Math101 · Independent Ontario learning library
Math101learn.math101.caFormula Library
A curated formula-library method that records meaning, conditions, units, derivations, and verification rather than isolated equations.
Precise definition
A formula library is a structured reference in which each formula is stored with variable definitions, domain and theorem conditions, units, a representative derivation or rationale, and a worked use. A formula is not valid merely because its symbols resemble a problem.
Notation and mathematical language
Examples include slope $m=(y_2-y_1)/(x_2-x_1)$ for $x_2\ne x_1$; quadratic roots $[-b\pm\sqrt{b^2-4ac}]/(2a)$ for $a\ne0$; circle area $\pi r^2$ for radius $r\ge0$; and compound growth $A=P(1+i)^n$ with rate and period aligned.
Conceptual picture
Grouping by structure—linear, quadratic, geometric, trigonometric, calculus, probability—makes selection easier than alphabetical memorization. Derivation links reduce memory load: distance comes from Pythagoras, midpoint from coordinate averaging, and exponent rules from repeated multiplication.
Conditions and key results
Each entry needs hypotheses. The Pythagorean theorem requires a right triangle; normal z-probabilities require a normal model; derivative rules require differentiability; finance rates require compatible periods. Symbols such as $r$ can mean radius or rate and must be redefined locally.
A reliable strategy
- For each formula, record name, exact statement, variables, units, and conditions.
- Add a one-line derivation or conceptual origin and one worked example.
- Index by problem structure and include warnings for common near-miss formulas.
- During practice, identify the structure before opening the library, then verify units and substitute back.
Fully worked example
Interpretation and application
A good library supports exams, engineering approximations, finance, and later courses. It cannot decide which model applies; that judgment comes from definitions, assumptions, and context.
Common mistakes
Verification and reasonableness
- Check dimensions or units on both sides.
- Derive the formula from a known principle when possible.
- Test with a simple case whose answer is known and verify the final result independently.
Practice
- What condition belongs with the slope formula?
- What condition belongs with the quadratic formula?
- What should accompany every finance rate?
Answers and brief solutions
- $x_2\ne x_1$.
- $a\ne0$.
- Its time period and compounding convention.
Further deduction
Add counterexamples to the library. Beside $\sqrt{a+b}\ne\sqrt a+\sqrt b$, record $a=b=1$. Beside the warning that correlation is not causation, record a plausible confounder. A formula sheet that includes boundary cases and nonexamples teaches selection better than a dense page of unqualified equations.
Dimensional analysis can reject a misremembered formula before any arithmetic. For constant acceleration, $d=v_0t+at^2/2$ has distance units because $v_0t$ is $(L/T)T=L$ and $at^2$ is $(L/T^2)T^2=L$. The incorrect expression $d=v_0+at^2/2$ attempts to add velocity and distance, so it cannot represent a physical displacement. Units do not prove a formula correct—many wrong expressions are dimensionally consistent—but they are a fast necessary check. A useful formula entry therefore records variable meanings, admissible units, assumptions such as constant acceleration, and a rearranged form only after algebraic verification. When converting units, attach conversion factors equal to one and cancel labels symbolically. This prevents applying a numerically familiar constant in incompatible centimetre, metre, hour, or second conventions.
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
What is the positive root of $x^2-5x+6=0$ that is larger?
- $(x-2)(x-3)=0$.
- The roots are 2 and 3, so the larger is 3.
End of lesson
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