08_Analytic_Geometry — what does it mean to draw a number?¶
Level: 101 · for anyone starting precalculus, or anyone who has ever plotted a point
Descartes' idea, and the first section of every precalculus book: two number lines at right angles turn every point of the plane into a pair of numbers and every pair of numbers into a point. After that a question about a picture is a question about arithmetic, and a question about an equation has a picture. This chapter follows that idea in the order the books take it, one page of the book at a time, with the book's own questions at the end of each lesson and a deck of flashcards beside it.
| # | Lesson | The question it answers |
|---|---|---|
| 1 | Rectangular coordinates | What do the two numbers of a point measure, what is a quadrant, and why is the word worth having? |
| 2 | The distance formula | How far apart are two points, and why does the formula not care which one comes first? |
| 3 | The midpoint formula | Which point is halfway, and why is "equally far from both ends" not enough to say so? |
| 4 | Graphs of equations: intercepts and symmetry | What is the graph of an equation, and how do intercepts and symmetry follow from that definition? |
| 5 | Lines: slope, equations, parallel and perpendicular | Why does a line have one slope, and why do perpendicular slopes multiply to −1? |
| 6 | Circles: standard form and general form | Why is a circle's equation the distance formula, and when is x² + y² + ax + by + c = 0 not a circle? |
| 7 | Distance in n dimensions, and the length of a vector | What happens to the distance formula with a third coordinate, and why is √(v·v) a length? |
| 8 | Distance is the four properties | What else counts as a distance, what shape is its circle, and why did geometry pick Pythagoras? |
The through-line¶
Lesson 1 sets up the plane. A point is two signed distances, x from the y-axis and y from the x-axis, so the Cartesian product ℝ² is not just a set of pairs but the plane itself. A quadrant is the pair of signs and nothing more, which is why the axes belong to none and why the word carries so far: the sign of a trigonometric function, the quarter turn that multiplying by i performs, and the symmetry of a graph are all statements about quadrants.
Lesson 2 measures in it. The distance between two points is the hypotenuse of a right triangle whose legs are the differences of their coordinates, so the formula is the Pythagorean theorem, and the squares in it erase the signs that lesson 1 took such care over: distance has no direction.
Lesson 3 finds the point halfway, one average per coordinate, and uses the distance formula to check both halves of "halfway". Lesson 4 turns an equation into a picture: its graph is the set of points that pass it, the same object as the solution set of a linear equation, and intercepts and symmetry are that definition applied. Lessons 5 and 6 are the two families of graphs the chapter ends with. A line has one slope because similar triangles share their ratios, and a circle's equation is the distance formula held fixed. That is the whole of the first chapter of Sullivan's Precalculus; the book's next chapter is functions.
All the flashcards in one file¶
precalculus_chapter_1_anki.txt holds every card of this chapter and of the geometry it assumes, 10_Geometry: 370 cards in nine decks, one per lesson. In Anki choose File → Import, pick the file, and click Import; the header lines put each card in its lesson's deck, under Math, with its tags. On a phone or a machine without a clone, download it from this link ↗ with the browser's "Save as".
Where this is taught¶
The opening section of the chapter on graphs in any precalculus book; the lessons here follow Michael Sullivan's Precalculus, but Stewart, Larson, Blitzer and the free Stitz and Zeager cover the same page in the same order. For which book to read, see Precalculus: a reading guide.
Po polsku, w skrócie¶
Pomysł Kartezjusza: dwie osie liczbowe pod kątem prostym zamieniają każdy punkt płaszczyzny w parę liczb, a każdą parę liczb w punkt. Od tej chwili pytanie o rysunek jest pytaniem o arytmetykę, a równanie ma swój obraz. Potem odległość i środek odcinka, wykres równania jako zbiór punktów, które je spełniają, prosta i jej nachylenie, okrąg jako wzór na odległość ze stałym r. Rozdział idzie w kolejności podręcznika do precalculusu, strona po stronie, z pytaniami z książki na końcu każdej lekcji i talią fiszek Anki obok.
Auf Deutsch: Stichwörter¶
Was es heißt, eine Zahl zu zeichnen: Koordinaten, Abstand, Mittelpunkt, Graphen, Geraden und Kreise, das erste Kapitel eines Precalculus-Buchs.
Stichwörter: analytische Geometrie, Koordinatensystem, Abstandsformel, Mittelpunkt, Graph einer Gleichung, Gerade, Steigung, Kreis, Vektor, Metrik.
A note on the code¶
Every point is a Python tuple of integers, so "is in quadrant II" is a check on two signs and every count is exact. Nothing is drawn: the one picture in the chapter is a grid of text cells, which is enough to see where four points land.