using only two digits (0 and 1) to represent all numbers. Understanding binary arithmetic and conversions is essential for computer science and digital electronics.
Basic Concepts
Key Points:
- Binary uses only two digits: 0 and 1 (bits)
- Each position represents a power of 2
- Reading from right to left: 2⁰, 2¹, 2², 2³, ...
- Example: 1101₂ = 1×2³ + 1×2² + 0×2¹ + 1×2⁰ = 13₁₀
Base Indicators:
The small subscript number after a value is called a base indicator — it tells you which number system the number is written in.
₂ = base 2 (binary)
₁₀ = base 10 (decimal)
₁₆ = base 16 (hexadecimal)
₈ = base 8 (octal)
Example: 1101₂ = 13₁₀ = D₁₆ = 15₈
(All represent the same value in different number systems)
Binary Arithmetic
Binary Addition
Rules for binary addition:
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 10 (carry the 1)
Example:
1010 (10)
+ 1100 (12)
= 10110 (22)
Binary Subtraction
Rules for binary subtraction:
0 - 0 = 0
1 - 0 = 1
1 - 1 = 0
0 - 1 = 1 (borrow 1)
Example:
1100 (12)
- 1010 (10)
= 0010 (2)
Binary Multiplication
Rules for binary multiplication:
0 × 0 = 0
0 × 1 = 0
1 × 0 = 0
1 × 1 = 1
Example:
1010 (10)
× 0011 (3)
= 11110 (30)
Step-by-Step Manual Calculation Guide
How to Add Binary Numbers
Let's learn how to add binary numbers using a simple method:
Example: Add 1101₂ + 1011₂
Step 1: Write numbers aligned right
1101
+ 1011
------
Step 2: Add each column from right to left:
1. 1 + 1 = 10 (write 0, carry 1)
2. 0 + 1 + 1(carry) = 10 (write 0, carry 1)
3. 1 + 0 + 1(carry) = 10 (write 0, carry 1)
4. 1 + 1 + 1(carry) = 11 (write both digits)
1101
+ 1011
------
11000
Final result: 11000₂ = 24₁₀
How to Subtract Binary Numbers
Let's learn binary subtraction using the borrowing method:
Example: Subtract 1101₂ - 0111₂
Step 1: Write numbers aligned right
1100
- 0111
------
Step 2: Subtract each column from right to left:
1. 1 - 1 = 0
2. 0 - 1 = borrow 1 from next column: 10 - 1 = 1
3. 0(after borrow) - 1 = borrow again: 10 - 1 = 1
4. 0(after borrow) - 0 = 0
1101
- 0111
------
0110
Final result: 0110₂ = 6₁₀
How to Multiply Binary Numbers
Binary multiplication is similar to decimal multiplication, but simpler because we only multiply by 0 or 1:
Example: Multiply 101₂ × 11₂
Step 1: Write numbers in traditional multiplication format
101
× 11
-----
Final result: 1111₂ = 15₁₀
Verification: 5₁₀ × 3₁₀ = 15₁₀
Helpful Tips for Manual Calculation
Addition Tips:
- • Write numbers neatly aligned
- • Always work right to left
- • Don't forget carried digits
- • Double-check your carries
Subtraction Tips:
- • Mark your borrows clearly
- • Remember: borrowing gives you 2 (10 in binary)
- • Check if borrowing is needed before each step
- • Verify your answer makes sense
Common Mistakes to Avoid
- • Don't treat binary numbers like decimal numbers
- • Don't forget that 1 + 1 = 10 in binary (not 2)
- • Always align numbers properly from the right
- • Don't forget to include leading zeros when necessary
- • When in doubt, convert to decimal to verify your answer
Number System Conversions
Binary to Decimal
Multiply each bit by its position value (power of 2) and sum.
1101₂ = (1×2³) + (1×2²) + (0×2¹) + (1×2⁰)
= 8 + 4 + 0 + 1
= 13₁₀
Decimal to Binary
Divide by 2 repeatedly and read remainders from bottom to top.
13 ÷ 2 = 6 remainder 1
6 ÷ 2 = 3 remainder 0
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1
13₁₀ = 1101₂
Hexadecimal (Base-16)
Uses digits 0-9 and letters A-F (A=10, B=11, ..., F=15).
Binary to Hex: Group by 4 bits
1101₂ = 0D₁₆ (13₁₀)
11110000₂ = F0₁₆ (240₁₀)
Real-Life Applications
Digital Electronics
Binary is used in digital circuits where voltage levels represent 0s and 1s. This forms the basis of all digital computing and electronic devices.
Data Storage
All data in computers is stored as binary. Text, images, videos, and programs are ultimately sequences of 0s and 1s. For example, the letter 'A' is stored as 01000001 in ASCII.
Tips & Best Practices
Quick Tips:
- • Memorize powers of 2 up to 2⁸
- • Practice binary addition tables
- • Use grouping for hex conversion
- • Double-check your carries
Watch Out For:
- • Forgetting to carry in addition
- • Mixing up number bases
- • Incorrect bit grouping
- • Forgetting leading zeros