How do you add two binary numbers?

Binary addition works similarly to decimal addition but uses only 0 and 1. The rules are: 0+0=0, 1+0=1, 0+1=1, and 1+1=0 (with a carry of 1). For example, adding 101 and 11 gives 1000. You align the columns from the right and carry over the 1 when the sum exceeds 1.

How do you subtract binary numbers?

Binary subtraction uses the rules: 0-0=0, 1-0=1, 1-1=0. When subtracting 1 from 0 (0-1), you must borrow 1 from the next higher-order bit, which makes the 0 become 10 (binary for 2), so 10-1=1. Alternatively, subtraction can be done using the two's complement method.

What is the difference between AND, OR, and XOR?

These are bitwise logic operations. AND results in 1 only if both bits are 1. OR results in 1 if at least one bit is 1. XOR (Exclusive OR) results in 1 only if the bits are different (one is 1, the other is 0). NOR is the opposite of OR.

What is a binary left shift (<<)?

A left shift moves all bits in a binary number to the left by a specified number of positions, filling the new empty spaces on the right with zeros. A left shift of one position is mathematically equivalent to multiplying the number by 2.

What is a binary right shift (>>)?

A right shift moves all bits to the right. The bits shifted off the right end are discarded. A right shift of one position is mathematically equivalent to dividing the number by 2 and ignoring the remainder.

What is one's complement?

The one's complement of a binary number is formed by simply flipping every bit. All 1s become 0s, and all 0s become 1s. For example, the one's complement of 1010 is 0101.

What is two's complement?

Two's complement is a method used by computers to represent negative numbers. It is calculated by first finding the one's complement (flipping the bits) and then adding 1 to the result. For example, to find the two's complement of 0101: flip to 1010, add 1 to get 1011.

How do you multiply binary numbers?

Binary multiplication is much like decimal long multiplication but simpler since the multipliers are only 0 or 1. You write down the original number shifted left for every 1 in the multiplier, and write 0 for every 0. Then, add all the partial products together.

Can this binary calculator handle negative binary numbers?

This calculator processes the magnitude of standard positive binary strings. To explore negative representations, use the Complements tab to convert positive binary numbers into their Two's Complement form, which is how computers handle negative binary numbers.

How is this binary calculator useful for homework and exams?

This tool covers every binary calculation tested in computer science and digital logic courses. Solve basic arithmetic, verify your bitwise logic truth tables, and test bit shift and complement problems instantly. The output shows the decimal equivalents, helping you understand the process completely.

Binary Calculator

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Binary Calculator Arithmetic · Logic · Shift

Result

—

Add, subtract, multiply, or divide binary numbers.

Binary Number 1

Binary Number 2

Quick Examples

Addition:

Subtraction:

Binary Reference Guide

Decimal Base: 10 (digits 0-9)

Binary Base: 2 (digits 0-1)

AND Rule: 1&1=1, else 0

OR Rule: 0|0=0, else 1

XOR Rule: 1^0=1, 0^1=1, else 0

1's Complement: Flip 1 to 0, 0 to 1

Calculations display both the final binary result and the decimal mapping. Logical and Shift operations output lengths reflecting their minimal required bits or matching bit lengths for clarity.

Binary Calculator: Addition, Subtraction, Bitwise & Shifts

Your complete guide to solving binary math problems, from basic addition and subtraction to logical operators like AND, OR, XOR, and Complements.

How to Use This Binary Calculator

Arithmetic

Solve standard math problems using base-2. Use this tab to perform binary addition, subtraction, multiplication, and division.

Example: Enter 1010 + 110 to calculate the sum. The step-by-step solver reveals both binary (10000) and decimal equivalents.

Bitwise Logic

Apply digital logic gates between two numbers. Support includes AND, OR, XOR, and NOR operations.

Example: 1100 AND 1010 results in 1000. It processes the strings bit by bit to determine the logical outcome.

Bit Shift

Shift a binary string left or right. Essential for understanding how computers multiply or divide by 2 internally.

Example: Left shift 101 by 2 positions yields 10100 (shifting adds trailing zeros).

Complements

Calculate one's complement (inverting bits) and two's complement (inverting bits and adding 1) effortlessly.

Example: Two's complement of 0110 evaluates to 1010. Essential for signed binary arithmetic.

Binary Calculator by Student Level

Middle School

Topics: Introduction to base-2 systems, binary addition, and decimal-to-binary conversion.

How to use: Use the Arithmetic tab. Enter simple binary sequences like 10 + 11. Read the decimal trace in the output to connect the new binary concepts to the base-10 math you already know.

Tip: Always check the decimal values. If you know that 10 is 2 and 11 is 3 in decimal, the sum should equal 5 (101).

High School (AP Computer Science / GCSE)

Topics: Logical operations (AND, OR, XOR), truth tables, one's complement, and binary shifts.

How to use: The Bitwise tab is your best friend. Create your truth table on paper first, then enter the binary numbers here to calculate and confirm your findings.

Tip: Pay attention to the string lengths when doing bitwise operations. Ensure you pad shorter strings with leading zeros to compare correctly.

College Level

Topics: Computer architecture, two's complement arithmetic, bit manipulation, overflow errors, and logical shifts.

How to use: Use the Complements tab to compute signed binary numbers and verify architecture homework. Apply the Shift tab for low-level memory assignment calculations.

Tip: When simulating processor operations, be mindful of standard bit widths (like 8-bit or 16-bit) as two's complement is reliant on the leading sign bit.

Using This Tool for Computer Science Assignments

Homework Problems

Solve binary arithmetic problems by hand first, then enter the values here to verify your answer. If the numbers differ, compare your steps. The most common homework mistakes involve forgetting to carry the 1 during addition or improperly borrowing during subtraction.

Case Studies & Programming

When developing low-level programs in C or Assembly, you frequently need to use bitwise masks. Enter your bit masks into the Bitwise tab and perform an AND or OR to ensure your mask correctly targets the right bits for your software project.

Quiz and Exam Preparation

Click the Random button to load an example. Try to evaluate the logic gate or shift manually, then check your result. Repeatedly converting and solving random binary values is the fastest way to build intuitive speed before an exam.

Problem Sets

For extensive truth table problem sets, use the Bitwise and Complements tabs back-to-back. The integrated calculator cuts down the repetitive manual tracking, freeing you to focus on the overarching digital logic theory.

Understanding Binary Systems

What is the binary number system?

The binary numeral system is a base-2 system used by digital computers to represent data. Unlike our everyday decimal system, which uses 10 digits (0-9), binary uses only two digits: 0 and 1. These represent the off and on states of electrical transistors in a computer's processor.

How do you explain binary to decimal conversion?

Each digit in a binary number represents a power of 2, increasing from right to left starting at 2°. For example, the binary number 1011 equates to (1 × 8) + (0 × 4) + (1 × 2) + (1 × 1) = 11 in decimal. This calculator provides the decimal conversion on every step to help reinforce this concept.

Why are bitwise operators important?

Bitwise operators evaluate numbers at their binary bit level. They are the fundamental building blocks of computer processing. For example, an AND operation checks if two bits are both true. Programmers use these operators to optimize code, manipulate memory directly, and design cryptographic systems.

Real World Uses of Binary Calculations

Where is binary math used in real life?

Every digital device relies on binary math. When you send a text, stream a video, or play a video game, the underlying hardware processes billions of binary additions, shifts, and logical operations every second. Graphics rendering heavily uses bitwise operators and bit shifts to calculate color blending.

How are complements used in processors?

Modern computers do not have dedicated hardware circuits for subtraction. Instead, they use two's complement. By converting a positive number to its two's complement, the processor simply adds the numbers together. Subtracting 5 from 10 becomes adding the two's complement of 5 to 10.

Real World Solved Examples

Each example below shows a real situation where binary math is used, followed by the step-by-step working you would see in this calculator.

Example 1 — Binary Addition (Memory Address Calculation)

A program stores data at memory address 1011 and needs to jump forward by 0110 bytes. What is the new address?

Inputs: 1011 + 0110

1011

+ 0110

Step 1 (bit 0): 1 + 0 = 1

Step 2 (bit 1): 1 + 1 = 0, carry 1

Step 3 (bit 2): 0 + 1 + carry 1 = 0, carry 1

Step 4 (bit 3): 1 + 0 + carry 1 = 0, carry 1

Step 5: carry 1 becomes new bit 4

= 10001

Result: 10001 in binary = 17 in decimal. The new memory address is 17.

Example 2 — Binary Subtraction (Remaining Disk Space)

A storage block has 1110 (14) units allocated. A file uses 0101 (5) units. How many units remain?

Inputs: 1110 − 0101

1110

− 0101

Step 1 (bit 0): 0 − 1 → borrow: 10 − 1 = 1

Step 2 (bit 1): 1 − 1 − borrow 1 = 1 → borrow again: 11 − 1 − 1 = 1

Step 3 (bit 2): 1 − 1 − borrow 1 = 1 → borrow: 11 − 1 − 1 = 1

Step 4 (bit 3): 1 − 0 − borrow 1 = 0

= 1001

Result: 1001 in binary = 9 in decimal. Nine storage units remain.

Example 3 — Two's Complement (CPU Subtraction)

A processor needs to calculate 10 − 3 without a subtraction circuit. It converts 3 to its two's complement and adds.

Input: Two's complement of 00000011 (3)

Step 1 — One's complement (flip all bits):

00000011 → 11111100

Step 2 — Add 1:

11111100 + 00000001 = 11111101

Step 3 — Add to 10 (00001010):

00001010 + 11111101 = 00000111 (ignore overflow bit)

= 00000111

Result: 00000111 = 7 in decimal. The CPU correctly computed 10 − 3 = 7 using only addition.

Example 4 — Bitwise AND (File Permission Check)

A Linux system stores file permissions as a binary mask. To check if a user has read permission (bit 2), the OS runs AND with the mask 0100.

Inputs: 0110 AND 0100

0110

AND 0100

bit 0: 0 AND 0 = 0

bit 1: 1 AND 0 = 0

bit 2: 1 AND 1 = 1

bit 3: 0 AND 0 = 0

= 0100

Result: 0100 ≠ 0, so read permission is granted. If the result were 0000, access would be denied.

Example 5 — Bitwise XOR (Simple Data Encryption)

A program encrypts the byte 1011 using the secret key 1101. XOR is used because applying the same key twice restores the original data.

Inputs: 1011 XOR 1101

1011

XOR 1101

bit 0: 1 XOR 1 = 0

bit 1: 1 XOR 0 = 1

bit 2: 0 XOR 1 = 1

bit 3: 1 XOR 1 = 0

Encrypted = 0110

Decrypt: 0110 XOR 1101 = 1011 ✓ (original restored)

Result: XOR with the same key acts as a toggle — encrypt and decrypt use the exact same operation.

Example 6 — Left Bit Shift (Fast Multiplication in Graphics)

A graphics engine needs to multiply a pixel index (0011 = 3) by 4 to find its byte offset in an RGBA buffer. Left-shifting by 2 is faster than multiplying.

Input: 0011 << 2

Original: 0011

Shift left 1: 0110 (×2 = 6)

Shift left 2: 1100 (×4 = 12)

= 1100

Result: 1100 = 12 in decimal. Pixel 3 starts at byte 12 in the buffer. Left shift by n = multiply by 2ⁿ.

Example 7 — Bitwise OR (Setting Configuration Flags)

A network driver stores settings as bits. The current config is 1001. The program needs to also enable bit 1 (value 0010) without touching other flags.

Inputs: 1001 OR 0010

1001

OR 0010

bit 0: 1 OR 0 = 1

bit 1: 0 OR 1 = 1

bit 2: 0 OR 0 = 0

bit 3: 1 OR 0 = 1

= 1011

Result: 1011. Bit 1 is now set while all original flags are preserved. OR is the standard way to turn on a specific bit.

Frequently Asked Questions

1. How do you add two binary numbers?: Binary addition works similarly to decimal addition but uses only 0 and 1. The rules are: 0+0=0, 1+0=1, 0+1=1, and 1+1=0 (with a carry of 1). For example, adding 101 and 11 gives 1000. You align the columns from the right and carry over the 1 when the sum exceeds 1.
2. How do you subtract binary numbers?: Binary subtraction uses the rules: 0-0=0, 1-0=1, 1-1=0. When subtracting 1 from 0 (0-1), you must borrow 1 from the next higher-order bit, which makes the 0 become 10 (binary for 2), so 10-1=1. Alternatively, subtraction can be done using the two's complement method.
3. What is the difference between AND, OR, and XOR?: These are bitwise logic operations. AND results in 1 only if both bits are 1. OR results in 1 if at least one bit is 1. XOR (Exclusive OR) results in 1 only if the bits are different (one is 1, the other is 0). NOR is the opposite of OR.
4. What is a binary left shift (<<)?: A left shift moves all bits in a binary number to the left by a specified number of positions, filling the new empty spaces on the right with zeros. A left shift of one position is mathematically equivalent to multiplying the number by 2.
5. What is a binary right shift (>>)?: A right shift moves all bits to the right. The bits shifted off the right end are discarded. A right shift of one position is mathematically equivalent to dividing the number by 2 and ignoring the remainder.
6. What is one's complement?: The one's complement of a binary number is formed by simply flipping every bit. All 1s become 0s, and all 0s become 1s. For example, the one's complement of 1010 is 0101.
7. What is two's complement?: Two's complement is a method used by computers to represent negative numbers. It is calculated by first finding the one's complement (flipping the bits) and then adding 1 to the result. For example, to find the two's complement of 0101: flip to 1010, add 1 to get 1011.
8. How do you multiply binary numbers?: Binary multiplication is much like decimal long multiplication but simpler since the multipliers are only 0 or 1. You write down the original number shifted left for every 1 in the multiplier, and write 0 for every 0. Then, add all the partial products together.
9. Can this binary calculator handle negative binary numbers?: This calculator processes the magnitude of standard positive binary strings. To explore negative representations, use the Complements tab to convert positive binary numbers into their Two's Complement form, which is how computers handle negative binary numbers.
10. How is this binary calculator useful for homework and exams?: This tool covers every binary calculation tested in computer science and digital logic courses. Solve basic arithmetic, verify your bitwise logic truth tables, and test bit shift and complement problems instantly. The output shows the decimal equivalents, helping you understand the process completely.