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Math calculator

Multiplication Calculator

Multiply two numbers instantly, or see long multiplication with partial products, carrying, decimals, and sign rules. Runs locally in your browser.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance
Input interpretation
Enter values to calculate.
Result
—
Assurance
Core
Declared partition coverage
PASS · 5/5 declared partitions (product, long, long-grid, digit-bound, invalid-domain) · Matrix
Numerical scope
Exact decimal-string identity (productStr) vs O2 mantissa×BigInt dual-checked against per-digit BigInt partials. Long vectors also require IUT work to contain those partials, or LONG_GRID_TRUNCATED at 25 digits. 48 significant digits per factor; 49-digit factors are rejected. This is not a whole-domain proof beyond the published vectors.
Known limitations
  • product or long mode; shared exact path; long steps independently reconstructed
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Product of two numbers; optional long-multiplication working with partial products for the standard algorithm.
Scope
Real decimal inputs parsed as exact decimal strings (integer mantissa × 10^−scale), not binary floating-point.
Verification
Engine tested · Source checked · v1.0.3 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance· View Manifest · CVP overview · Specification
Versions
Calculation 1.0.3 · CVP protocol 1.0.0-proposed · Evidence 2026-09-16.sources-no-wolfram
Verification revision
2026-09-16.sources-no-wolfram · 6/6 property · digest 7a9319794cf7
Legacy regression
15/15 tests · Production surface contract 4/4
Trust layers
Verification VERIFIED · Production CURRENT · overall VERIFIED
Reference
O1 model · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
12/12 golden · 5/5 CVP boundary · 4/4 invalid · 6/6 property · 2/2 metamorphic · 2/2 round-trip · 3/3 cross-interface · 4/4 CVP contract · Manifest
Sources
Sources
Evidence
10 legacy golden · 5 legacy boundary · legacy regression suite · 12/12 oracle-backed golden · 4/4 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.0.3 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas & properties

Core product identity plus algebraic properties of multiplication.

Productproduct = multiplicand × multiplier
Repeated addition — for non-negative integer multipliera × n = a + a + … + a (n times), n ∈ ℤ≥0
Commutativea × b = b × a
Distributivea × (b + c) = a×b + a×c
iSymbols: ×, ·, or *. Decimal model: parse each factor as an exact decimal (integer mantissa + scale), multiply mantissas, restore scale (e.g. 45.2 × 0.21 → 452 × 21 × 10⁻³ = 9.492). Sign: same signs → positive; different signs → negative. Product and Long modes share this exact path.

How to use

1

Choose a mode

Product for a quick answer; Long multiplication for the standard algorithm with work shown.

2

Enter multiplicand and multiplier

Whole numbers, decimals, and negatives are supported.

3

Read the product and steps

Long mode shows a digit grid you can step through: each partial product, carry, shift, decimal placement, and sign.

Example calculations

Common configurations with formula and result.

ϟ

Basic

8 × 6

repeated addition
48
ϟ

Long multiplication

2552 × 64

10208 + 15312×10
163,328
ϟ

Decimals

45.2 × 0.21

3 decimal places
9.492
ϟ

Signs

−5 × 11

unlike signs
−55

Quick examples

Common values at a glance.

MultiplicandMultiplierProduct
7856
1212144
2345613,104
255264163,328
45.20.219.492
−511−55
1.51725.5
i In 234 × 56 = 13,104, 234 is the multiplicand, 56 the multiplier, and the rows under the line are partial products.

Multiplication calculator specification

Version 1.0.3 · Engine tested

Calculation status

Review policy · Evidence

Definition
Multiplication finds the product of a multiplicand and a multiplier. For a non-negative integer multiplier n, it is repeated addition: a × n means a added to itself n times. Long multiplication is the standard place-value algorithm with partial products and carrying.
What it calculates
Product of two numbers; optional long-multiplication working with partial products for the standard algorithm.
Inputs
  • Multiplicand
  • Multiplier
Outputs
  • Product
  • Long-multiplication steps (partial products, carries, decimal placement, sign)
Formula
product = a × b
Assumptions
  • Real decimal inputs parsed as exact decimal strings (integer mantissa × 10^−scale), not binary floating-point.
  • Product mode and Long multiplication mode use the same exact mantissa multiplication; Long adds partial-product steps.
Units
  • Same as the product of the input units (dimensionless if pure numbers)
Boundary conditions
  • Non-numeric input → error
  • Scientific notation (e.g. 1.2e3) → error; use plain decimals only
  • Product mode: up to 48 significant digits per factor
  • Long visual grid: up to 24 significant digits per factor; larger factors still return an exact product with LONG_GRID_TRUNCATED (working steps omitted — never silently truncated)
Example
2552 × 64 = 163328
Validation cases

7 published on this page · 15/15 tests · Production surface contract 4/4 · View evidence

  • 8 × 6 → 48
  • 2552 × 64 → 163328
  • 234 × 56 → 13104
  • 45.2 × 0.21 → 9.492
  • −5 × 11 → −55
  • 0.1 × 0.2 → 0.02
  • 0 × 99 → 0
Sources
  • OpenStax Prealgebra 2e — Multiply Whole Numbers — 1.4 Product of two numbers
    Supports: Standard multiplication as the product of a multiplicand and a multiplier.
  • CalculatorX mathematical conventions — Exact decimal place-count multiplication
    Supports: Long-multiplication algorithm with partial products; exact decimal mantissa × BigInt; sign rules for negatives.
Calculation version
1.0.3

Background

Interpretation and common distinctions.

What is multiplication?

Multiplication combines equal groups. For a non-negative integer multiplier n, it is repeated addition:

a × n = underbracea + a + ⋯ + a(n times), n ∈ ℤ(≥ 0)

Example: 3 × 4 = 3+3+3+3 = 12. This interpretation does not apply to fractional multipliers (e.g. 8 × 0.5 is not “8 added half a time”).

The result is the product. Common symbols: ×, ·, or *.

product = multiplicand × multiplier

How to do long multiplication

  1. Stack numbers, aligning place values (usually the longer number on top).
  2. Multiply the top number by the ones digit of the bottom number; write the partial product.
  3. Multiply by the tens digit, writing the next partial product shifted one place left, and so on.
  4. Add the partial products to get the product.

Example: 2552 × 64

  • 2552 × 4 = 10208
  • 2552 × 6 = 15312 (shift one place)
  • Sum → 163,328

Decimals

CalculatorX uses exact decimal arithmetic (not binary float):

  1. Parse each factor as an integer mantissa and a decimal scale (e.g. 45.2 → 452 × 10⁻¹).
  2. Multiply mantissas with BigInt.
  3. Restore the combined scale (e.g. 45.2 × 0.21 → 452 × 21 × 10⁻³ = 9.492).

So 0.1 × 0.2 is exactly 0.02, never a binary rounding artifact. Product and Long modes share this path.

Negative numbers

Ignore signs while multiplying absolute values, then:

  • Same signs → positive product
  • Different signs → negative product

Properties

  • Commutative: a × b = b × a
  • Associative: (a × b) × c = a × (b × c)
  • Distributive: a(b+c) = ab + ac
  • Identity: a × 1 = a; zero: a × 0 = 0

Other calculators in this family: Division, Addition (sum), Fractions, Percentage .

÷ Division › + Addition (sum) › a/b Fractions › % Percentage ›
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Frequently asked questions

Key distinctions behind the calculation.

What is multiplication?

It is one of the four basic arithmetic operations. For a non-negative integer multiplier n it means repeated addition: 3 × 4 = 3+3+3+3 = 12. Fractional multipliers use the exact decimal (scale) rule instead.

What are the parts of a multiplication problem?

The multiplicand is multiplied by the multiplier; the result is the product. In 5 × 7 = 35, 5 is the multiplicand, 7 the multiplier, 35 the product.

How do you do long multiplication?

Stack the numbers by place value. Multiply the top number by each digit of the bottom number from right to left, writing partial products shifted left for each higher place. Add the partial products.

How do you multiply decimals?

Ignore decimals and multiply as whole numbers (exact mantissa × BigInt), then put a decimal in the product with as many places as the total decimal places in both factors. Example: 0.1 × 0.2 = 0.02 exactly. Plain decimals only — scientific notation like 1.2e3 is not supported here.

How do signs work?

Positive × positive or negative × negative → positive. One positive and one negative → negative product.

What is the commutative property?

Order does not matter: a × b = b × a.

What is the distributive property?

a × (b + c) = a×b + a×c. Useful for mental math and algebra.

Where is multiplication used in real life?

Totals (price × quantity), area (length × width), scaling recipes, unit conversions, and school word problems.