Binary Arithmetic Calculator with Visual Steps

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Category Math & Algebra

Binary A

Binary B

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About

Binary arithmetic is the foundation of all digital computing. While computers process these operations instantly using logic gates, students of computer science must understand the manual algorithms. This includes handling the "Carry" bit in addition and the "Borrow" concept in subtraction.

This tool goes beyond simple calculation by visualizing the vertical layout (long arithmetic). It explicitly shows the carry row for addition and the intermediate partial products for multiplication. This visualization aids in debugging manual calculations and understanding how the Arithmetic Logic Unit (ALU) processes data at the hardware level.

Formulas

Binary addition follows the rule: 1 + 1 = 10 (result 0, carry 1).

S_i = A_i &xor; B_i &xor; C_in

C_out = (A_i ∧ B_i) ∨ (C_in ∧ (A_i &xor; B_i))

Reference Data

A	B	Sum (+)	Carry	Difference (−)
0	0	0	0	0
0	1	1	0	1 (Borrow 1)
1	0	1	0	1
1	1	0	1	0

Frequently Asked Questions

When subtracting 1 from 0, you must borrow from the next higher column. The 0 becomes "10" (decimal 2). Then, 2 - 1 = 1. The column you borrowed from is reduced by 1.

Overflow occurs when the result of an arithmetic operation exceeds the number of bits allocated to store it. For example, adding 1 to 1111 (15) in a 4-bit system results in 10000 (16), but if only 4 bits are stored, the result appears as 0000.

This calculator treats inputs as unsigned integers for simplicity. In signed arithmetic (Two's Complement), the leading bit indicates the sign (0 for positive, 1 for negative).