Checking the answer for physical/logical adequacy

1. Order-of-Magnitude Checks

Order-of-Magnitude Checks

An order-of-magnitude check estimates whether a computed result falls within a plausible range by comparing it to known reference values or physical constraints. This technique identifies catastrophic errors (such as misplaced decimal points or unit conversion mistakes) before detailed verification.

Intuition: If you calculate the mass of a car as 50000000 kg or 0.005 kg, both answers are immediately suspect because typical cars weigh around 1000–2000 kg.

Core Rules:

• Compare the result to familiar benchmarks (e.g., human height is roughly 1–2 m, not 100 m or 0.01 m)
• Check units and powers of ten separately; a factor of 1000 error often signals unit confusion
• Reject answers that violate known physical bounds (e.g., speed exceeding light speed, negative absolute temperature)
• Use rough mental arithmetic to verify the calculation's scale

Consequence: This method catches gross errors instantly, preventing propagation into downstream calculations.

Example: Computing Earth's radius as 6400000 m is reasonable (order $10^6$ m), but 64 m would be absurd.

2. Sign Checks

Sign Checks

A sign check verifies whether the algebraic sign (positive or negative) of a computed quantity aligns with the physical or logical context of the problem. Many real-world variables have inherent sign constraints that must be respected.

Intuition: If you calculate a person's age as $-5$ years or a distance as $-12$ meters (when measuring separation), the negative sign signals an error in setup or computation.

Core Rules:

• Identify physically non-negative quantities: mass, distance, absolute temperature, population count, time duration (in most contexts)
• Recognize directional quantities: velocity, acceleration, force, profit/loss, where sign indicates direction or surplus/deficit
• Verify that the sign matches the problem's stated conditions (e.g., "moving left" implies negative velocity if right is positive)
• Check whether sign changes occur at expected transitions (e.g., profit turning to loss)

Consequence: Sign errors often reveal algebraic mistakes, incorrect coordinate system choices, or misapplied formulas.

Example: Calculating gravitational force as $+500$ N when it should attract (negative direction) indicates a sign error in the force law.

3. Boundary Condition Checks

Boundary Condition Checks

A boundary condition check tests a mathematical model or formula by substituting extreme or limiting input values (e.g., zero, infinity, maximum capacity) and verifying that the output behaves as physically or logically expected. This exposes formula errors and domain violations.

Intuition: If a population growth model predicts negative population when time $t = 0$ (initial condition), the model is fundamentally flawed.

Core Rules:

• Test $t = 0$ (initial time): verify the model returns known starting values
• Test limits as variables approach infinity or zero: ensure asymptotic behavior matches reality (e.g., friction force vanishes when velocity is zero)
• Check maximum/minimum physical constraints (e.g., tank volume at full capacity, temperature at absolute zero)
• Confirm the model does not produce undefined operations (division by zero, negative square roots in real contexts)

Consequence: Boundary failures reveal incorrect assumptions, missing terms, or inapplicable formulas.

Example: For $v(t) = v_0 + at$, setting $t = 0$ gives $v(0) = v_0$, confirming initial velocity is preserved.

4. Applications in Engineering and Finance

Applications in Engineering and Finance

In engineering stress analysis and financial modeling, adequacy checks prevent catastrophic design failures and absurd forecasts by filtering computationally correct but physically/economically impossible results.

Intuition: A bridge design predicting tensile stress of 100000000 Pa in a steel cable (far exceeding material strength) must be recalculated, even if the math is algebraically correct.

Core Rules in Engineering:

• Verify stresses remain below material yield limits (e.g., steel yields near 250000000 Pa)
• Check that deflections, temperatures, and loads fall within operational ranges
• Ensure safety factors are positive and greater than 1

Core Rules in Finance:

• Reject negative prices, interest rates below -100%, or probabilities outside [0, 1]
• Test revenue projections against market size (e.g., claiming 200% market share is impossible)
• Verify exponential growth models do not predict infinite wealth in finite time

Consequence: These checks act as sanity filters, catching unit errors, formula misapplications, and unrealistic assumptions before implementation.

Example: A loan model yielding monthly payment of -500 dollars signals a sign error in the amortization formula.