Calculator Inputs
Use direct comparison, series truncation, or a step-size error model. Results appear above this form after submission.
Example Data Table
| Mode | Input summary | Approximation | Absolute error | Relative error | Notes |
|---|---|---|---|---|---|
| Direct | Exact = 3.1415926536, decimal truncation to 2 places | 3.14 | 0.0015926536 | 0.0005069574 | Shows chopped decimal truncation. |
| Direct | Exact = 12.98765, decimal truncation to 2 places | 12.98 | 0.00765 | 0.0005890211 | Useful for fixed decimal storage. |
| Direct | Exact = 0.0049876, truncation to 3 significant figures | 0.00498 | 0.0000076 | 0.0015237790 | Highlights scale-aware precision control. |
| Series | e^x at x = 1 with 4 retained terms | 2.6666666667 | 0.0516151618 | 0.0189881569 | Approximation improves as more terms are kept. |
| Series | sin(x) at x = 0.7 with 3 retained terms | 0.6442339167 | 0.0000162294 | 0.0000251925 | Alternating series can converge rapidly. |
| Step-size | C = 1.6, h = 0.25, p = 2 | Estimated |E(h)| = 0.1 | 0.1 | N/A | Model-based truncation estimate. |
Formula Used
1) Absolute truncation error
|E| = |Exact value − Approximate value|
2) Relative truncation error
Relative error = |E| / |Exact value|
3) Percentage error
Percentage error = Relative error × 100
4) Decimal truncation
Approximate value = chopped value after chosen decimal places
5) Significant-figure truncation
Approximate value = chopped value after chosen significant figures
6) Step-size truncation model
|E(h)| ≈ C × hp, where C is a constant, h is the step size, and p is the method order.
7) Series remainder idea
For a truncated Maclaurin series, the omitted next terms create the truncation error. The calculator also estimates a remainder bound when the selected function allows it.
How to Use This Calculator
Step 1: Choose the calculation mode that matches your problem. Use direct comparison for known exact values, series mode for Maclaurin approximations, and step-size mode for numerical methods.
Step 2: Enter the required inputs. For direct comparison, decide whether the approximation is manual, chopped by decimal places, or chopped by significant figures.
Step 3: Click Calculate Truncation Error. The result section will appear below the header and above the form.
Step 4: Review the absolute, relative, and percentage error values. Use the Plotly graph to see how the approximation behaves as retained precision or step size changes.
Step 5: Export the result and example table using the CSV or PDF buttons for documentation, reporting, or classroom use.
FAQs
1) What is truncation error?
Truncation error is the difference caused by stopping an infinite process early. It appears when you cut off digits, stop a series, or replace calculus with a discrete numerical formula.
2) How is truncation different from rounding?
Truncation simply cuts off extra digits or terms. Rounding adjusts the kept part based on the next digit or term. Truncation usually introduces a one-sided loss, while rounding may reduce bias.
3) Why is relative error useful?
Relative error scales the absolute gap by the exact value. This makes comparisons fair when values have very different sizes, such as comparing errors in 0.01 and 10,000.
4) When should I use the step-size model?
Use the step-size model when a numerical method has a known order, such as first-order or second-order finite differences. It helps predict how error changes as the step size decreases.
5) What does the order p mean?
The order p tells how strongly error depends on step size. If p is larger, halving the step size reduces truncation error more aggressively, assuming the error model remains valid.
6) Why does the series mode include a remainder bound?
A remainder bound estimates the maximum size of the omitted terms. It helps you judge whether the retained terms are enough for your required precision.
7) Can truncation error ever be zero?
Yes. It can be zero when the approximate value exactly matches the exact value, or when a series or method reproduces the target value exactly for that input.
8) Does a smaller absolute error always mean a better approximation?
Usually yes, but context matters. If the true value is very small or very large, relative or percentage error may give a more meaningful view of practical accuracy.