截断数

Truncation removes digits from a number beyond a specified position without rounding the remaining digits. For decimal numbers, truncation toward zero removes the unwanted fractional digits while keeping the number closer to zero.

Applying truncation requires following a simple, strict rule:

Truncation to a whole number simply removes the entire fractional portion of the value, leaving only the integer.

• 2.99 → 2
• 2.01 → 2
• −2.99 → −2
• −2.01 → −2

Notice that for negative numbers, truncation removes the fraction, meaning the result moves closer to zero. This is critically different from the mathematical floor function.

To truncate a number to a specific number of decimal places, you simply keep that many digits after the decimal point and delete the rest.

Example for 12.9876:

• 1 decimal place: 12.9876 → 12.9
• 2 decimal places: 12.9876 → 12.98
• 3 decimal places: 12.9876 → 12.987
• 4 decimal places: 12.9876 → 12.9876

Example for −12.9876 (The sign does not change):

• 1 decimal place: −12.9876 → −12.9
• 2 decimal places: −12.9876 → −12.98
• 3 decimal places: −12.9876 → −12.987

Truncation toward zero moves negative numbers strictly toward zero.

Because the negative sign remains unchanged and the fractional part is simply deleted, the resulting value is technically greater than the original (e.g. −3 is greater than −3.9).

• −3.9 → −3
• −3.5 → −3
• −3.1 → −3
• −3.99 to 1 decimal place → −3.9
• −3.99 to 2 decimal places → −3.99

This is a critical distinction: truncation behaves identically to the ceiling function for negative values, whereas it behaves like the floor function for positive values.

For positive numbers, the fractional digits are simply removed, meaning the number moves toward zero (downward on the number line).

• 3.9 → 3
• 3.99 → 3
• 15.678 → 15.67
• 123.4567 → 123.45

If the number is already an integer (a whole number), truncation leaves it completely unchanged because there are no fractional digits to remove.

• 5 → 5
• 0 → 0
• −5 → −5
• 100 → 100

Truncating zero or any variation of zero simply results in zero.

• 0 → 0
• 0.0 → 0
• −0.5 → 0 (when truncating toward zero to a whole number)

Truncation can also apply after evaluating a fraction into its decimal form.

Example 1: 7 / 3 = 2.333...
Truncate to 2 decimal places: 2.33

Example 2: 11 / 4 = 2.75
Truncate to 1 decimal place: 2.7

Example 3: −7 / 3 = −2.333...
Truncate toward zero to 2 decimal places: −2.33

The mathematical concept behind truncating toward zero to a specific number of decimal places (n) involves scaling the number, dropping the fraction, and scaling back.

This works by:

1. Multiplying by 10ⁿ to move the target decimal place to the integer position.
2. Truncating toward zero to drop the remaining fractional portion.
3. Dividing by 10ⁿ to restore the original decimal placement.

Positive Example

Truncate 12.987 to 2 decimal places:

• 12.987 × 100 = 1298.7
• trunc(1298.7) = 1298
• 1298 / 100 = 12.98

Negative Example

Truncate −12.987 to 2 decimal places:

• −12.987 × 100 = −1298.7
• trunc(−1298.7) = −1298
• −1298 / 100 = −12.98

Mathematically, truncation toward zero for a real number x can be defined piecewise using the floor and ceiling functions:

Alternatively, using the sign function (sgn or sign) and absolute value (|x|):

Both formulas mathematically enforce the "move toward zero" rule for both positive and negative values.

Truncation and rounding are fundamentally different operations. Truncation simply ignores the discarded digits entirely. Rounding uses the discarded digits to decide whether the retained digits should be increased.

Example 1: 2.987

• Truncate to 2 decimal places: 2.98
• Round to 2 decimal places: 2.99

Example 2: 2.983

• Truncate to 2 decimal places: 2.98
• Round to 2 decimal places: 2.98

Example 3: −2.987

• Truncate toward zero: −2.98
• Round to nearest hundredth: −2.99

Truncation does not imply that it is a type of nearest-value rounding; it is a separate cutoff mechanism.

Truncation moves values toward zero. The floor function moves values toward negative infinity.

For positive numbers, they act identically. For negative numbers, they move in opposite directions. This makes the negative-number difference extremely clear.

Truncation moves toward zero. The ceiling function moves toward positive infinity.

For negative values, truncation and ceiling often produce the same integer result because moving toward positive infinity from a negative decimal leads to zero. For positive numbers, they differ.

No. Truncation toward zero and mathematical floor rounding (often informally called "rounding down") are different operations for negative numbers.

Consider the value −2.7:

• Truncation → −2
• Floor (Rounding Down) → −3

Because the term "round down" can be ambiguous, always clarify whether you mean moving strictly toward negative infinity (floor) or stripping digits toward zero (truncate).

Round Half Up considers the next digit to decide whether to change the retained value, pushing exact ties away from zero.

For 2.65 to 1 decimal place:

• Truncation → 2.6
• Round Half Up → 2.7

For −2.65 to 1 decimal place:

• Truncation → −2.6
• Round Half Up → −2.7

Banker's Rounding is a nearest-value rule with special handling for exact halfway cases. Truncation simply removes digits without any tie-breaking rules.

For 2.65 to 1 decimal place (exact halfway, retaining an even digit):

• Truncation → 2.6
• Banker's rounding → 2.6

For 2.75 to 1 decimal place (exact halfway, retaining an odd digit):

• Truncation → 2.7
• Banker's rounding → 2.8

Visually, truncation always pulls a non-integer value inward toward zero on the number line.

• For positive values (2.8 → 2), it pulls the value to the left (toward zero).
• For negative values (−2.8 → −2), it pulls the value to the right (toward zero).

Compare this to floor, which always moves to the left (−2.8 → −3) regardless of the sign.

Truncation is highly specific and should be used depending on the requirements of the calculation. You should use truncation when:

• Displaying a fixed number of decimal places without rounding.
• Removing extra precision entirely.
• Processing measurements where the specified cutoff is intentional.
• Performing certain programming and data-processing tasks.
• Implementing controlled numerical display where increasing the displayed value would be undesirable.

Truncation is not always better or worse; it simply serves distinct analytical needs compared to standard rounding.

Concrete examples where truncation is preferred include:

• Financial display: Some specific financial displays explicitly require truncating partial cents downward to prevent showing inflated balances, though actual arithmetic often requires specific rounding rules.
• Measurement data and sensor values: Dropping excessive precision from sensor noise before storage to save space.
• Digital displays: Progress bars showing 99.9% complete will often truncate to 99% to prevent prematurely displaying 100%.
• Data reporting: Statistics and data preprocessing often employ truncation to bucket demographic data securely.
• Scientific calculations: Environments where a fixed cutoff is required by a strict protocol.

Programming languages provide different functions specifically intended for truncating values.

Truncation in Python

Use the math.trunc() function from the math module to truncate a number toward zero. Do not confuse it with math.floor().

Truncation in JavaScript

Use the Math.trunc() function to easily remove the fractional digits of a number toward zero.

Truncation in Java

In Java, explicitly casting a floating-point number to an integer performs mathematical truncation.

Truncation in C#

Use Math.Truncate() which removes the fractional part toward zero.

Truncation in Excel

Use the TRUNC(number, num_digits) function. By default, it truncates to a whole number, but you can easily specify decimal places.

Truncation in SQL

Different database systems use different functions. PostgreSQL and Oracle offer a native TRUNC() function, whereas SQL Server often relies on ROUND(value, decimals, 1) to force truncation instead of rounding.

• Thinking truncation means nearest rounding: Correction — Truncation ignores discarded digits entirely and never rounds up.
• Thinking truncation always moves downward: Correction — It moves toward zero. For negative numbers, this means the value technically moves upward (e.g. −3.9 becomes −3).
• Confusing truncation with floor: Correction — Floor moves toward negative infinity. Truncation moves toward zero.
• Confusing truncation with ceiling: Correction — Ceiling moves toward positive infinity. Truncation moves toward zero.
• Forgetting negative numbers move toward zero: Correction — −2.9 truncates to −2, not −3.
• Using truncation when nearest rounding is required: Correction — Do not use truncation to approximate values; use it only when a hard cutoff is desired.
• Assuming discarded digits affect the retained digits: Correction — Truncation never adjusts the retained digit based on what was removed.
• Confusing decimal display formatting with mathematical truncation: Correction — UI components that hide decimals sometimes apply rounding under the hood. Verify your display layer.
• Using truncation in financial calculations without checking the required rules: Correction — Financial math often mandates standardized rounding rules to avoid destroying value.

Truncation toward zero removes the fractional part without rounding the remaining value.

Test your knowledge with these examples. Click to reveal the truncated answers.