Domain: D00 — Financial Mathematics, Statistics, and Data Foundations
Family: D00-F01 — Mathematical Language and Quantitative Reasoning
Audience: a curious beginner who may not have studied finance or programming
Delivery: plain-language lesson, hand-worked arithmetic, and a small interactive lab
The one-sentence idea
Use three different languages for change without mixing them up.
Why this matters in money
A rate moving from 8% to 9% can be described as a 1 percentage-point move, 100 basis points, or a 12.5% relative increase. All three can be correct, but they answer different questions.
What you will be able to do
- calculate relative percentage change
- separate percentage points from percent change
- translate percentage points into basis points
What you need first
- A03 ratios and percentages
- the idea of an old value and a new value
You do not need to program for this lesson. We use ordinary words and small numbers first. Later domains may show implementation details after the idea is comfortable.
Words to know
| Term | In simple words |
|---|---|
| Percentage change | The change measured relative to the old value. |
| Percentage point | The direct subtraction of two percentages. |
| Basis point | One hundredth of a percentage point; 100 basis points equal 1 percentage point. |
| Old value | The reference value used as the denominator for relative change. |
Scope and simple boundary
We compare two positive values. If the old value is zero, relative percentage change is undefined; a separate level change may still be meaningful.
Definition contract
| Decision | Our simple choice | What we do not claim | Check |
|---|---|---|---|
| Relative change | (new - old) / old × 100. | Do not use the new value as the denominator unless that is explicitly the chosen convention. | The example shows the old value as the base. |
| Percentage points | new percentage - old percentage. | Do not call this a percent change. | 8% to 9% is +1 percentage point. |
| Basis points | percentage-point change × 100. | Do not multiply a raw price change by 100 and call it basis points. | +1 percentage point = +100 basis points. |
Input contract
| Name | Kind | Unit | Empty? | Meaning |
|---|---|---|---|---|
| old | number | same unit | No | Reference value; must be non-zero for relative change. |
| new | number | same unit | No | New value measured on the same basis. |
| old_rate | rate | % | No | Old percentage rate for the rate illustration. |
| new_rate | rate | % | No | New percentage rate for the rate illustration. |
All values in one example must use the same time period, currency, and scale unless the lesson explicitly shows a conversion. A missing or impossible input is a question to resolve, not a number to guess.
Output contract
| Output | Kind | Unit | Meaning | Safety rule |
|---|---|---|---|---|
| relative_change | percentage | % | Change relative to old. | Undefined when old is zero. |
| point_change | percentage points | pp | Direct difference between two rates. | Same rate definition on both dates. |
| basis_point_change | basis points | bp | Point change multiplied by 100. | 100 bp = 1 pp. |
The rule in everyday language
relative change = (new - old) / old × 100; basis points = percentage-point change × 100
Think of the rule as a sentence. Say what each number means before you put it into the sentence.
Symbol table
| Name | Meaning | Unit |
|---|---|---|
| old | starting value | unit |
| new | ending value | unit |
| pp | percentage-point difference | percentage points |
| bp | basis points | bp |
Four small steps
- Ask whether the values are levels or rates.
- For levels, keep the old value as the denominator.
- For rates, subtract the two percentages first.
- Only then translate percentage points into basis points.
Worked example (synthetic teaching numbers)
A share price moves from 100 to 110: relative change = (110 - 100) / 100 × 100 = 10%. A lending rate moves from 8% to 9%: the move is 1 percentage point, or 100 basis points. Its relative change is 1 / 8 × 100 = 12.5%, which is a different description.
Independent check
If old = 0, (new - old) / old asks us to divide by zero. Report “relative change undefined” and show the level change separately instead of inventing an infinite percentage.
The numbers above are synthetic and author-derived for learning. They are not an observation about a named company, market, customer, or investment.
Simple playground
Open the small guided lab. Choose a number, press Step, and read the explanation under the result. The lab is designed to show one idea at a time, not to replace the lesson.
What to notice: the input, the rule, the check, and the safe interpretation are shown in that order. Open the full-size visual.
Where this is useful
- interest-rate communication
- price and revenue changes
- fee and spread comparisons
What this does not tell you
These are descriptions of change, not explanations of why the change happened or predictions of what comes next.
This is educational content, not investment, tax, legal, accounting, or regulatory advice. A simple calculation can be correct and still be the wrong calculation for a real decision.
Historical-example decision
Not useful for this lesson. A named company would add a story but would not teach the primitive more clearly than the small synthetic numbers above. A real case would also need a verified entity, period, unit, and publication right. The lesson therefore keeps its arithmetic transparent and synthetic.
Related lessons
Evidence boundary
The plain-language definitions and measurement cautions are supported by INVESTOR, NIST. The formula wording, examples, and lab behavior are author-derived teaching choices. See the claim ledger and references for the boundary.
Optional verification implementation
You do not need code to learn this lesson. The package now includes matching Python and TypeScript verification façades, a shared worked-example fixture, and parity tests. They reproduce the lesson’s frozen rule and remain optional for nontechnical learners.
Enhancement studio: draw, compare, explain
This additive studio does not replace the beginner lesson above. It gives you two more drawings, a decision comparison, and short practice prompts so you can explain the idea without copying a formula or writing code.
Drawing 1 — name, apply, check
Read left to right: name what the data means, apply the narrow lesson rule, then use an independent check. Open the full-size concept anatomy.
Choose the right idea
| Decision | This lesson | Closest next or comparison | Why the difference matters |
|---|---|---|---|
| Main question | Use three different languages for change without mixing them up. | Exponents, Roots, and Logarithms | Choose the question before choosing the arithmetic. |
| Safe rule | relative change = (new - old) / old × 100; basis points = percentage-point change × 100 | Uses its own input and boundary contract. | Neighboring lessons can use the same numbers but answer different questions. |
| Required check | If old = 0, (new - old) / old asks us to divide by zero. Report “relative change undefined” and show the level change separately instead of inventing an infinite percentage. | Re-check its own unit, time, denominator, or schema. | A correct answer to the wrong question is still wrong. |
| Stop condition | Do not use the new value as the denominator unless that is explicitly the chosen convention. | Move only when its prerequisites are satisfied. | Unknown meaning is a reason to pause, not to guess. |
Drawing 2 — common-mistake clinic
The left side states the safe interpretation; the right side shows the mistake that often produces a believable but misleading result. Open the full-size mistake comparison.
Explain it back without code
- Name it: What does the first input or observation mean?
Answer: Reference value; must be non-zero for relative change. - Choose it: Which rule belongs to this question?
Answer: relative change = (new - old) / old × 100; basis points = percentage-point change × 100 - Challenge it: What check could make you stop?
Answer: If old = 0, (new - old) / old asks us to divide by zero. Report “relative change undefined” and show the level change separately instead of inventing an infinite percentage.
If your explanation leaves out the unit, period, denominator, grain, or availability time that the lesson needs, it is not complete yet.
Related concepts and learning handoff
- Governed glossary: Percentage change, Percentage point, Basis point, Old value. Browse the full financial glossary when a term is unfamiliar.
- Continue with: Exponents, Roots, and Logarithms.
- Evidence boundary: all displayed numbers remain synthetic teaching data; the drawings do not claim a market observation, forecast, or investment result.
Rendered from the canonical Mermaid sources linked by this article.
Percentage Change, Percentage Points, and Basis Points — four-part map
This diagram shows the learner's path from a named input to a safe explanation.
Takeaway: the check is not an afterthought. It tells the learner whether the answer belongs to the question that was asked.
ReferencesPrimary sources and evidence notesExpand the source trail, evidence role, and limitations behind the engineering choices.
Expand the source trail, evidence role, and limitations behind the engineering choices.
The family uses a small shared evidence set. Each source has a limited role; the examples and lab are clearly author-derived synthetic teaching material.
INVESTOR — Basis Point
- Organization or authors: U.S. Securities and Exchange Commission
- Source type: Official investor education glossary
- Publication or effective date: Current web edition
- Version: current web edition accessed for this build
- URL: https://www.investor.gov/introduction-investing/investing-basics/glossary/basis-point
- Accessed: 2026-08-10
- Jurisdiction: general educational or measurement context
- Supports: One basis point is one hundredth of a percentage point.
- Limitations: It does not prescribe how a learner should choose a denominator for percentage change.
NIST — NIST/SEMATECH e-Handbook of Statistical Methods
- Organization or authors: NIST/SEMATECH
- Source type: Official statistical handbook
- Publication or effective date: 2022 current web edition
- Version: current web edition accessed for this build
- URL: https://www.itl.nist.gov/div898/handbook/
- Accessed: 2026-08-10
- Jurisdiction: general educational or measurement context
- Supports: Definitions, careful measurement language, and the difference between a calculation and a claim.
- Limitations: It does not choose this family's teaching examples or financial conventions.
Author-derived and synthetic boundary
The formulas are standard classroom definitions expressed in plain language. The worked values, visual labels, and playground scenarios are synthetic teaching inputs created for this package. They do not establish a company fact, market outcome, or investment result.
Full dependency-light reference implementations in both supported languages.
import { runTopic as runD00Topic, type D00Input, type D00Output } from "../../../../shared/typescript/d00Engine.ts";
/** Run the canonical D00-F01-A04 calculation. */
export function percentageChangePercentagePointsAndBasisPoints(input: D00Input): D00Output {
return runD00Topic("D00-F01-A04", input);
}
The embedded lab now expands to its full document height, keeping the article as the only scroll surface.