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Free Mathematics tool

Truncation Calculator

Remove digits beyond a selected place without using a nearest-value threshold.

Calculator

Calculate with Truncation

Use 0 for whole numbers, positive values for decimal places, or negative values for tens and larger places. Digits after the selected place are removed, moving the result toward zero.

Truncation result

Your result will appear here.

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Overview

What is a truncation calculator?

Truncation cuts off all digits beyond a selected place and keeps the remaining prefix. Numerically this is directed rounding toward zero: 12.349 truncates to 12.34 at hundredths, while −12.349 truncates to −12.34. The discarded digits never increase the retained digit.

Important terms

Truncation
Removal of digits beyond a selected position without a nearest-neighbor decision.
Toward zero
The direction that reduces absolute value.
Retained prefix
The sign and digits kept through the target place.
Discarded suffix
All digits removed after the target place.

When this method is useful

  • Limiting a displayed value without rounding.
  • Matching integer conversion or ROUND_DOWN behavior.
  • Separating truncation from floor for signed inputs.
  • Studying the error introduced by simply cutting off digits.
Calculator explainer

Truncation Calculator

Cut off digits and move toward zero

Primary formulaTₚ(x) = sign(x) · floor(|x|10ᵖ) / 10ᵖ
Worked example−12.349 → −12.34 at hundredths
retained prefix−12.34discarded suffix9

No carry: the result is −12.34.

Formula

Formula and variables

Tp(x)=sgn⁡(x)⌊∣x∣10p⌋10pT_p(x)=\operatorname{sgn}(x)\frac{\lfloor |x|10^p\rfloor}{10^p}
x
the original signed value
p
decimal places
|x|
the non-negative magnitude
Tₚ(x)
the truncated result

The magnitude is scaled, its fractional part is removed with floor, then the original sign and scale are restored. Using magnitude is what makes negative values move toward zero rather than negative infinity.

Domain note: Places must be an integer from −20 through 50. Truncation does not choose the nearest value and can differ substantially from ordinary rounding when the discarded part is large.

Formula visual

How the rule makes its decision

magnitude|x|10ᵖ→ floor →drop fraction⌊|x|10ᵖ⌋→restore signTₚ(x)
How it works

How to use this calculator

  1. 1

    Enter the signed value.

  2. 2

    Choose the final retained place.

  3. 3

    Scale the magnitude by 10ᵖ.

  4. 4

    Remove the entire fractional part without inspecting its digits.

  5. 5

    Restore the scale and original sign, then review the signed difference.

Worked examples

Step-by-step examples

Example 1

Problem: Truncate −12.349 to two decimal places.

Substitution: |−12.349| × 100 = 1234.9; floor = 1234

Result: −12.34

Restoring the sign makes the result less negative and therefore toward zero.

Example 2

Problem: Truncate 98.999 to one decimal place.

Substitution: floor(989.99) = 989

Result: 98.9

The discarded 99 does not carry into the tenths digit.

Example 3

Problem: Truncate 8,746 to hundreds.

Substitution: p = −2; floor(87.46) = 87

Result: 8,700

Tens and ones are removed.

Worked-example visual

Worked rounding example

1|−12.349| × 1001234.92truncate12343restore − and ÷100−12.34
Method knowledge

Rules, edge cases, and related ideas

No midpoint

Truncation never compares the discarded part with one half.

Magnitude cannot grow

The absolute value of the result is no larger than the input.

Idempotence

Truncating an already truncated result at the same place changes nothing.

Edge cases and limitations

  • Zero remains zero.
  • An exact target-place value remains unchanged.
  • A value smaller than the selected unit may truncate to zero.
  • Negative values move toward zero, unlike floor.

How this differs from a related concept

Truncation and Round Down share the same toward-zero result in this decimal-place setting. Ordinary rounding uses the discarded digits to select a nearest neighbor, while floor differs from truncation for negative values.

Interpretation

Understand the result

Unless the input was already exact, the result has smaller absolute value. The difference reports truncated minus original and can be used to measure the cut-off error.

Common mistakes

  • Increasing a retained digit because the next digit is 5 or more.
  • Using mathematical floor for a negative input.
  • Dropping the sign.
  • Confusing decimal places with significant figures.
  • Assuming trailing written zeros change the numerical result.
FAQ

Frequently asked questions

Is truncation the same as rounding down?

In this calculator, yes: both mean directed rounding toward zero at the selected place.

What happens to −3.99 at whole numbers?

It truncates to −3, not −4.

Does a discarded 9 cause a carry?

No. Truncation simply removes the discarded suffix.

Can I truncate to tens?

Yes. Enter −1 decimal place.

Is truncation always a good approximation?

Not necessarily. It introduces a one-direction magnitude bias and can be nearly one full target unit from the original.

References

References & technical sources

OracleJava RoundingMode — DOWNDefinition of toward-zero rounding as truncation, magnitude behavior, signed examples, and IEEE roundTowardZero correspondence. Verified · Accessed 2026-09-29.Python Software Foundationdecimal — ROUND_DOWN and quantizeExact decimal quantization and toward-zero ROUND_DOWN behavior at a fixed exponent. Verified · Accessed 2026-09-29.
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