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

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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: OpenStax Prealgebra 2e — Multiply Whole Numbers — 1.4 Product of two numbers
CalculatorX mathematical conventions — Exact decimal place-count multiplication
Sources
Evidence: 10 legacy golden · 5 legacy boundary · legacy regression suite · 12/12 oracle-backed golden · 4/4 invalid · Artifact integrity PASS
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

Choose a mode

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

Enter multiplicand and multiplier

Whole numbers, decimals, and negatives are supported.

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.

Basic

8 × 6

repeated addition

Long multiplication

2552 × 64

10208 + 15312×10

163,328

Decimals

45.2 × 0.21

3 decimal places

9.492

Signs

−5 × 11

unlike signs

−55

Quick examples

Common values at a glance.

Multiplicand	Multiplier	Product
7	8	56
12	12	144
234	56	13,104
2552	64	163,328
45.2	0.21	9.492
−5	11	−55
1.5	17	25.5

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

Engine tested 15/15 tests · Production surface contract 4/4
Named expert review Not performed
Calculation version 1.0.3

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

Stack numbers, aligning place values (usually the longer number on top).
Multiply the top number by the ones digit of the bottom number; write the partial product.
Multiply by the tens digit, writing the next partial product shifted one place left, and so on.
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):

Parse each factor as an integer mantissa and a decimal scale (e.g. 45.2 → 452 × 10⁻¹).
Multiply mantissas with BigInt.
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 Calculator, Fractions Calculator, Scientific Notation Calculator, Subtraction Calculator, Sum Calculator . Explore all Basic Arithmetic.

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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.

Multiplication Calculator

Formulas & properties

How to use

Choose a mode

Enter multiplicand and multiplier

Read the product and steps

Example calculations

Basic

Long multiplication

Decimals

Signs

Quick examples

Multiplication calculator specification

Background

What is multiplication?

How to do long multiplication

Decimals

Negative numbers

Properties

Related tools

Frequently asked questions