About

Long arithmetic is the cornerstone of numerical literacy, teaching not just how to get an answer, but the logical structure of our base-10 number system. Whether you are a student verifying homework or a professional needing a sanity check on manual calculations, seeing the process is as vital as the result. This tool visualizes the carry and borrow mechanisms that occur in standard algorithms.

Unlike standard calculators that hide the complexity, this utility breaks down the operation into atomic steps. It supports Addition, Subtraction, Multiplication, and Long Division. Crucially, for division, it supports different notational standards, accommodating the visual differences between American (bracket) and European (right-angle) styles.

Formulas

The general algorithm for column addition relies on processing digits from right to left (least significant to most significant). For two integers A and B:

S_i = (A_i + B_i + C_i) mod 10

Where C is the carry value defined as:

C_i+1 = floor(A_i + B_i + C_i10)

Long division follows the recursive relationship:

Dividend = Divisor × Quotient + Remainder

Reference Data

Operation	Key Concept	Visual Element	Complexity (Big O)
Addition	Regrouping (Carrying)	1 above column	O(n)
Subtraction	Decomposition (Borrowing)	Strike-through	O(n)
Multiplication	Partial Products	Staggered Rows	O(n²)
Division	Quotient & Remainder	Bus Stop / Bracket	O(n²)
Modulo	Remainder Only	r	O(n²)
Exponentiation	Repeated Multiplication	Superscript	O(log n)

Frequently Asked Questions

Mathematics is universal, but notation evolves locally. In the US and UK, the 'Bus Stop' method (using a bracket √) is common. In many parts of continental Europe and Latin America, the divisor is placed to the right of the dividend separated by a vertical bar or a right angle. This tool allows you to toggle between these visual styles.

The tool calculates the integer quotient and the final remainder. It does not automatically continue into decimals (e.g., 10 ÷ 3 will show 3 with a remainder of 1, rather than 3.333). This is to preserve the step-by-step integer arithmetic logic taught in schools.

Currently, the Column Method visualizer is optimized for non-negative integers (natural numbers &Nopf;). Standard column algorithms are typically taught using absolute values, with the sign applied to the final result separately.