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Math calculator

Division Calculator

Divide numbers with quotient, remainder, and decimal results for quick arithmetic checks. Runs locally in your browser. Free online tool — no sign-up.

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 · 7/7 declared partitions (quotient-normal, quotient-rounding, quotient-boundary, long-normal, long-large-integer, long-boundary, invalid-domain) · Matrix
Numerical scope
≤2 ULP vs O3 applies to the published tabulated quotient vectors (1/3, 100/7, 1.005/1, 1/1e-300, (2^53−1)/7). Long-division vectors are exact integer q,r (Python int divmod). This is not a whole-domain guarantee.
Known limitations
  • quotient or long integer division
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Decimal quotient a÷b (IEEE-754 binary64, then rounded for display), or integer long division with quotient, remainder, and steps.
Scope
Quotient mode divides in IEEE-754 binary64, then rounds the display with Number.prototype.toFixed (round half away from 0 on that binary64 value). Trailing zeros may be stripped.
Verification
Engine tested · Source checked · v1.0.5 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance· View Manifest · CVP overview · Specification
Versions
Calculation 1.0.5 · CVP protocol 1.0.0-proposed · Evidence 2026-09-08.exact-contract
Verification revision
2026-09-08.exact-contract · 8/8 property · digest e5997fd9b998
Legacy regression
34/34 tests · Production surface contract 3/3
Reference
O1 model · O3 expected_values · O3 numerical_behavior · 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
10/10 golden · 9/9 CVP boundary · 9/9 invalid · 8/8 property · 2/2 metamorphic · 2/2 round-trip · 9/9 O3 · 2/2 cross-interface · 1/1 CVP contract · Manifest
Sources
Sources
Evidence
25 legacy golden · 9 legacy boundary · legacy regression suite · 10/10 oracle-backed golden · 9/9 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.0.5 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

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

Formulas

Core equations used by this calculator.

Quotientq = a ÷ b
Displayed decimalround_toFixed(a ÷ b, n) after IEEE-754 binary64 division
Long divisiona, b ∈ ℤ≥0, b > 0; q = ⌊a/b⌋; r = a − bq; 0 ≤ r < b
Checkb·q + r = a
iNotation: a ÷ b, a/b, or a fraction bar. Division by zero is undefined. Quotient mode shows a rounded IEEE-754 binary64 decimal, not an exact repeating expansion. Long division is exact BigInt Euclidean division for non-negative integers, limited to 48 digits to bound computation and long-division step output. Decimal places must be an integer 0–16; out-of-range values are rejected.

How to use

1

Choose a mode

Quotient for a decimal answer. Long division for integer quotient, remainder, and written steps.

2

Enter dividend and divisor

Dividend is the number being divided; divisor is what you divide by (must not be zero).

3

Read the result

Long division also prints step-by-step work under the form and verifies b·q + r = a.

Example calculations

Common configurations with formula and result.

ϟ

Even split

8 ÷ 4

8 / 4
2
ϟ

With remainder

100 ÷ 7

14 R 2
14 R 2 (7×14+2=100)
ϟ

Long division

487 ÷ 32

15 R 7
15 R 7 (32×15+7=487)
ϟ

Decimal quotient

100 ÷ 7

100/7
14.28571429 (8 decimal places, rounded)

Division calculator specification

Version 1.0.5 · Engine tested

Calculation status

Review policy · Evidence

Definition
Division asks how many times one number (the divisor) fits into another (the dividend). The result is the quotient. When the fit is not exact for integers, a remainder is left: dividend = divisor × quotient + remainder.
What it calculates
Decimal quotient a÷b (IEEE-754 binary64, then rounded for display), or integer long division with quotient, remainder, and steps.
Inputs
  • Dividend a
  • Divisor b ≠ 0
  • Mode: Quotient or Long division
  • Optional decimal places n (Quotient mode, default 8; must be an integer 0–16)
Outputs
  • Rounded decimal quotient (Quotient mode) plus exact flag
  • Quotient and remainder (Long division)
  • Step-by-step long division work
Formula
q = a/b (binary64); display = toFixed(q, n). Integers: q = ⌊a/b⌋, r = a − bq, 0 ≤ r < b.
Assumptions
  • Quotient mode divides in IEEE-754 binary64, then rounds the display with Number.prototype.toFixed (round half away from 0 on that binary64 value). Trailing zeros may be stripped.
  • The displayed decimal is not claimed to be an exact repeating expansion.
  • REST field exact is true iff ieee_value equals the real number a÷b for those binary64 operands. It is not display round-trip and not integer divisibility. Example: (2^53−1)÷7 has remainder 3, so exact is false even when toFixed round-trips.
  • REST field display_exact is true iff Number(ieee_value.toFixed(n)) === ieee_value (display did not change the binary64 value).
  • Long division uses exact BigInt Euclidean division for non-negative integers: a, b ∈ ℤ≥0, b > 0. Integers are limited to 48 decimal digits to bound computation and long-division step output (a CalculatorX product contract, not a JavaScript BigInt limit).
  • Remainder satisfies 0 ≤ r < b in long division (verified as b·q + r = a in BigInt). Long-mode exact is true iff remainder is 0.
  • Quotient mode does not silently round plain integer tokens outside ±(2^53−1); those inputs return INTEGER_OUT_OF_SAFE_RANGE. Send large integers as decimal strings and use Long division for an exact result.
  • REST/JSON reports ieee_value (raw binary64, or null when the long-division operands do not fit in a safe integer), decimal_value (display string), precision, rounding, exact (mathematical vs binary64 operands), display_exact (toFixed round-trip), exact_value (a/b when both are integers), integer_quotient/remainder for integer quotient-mode operands, and integer_engine=bigint in long mode.
Units
  • Unitless. This calculator operates on unitless numeric values. If the inputs represent physical quantities, units divide as unit(a) / unit(b).
Boundary conditions
  • b = 0 → DIVISION_BY_ZERO
  • Non-finite a or b → INVALID_NUMBER / MISSING_REQUIRED_INPUT
  • Non-finite binary64 quotient (overflow) → INVALID_NUMBER
  • Long mode: negatives or non-integers → INVALID_INPUT (use Quotient mode)
  • Long mode: more than 48 digits → INTEGER_TOO_LONG (product limit to bound computation and step output)
  • Quotient mode: plain integer token outside ±(2^53−1) → INTEGER_OUT_OF_SAFE_RANGE
  • decimals missing or blank → default 8; otherwise must be an integer in 0…16 or DECIMAL_PLACES_OUT_OF_RANGE (no silent clamp)
Numerical precision
  • Quotient-mode engine value is IEEE-754 binary64 a/b. That raw quotient is the published REST number (ieee_value / value).
  • Display rounding uses JavaScript Number.prototype.toFixed(n): round half away from 0 on the binary64 value, then optional trailing-zero strip.
  • exact means ieee_value equals the real a÷b of the binary64 operands. display_exact means the chosen toFixed(n) round-trips to that same binary64 value.
  • Because the input to toFixed is already binary64, values such as 1.005/1 at 2 places may display 1.00 (1.005 is not exact in binary64 as an input; dividing by 1 is mathematically exact in binary64).
  • 100÷7 at 8 places displays 14.28571429 (rounded). The exact rational is 100/7 = 14.285714… repeating. exact is false; display_exact is false at 8 places.
  • (2^53−1)÷7 ≈ 1286742750677284.428…; ieee_value is 1286742750677284.5, exact is false, display_exact at 8 places is true.
  • 1÷3 at 8 places displays 0.33333333; 2÷3 at 8 places displays 0.66666667.
  • Long division is exact integer arithmetic (JavaScript BigInt), so 9007199254740993 ÷ 1 = 9007199254740993 R 0.
  • Interactive UI, shared URLs, and REST use the same divide engine.
Example
100 ÷ 7 at 8 decimal places → 14.28571429 (rounded); long 100 ÷ 7 → 14 R 2; long 9007199254740993 ÷ 1 → 9007199254740993 R 0
Validation cases

16 published on this page · 34/34 tests · Production surface contract 3/3 · View evidence

  • 8 ÷ 4 → 2
  • 100 ÷ 7, decimals=8 → 14.28571429 (rounded)
  • 100 ÷ 7 (long) → 14 R 2
  • 487 ÷ 32 (long) → 15 R 7
  • 15 ÷ 2 (long) → 7 R 1
  • 0 ÷ 7 → 0
  • 1 ÷ 3, decimals=8 → 0.33333333
  • 2 ÷ 3, decimals=8 → 0.66666667
  • 9007199254740993 ÷ 1 (long) → 9007199254740993 R 0
  • 9007199254740991 ÷ 7, decimals=8 → ieee_value 1286742750677284.5; exact false; display_exact true
  • 100 ÷ 0 → DIVISION_BY_ZERO
  • 100 ÷ 7, decimals=100 → DECIMAL_PLACES_OUT_OF_RANGE
  • 9007199254740993 ÷ 1 (quotient) → INTEGER_OUT_OF_SAFE_RANGE
  • −10 ÷ 3 (long) → INVALID_INPUT
  • 1.5 ÷ 2 (long) → INVALID_INPUT
  • 49-digit integer ÷ 1 (long) → INTEGER_TOO_LONG
Sources
Calculation version
1.0.5

Background

Interpretation and common distinctions.

What division is

Division is one of the four basic arithmetic operations (with addition, subtraction, and multiplication). It asks how many times one number fits into another.

Example: 2 goes into 8 four times, so 8 ÷ 4 = 2.

Common notation:

8 ÷ 4 = 2, 8/4 = 2, 8/4 = 2

Parts of a division problem

Part Role Example in 487 ÷ 32 = 15 R 7
Dividend Number being divided 487
Divisor Number you divide by 32
Quotient How many times the divisor fits 15
Remainder What is left over 7

Think of the dividend as a total (for example 20 apples) and the divisor as how many equal groups you want. The quotient is how many go in each group; the remainder is what cannot form another full group.

Modes on this calculator

Quotient

Computes the IEEE-754 binary64 value a ÷ b, then rounds the display to the selected number of decimal places. Works with negatives and decimals. The main result is the rounded decimal, not an exact repeating expansion.

Long division

For non-negative integers a, b ∈ ℤ(≥ 0) with b > 0:

q = ⌊ a/b ⌋, r = a − bq, 0 ≤ r < b

and prints place-value steps under the form. Arithmetic is exact BigInt (not binary64), so integers larger than 2⁵³−1 stay exact. The 48-digit cap is a CalculatorX product limit (to bound computation and step output), not a JavaScript BigInt limit.

Plain integer tokens outside ±(2⁵³−1) are rejected in Quotient mode rather than silently rounded. Decimal places must be an integer in 0… 16; values such as 100 are a validation error, not a silent clamp to 16.

How to do long division (overview)

  1. Write the dividend under the bracket and the divisor to the left.
  2. From the left, see how many times the divisor fits into the leading part of the dividend; write that digit in the quotient.
  3. Multiply, subtract, bring down the next digit.
  4. Repeat until every digit is used. The last subtraction result is the remainder (if you stop without decimals).

Example: 100 ÷ 7

100 = 7 × 14 + 2 implies 100 ÷ 7 = 14 R 2

Continuing with zeros after the decimal produces 14.285714… (repeating).

Example: 487 ÷ 32

487 = 32 × 15 + 7 implies 487 ÷ 32 = 15 R 7

Tips

  • Division by zero is undefined — the tool will reject a divisor of 0.
  • Prefer long division when checking homework that asks for a remainder, or when the integers may exceed 2⁵³−1.
  • Prefer Quotient when you need a decimal for science, engineering, or everyday calculations.
  • Decimal places are 0–16. Out-of-range values are rejected; they are not silently limited.
  • Multiplication undoes division: if a ÷ b = q, then b × q = a when the remainder is 0.

Related: Multiplication, Sum, Subtraction, Fractions.

Other calculators in this family: Fractions Calculator, Multiplication Calculator, Scientific Notation Calculator, Subtraction Calculator, Sum Calculator . Explore all Basic Arithmetic.

→ Fractions Calculator Add, subtract, multiply, or divide fractions and mixed numbers with step-by-step LCD work. Free online tool — no sign-up. Runs locally in your browser. › → Multiplication Calculator Multiply two numbers instantly, or see long multiplication with partial products, carrying, decimals, and sign rules. Runs locally in your browser. › → Scientific Notation Calculator Convert numbers to scientific, E, and engineering notation. Free online tool — no sign-up. Mobile-friendly and privacy-first. Runs locally in your browser. › → Subtraction Calculator Subtract minuend − subtrahend to get the difference. Free online tool — no sign-up. Clear formulas and worked examples. Interactive calculations run locally in your browser. › → Sum Calculator Add a list of numbers (sum), see the mean, or work long addition with place-value steps. Paste values separated by commas, spaces, or new lines — decimals and negatives. ›
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Frequently asked questions

Key distinctions behind the calculation.

Is the decimal quotient exact?

The REST field exact means the binary64 ieee_value equals the real number a÷b for those binary64 operands — not that the displayed decimal is a repeating expansion, and not that toFixed round-tripped. 100÷7 and (2^53−1)÷7 are both exact=false. display_exact is the separate toFixed round-trip flag.

What are the parts of a division problem?

Dividend (number being divided), divisor (number you divide by), quotient (result), and sometimes a remainder (what is left).

What does 14 R 2 mean?

Quotient 14 with remainder 2. For example 100 ÷ 7 = 14 R 2 because 7×14+2=100.

When should I use long division?

When you want an exact integer quotient and remainder, including integers larger than 2^53−1. Use Quotient mode for a decimal answer.

Can I divide integers larger than JavaScript can store exactly?

In Long division, yes: the engine uses BigInt, so values such as 9007199254740993 ÷ 1 stay exact. The published limit is 48 decimal digits — a CalculatorX product bound on computation and step output, not a JavaScript BigInt limit. In Quotient mode a plain integer outside ±(2^53−1) is rejected rather than silently rounded.

What if I enter 100 decimal places?

Decimal places must be an integer between 0 and 16. Values outside that range return a validation error; the engine does not silently clamp 100 to 16.

Can the divisor be zero?

No. Division by zero is undefined.

What is the full division formula?

Dividend = Divisor × Quotient + Remainder.

How is division written?

Common forms are a ÷ b, a/b, and the fraction bar a⁄b.