Singapore School Mathematics Operating Manual · Chapter 28
Some problems are forgiving. Small input changes produce small output changes.
Other problems are fragile. A tiny measurement error, rounding change or parameter shift can create a much larger change in the answer. Two nearly identical data sets can lead to very different reconstructed values. A denominator close to zero can explode a result. A decision can flip because an uncertainty interval crosses a threshold.
This chapter develops the school-level ideas behind conditioning, ill-posedness, sensitivity and stability. The aim is not to turn ordinary Mathematics into advanced numerical analysis. It is to give learners a precise way to ask: if the input changes slightly, should I expect the answer to change slightly too?
1. Sensitivity asks how strongly outputs respond to input changes
For y = 2x + 3, increasing x by 0.01 changes y by 0.02.
The response is controlled and predictable.
For y = 1/x near x = 0, the same 0.01 input change can have a huge effect.
Sensitivity depends on both the formula and the operating region.
2. A stable problem tolerates small perturbations
If a rectangle length changes from 10.00 cm to 10.01 cm while width stays 5 cm, area changes from 50.00 cm² to 50.05 cm².
The output change is modest and proportional to the input change.
This is a well-behaved local relationship.
3. A sensitive problem can amplify uncertainty
Consider y = 1/(x−2).
At x=3, y=1.
At x=2.01, y=100.
At x=2.001, y=1000.
Near the forbidden boundary x=2, tiny input changes create enormous output changes.
4. Domain boundaries often create poor conditioning
Denominators near zero, logarithm arguments near zero, square-root boundaries and near-tangent intersections can all create sensitive behaviour.
The closer the problem sits to a structural boundary, the more carefully input uncertainty should be treated.
5. Sensitivity is not the same as an algebra mistake
A solver can perform every step correctly and still obtain an answer that is highly unstable because the underlying problem is sensitive.
Mathematical correctness and robustness are different properties.
6. Relative change is often more informative than absolute change
A change of 1 unit is enormous when the original value is 2 and negligible when the original is 1,000,000.
Relative sensitivity compares output percentage change with input percentage change.
This is useful when quantities operate at different scales.
7. Multiplication by a fixed constant has predictable conditioning
If y = kx, then relative changes in x produce the same relative changes in y, provided x and y are non-zero.
A 1% increase in x gives a 1% increase in y.
The multiplier changes absolute scale but not relative sensitivity.
8. Squaring doubles relative sensitivity approximately
If y=x² and x changes by a small percentage, y changes by roughly twice that percentage.
For x=10, a 1% increase to 10.1 gives y from 100 to 102.01, about 2.01% increase.
This is controlled sensitivity, not instability.
9. High powers amplify relative error more strongly
For y=x⁵, a small relative input error can produce about five times the relative output error locally.
This matters in scaling problems where volume, energy or other quantities depend on powers.
10. Subtraction of nearly equal numbers can lose relative information
Suppose two measured values are 100.01 and 100.00.
Their difference is 0.01.
If each measurement has uncertainty of ±0.02, the difference is dominated by uncertainty.
Subtracting large nearly equal quantities can therefore produce a small result with high relative uncertainty.
11. Cancellation can make an inverse problem fragile
If a hidden quantity is recovered as A−B and A and B are almost equal, tiny errors in either can strongly affect the difference.
The reconstruction may be mathematically defined yet practically unstable.
12. Parallel or nearly parallel lines illustrate geometric sensitivity
Two non-parallel lines intersect at one point.
If their gradients are nearly equal, the intersection may lie very far away.
Small changes in either gradient can move the intersection substantially.
Near-parallel geometry is a classic source of unstable reconstruction.
13. Tangency is a critical sensitive state
When a line is tangent to a curve, a small parameter change can create two intersections or none.
The repeated-root case lies exactly at a regime boundary.
This links conditioning directly to Thresholds and Regime Changes.
14. A discriminant near zero signals root sensitivity
For a quadratic, Δ close to zero means the two roots are close to merging.
Small coefficient perturbations may change the classification from two real roots to no real roots.
Near the boundary, exact coefficient handling and uncertainty matter more.
15. Repeated roots can be more sensitive than separated roots
The polynomial (x−1)² has a repeated root at 1.
A small constant perturbation, (x−1)²−0.0001=0, produces roots 0.99 and 1.01.
Another perturbation, (x−1)²+0.0001=0, removes real roots entirely.
The repeated-root state is structurally delicate.
16. Measurement uncertainty should be propagated through the model
If length L lies in [9.9,10.1] and width W in [4.9,5.1], area is not exactly 50.
For positive dimensions, area lies between 9.9×4.9 and 10.1×5.1.
The output interval reveals how input uncertainty affects the result.
17. Bounds provide robust sensitivity analysis
When exact error formulas are unnecessary, evaluate worst-case endpoint combinations consistent with the model.
This gives guaranteed output bounds rather than one fragile central estimate.
18. Rounding before the final step can amplify error
If a rounded intermediate value later enters a high-power or near-zero denominator calculation, the small early rounding can become much larger.
Preserving exact form or extra digits until the end can protect stability.
19. More calculator digits do not remove input uncertainty
If a measurement is only known to three significant figures, a twelve-digit calculator output is not twelve-digit knowledge.
Numerical precision of computation cannot exceed information quality of inputs and assumptions.
20. Condition number is a formal version of sensitivity
In advanced Mathematics, a condition number measures how much relative input error can be amplified in the output.
A small condition number suggests stability; a large one suggests sensitivity.
School learners do not need the full formal machinery to use the underlying question.
21. “Ill-posed” means the problem fails a basic reliability condition
A classical well-posed problem should have a solution, the solution should be unique, and it should depend continuously on the data.
If one of these fails, the problem can be called ill-posed in the classical sense.
This is enrichment terminology, but it unifies several school-level situations.
22. Non-existence is one form of ill-posedness
Conflicting exact constraints can produce no solution.
For x+y=5 and x+y=7, no exact pair exists.
The issue is not sensitivity; the feasible set is empty.
23. Non-uniqueness is another form
x+y=5 alone gives infinitely many real pairs.
The data do not identify one hidden state.
More independent information is required.
24. Instability is a third form
A unique answer may exist but respond wildly to tiny input changes.
This is the conditioning problem.
Existence, uniqueness and stability are separate tests.
25. School inverse problems often hide these three questions
Can the hidden quantity be reconstructed?
Is the reconstruction unique?
Would a small data change produce a similar reconstructed value?
This connects directly to Inverse Problems and Hidden Quantities.
26. A best-fit model can regularise noisy data
When exact equations conflict because measurements contain noise, a least-squares line chooses parameters that minimise total squared residual error.
This does not make every point exact. It replaces impossible exact reconstruction with a defined approximation criterion.
27. Regularisation means adding structure to stabilise a problem
In advanced Mathematics, unstable inverse problems are often stabilised by adding constraints or preferences.
At school level, familiar examples include restricting a domain, requiring positive lengths, choosing a simple model family, or using a best-fit line rather than forcing exact passage through noisy points.
28. Adding a valid constraint can improve identifiability
x²=25 has two real solutions.
Adding x>0 gives one.
The added domain knowledge stabilises the interpretation by removing a symmetry branch.
29. Adding an invalid constraint creates false certainty
A solver must not invent “x is positive” merely because one positive answer is preferred.
Regularisation is legitimate only when the added assumption belongs to the actual problem or model.
30. Threshold decisions need robustness margins
If a result is 100.001 and the threshold is 100, the classification may be mathematically above the threshold but practically fragile if input uncertainty is ±0.1.
A robust decision asks whether the entire plausible output range remains on one side.
31. Stable answers survive reasonable perturbations
If every plausible input change still produces the same qualitative conclusion, the decision is stable.
If tiny plausible changes flip the answer, communicate uncertainty rather than false confidence.
32. Numerical methods have stability as well as convergence
An iterative method may converge in exact arithmetic but behave poorly under finite precision or certain starting values.
Convergence asks what happens ideally; numerical stability asks whether computation preserves the intended behaviour under small perturbations.
The Iteration and Convergence chapter handles the main school-level numerical route.
33. Representation choice can improve stability
Two algebraically equivalent formulas can behave differently numerically.
For example, a formula involving subtraction of nearly equal large quantities may be less stable than a reformulated equivalent expression.
This is one reason representation is not only cosmetic.
34. Estimation can expose instability
If tiny input adjustments change the output’s order of magnitude, the problem is highly sensitive.
A quick estimate with upper and lower inputs can reveal this before detailed calculation.
35. Sensitivity analysis should target important parameters
Do not vary every number blindly.
Identify the inputs that are uncertain, estimated, rounded or close to critical boundaries.
Test those first.
36. A practical conditioning audit
Ask:
Does a unique solution exist? Is the answer close to a domain boundary or threshold? Are we subtracting nearly equal numbers? Is a denominator close to zero? Could small measurement error be amplified? Are roots nearly repeated? Would a small parameter change alter the qualitative regime? Can bounds or a different representation produce a more stable conclusion?
37. Independent practice
1. Compare y=1/(x−2) at x=2.1 and x=2.01.
2. Explain why a denominator near zero is a sensitivity warning.
3. A length is 10.0±0.1 cm and width is exactly 5 cm. Bound the rectangle area.
4. Why can subtracting 100.01−100.00 be fragile if both measurements have uncertainty ±0.02?
5. Give one example of non-uniqueness.
6. Give one example of non-existence.
7. Why is a repeated quadratic root a sensitive state?
8. What does it mean for a qualitative decision to be robust?
9. Why do extra calculator digits not remove measurement uncertainty?
10. How can a valid additional constraint improve an inverse problem?
11. Why should an invented constraint be avoided?
12. Name the three classical well-posedness requirements.
38. Worked answers
1. At 2.1, y=10. At 2.01, y=100. A 0.09 input change multiplies output by 10 because the denominator approaches zero.
2. Reciprocal-type outputs can become very large, so tiny denominator changes can strongly alter the result.
3. L∈[9.9,10.1], so A∈[49.5,50.5] cm².
4. The 0.01 difference is smaller than the uncertainty in either measurement, so its sign and magnitude are not reliable.
5. x+y=5 over real numbers has infinitely many solutions.
6. x+y=5 and x+y=7 as simultaneous exact constraints have no solution.
7. Small coefficient perturbations can split it into two roots or remove real roots, changing the regime.
8. The conclusion remains unchanged across all reasonable input perturbations or uncertainty ranges.
9. Computation can be precise while the original data are not; numerical display does not create new information.
10. It can remove ambiguous branches or add independent information, improving uniqueness.
11. It manufactures certainty unsupported by the original problem.
12. Existence, uniqueness and continuous/stable dependence on the data.
39. Batch 07 as one reconstruction-and-stability layer
Inverse Problems asks what hidden state produced the observable result.
Symmetry and Case Reduction asks which candidate states are genuinely distinct.
Thresholds and Regime Changes asks where the system changes behaviour.
This chapter asks whether the resulting answer is stable enough to trust.
Return to the BTT Mathematics Hub for Batch 07.