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Long Multiplication Calculator

Perform long multiplication with clear partial products, aligned steps, and carry explanation. Enter two integers to get the final product instantly.

Calculate Product

Inputs

Your Results

699678
final product

Partial Products

1234 × 5 (×10^2)
617000
1234 × 6 (×10^1)
74040
1234 × 7
8638

Key Concepts

Place Value

Each digit’s position represents powers of ten; shifting adds trailing zeros.

Partial Products

Multiply the multiplicand by each digit of the multiplier, then sum.

Carry Handling

Carries propagate left; we show the carry trail for transparency.

Big Integers

Algorithm works on digit strings, enabling very large integer inputs.

How to Calculate

Algorithm (Manual Method)

1) For each digit d of multiplier (right → left), compute multiplicand × d.
2) Shift the partial product by its position (append zeros).
3) Sum all partial products to get the final product.

Calculation Steps:

  1. 1
    Normalize inputs
    Remove non-digits and leading zeros (keep one zero if needed).
  2. 2
    Create partial products
    Multiply multiplicand by each digit of multiplier.
  3. 3
    Apply shifts
    Append zeros based on the digit position.
  4. 4
    Sum all partials
    Add the shifted results to obtain the final product.

Important Considerations

⚠️ Input Rules

Only non-negative integers are supported in this version; decimals and negatives are not handled.

Input Sanitization

We strip non-digits and collapse leading zeros to keep calculations stable.

Large Numbers

Performance is linear in digits; extremely long inputs may be slower on low-end devices.

Educational Aid

Use the carry trail to verify each manual step when learning.

Rounding

All operations are exact integer arithmetic; no rounding is applied.

Example Cases

Case 1

Inputs: 1234 × 567
Product: 699,678

Demonstrates three partial products aligned by place value.

Case 2

Inputs: 8005 × 902
Product: 7,220,510

Zeros in inputs create larger shifts; useful for teaching alignment.

Tips

  • Break long inputs into chunks to verify each partial product.
  • Use grouping (e.g., thousands separators) when checking results.
  • If a digit is 0, its partial product is 0; still include the shift.

Frequently Asked Questions

What is long multiplication?
Long multiplication is a method to multiply multi-digit integers by breaking the process into partial products and summing them.
Does this support big integers?
Yes. The algorithm operates on digit strings, so it can handle arbitrarily long non-negative integers.
Are decimals or negatives supported?
This version focuses on non-negative integers without decimals. You can simulate powers of ten by adding zeros.
What is a partial product?
It is the result of multiplying the multiplicand by one digit of the multiplier, shifted by its place value (trailing zeros).
How are carries shown?
Each step shows a carry trail to illustrate how carries accumulate across digits during multiplication.

Related Calculators

Long Multiplication Calculator - Steps & Partial Products