MAT336: History of Mathematics

Spring 2026

Study Guide: Quiz 5

Questions may be added or modified before March 8.

Plimpton 322 — Inca Mathematics — Maya Mathematics

Note: This is a study guide. The quiz will consist of three or four questions covering the material below. If you understand the ideas and facts in the non-computational questions and can work through the computational problems, you will be well prepared.

Part 1: Quiz-Style Questions

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The Incas

  1. What is a quipu? Briefly describe its physical structure.
  2. When and where was the Inca Empire located? Give approximate centuries and geographic region.
  3. Name two features of Inca infrastructure that allowed them to govern a vast and mountainous empire.
  4. The Incan quipu used knots to record numbers. Which of the four characteristics (additive, alphabetic, multiplicative, positional) best describes the numerical structure of the quipu? Justify your answer.
  5. The Inca road network extended over 30,000 km. If relay stations were placed every 12 km:
    1. Approximately how many stations would be needed?
    2. What does this suggest about the administrative organization of the empire?

The Maya

  1. What time period (in centuries) and geographic region corresponds to Maya civilization?
  2. Write the number 425 in the Mayan base-20 positional system. Show your work by indicating how many units belong in each position.
    (Recall: vertical notation, bottom position = 1s, second position = 20s, third position = 400s.)
  3. The Haab (solar) calendar has 365 days and the Tzolk’in (sacred) calendar has 260 days. How many days pass before both calendars return to the same combination? Justify your answer.
    (You do not need to compute the final number; for example, 3·4·71 is an acceptable format.)
  4. The Long Count calendar tracks days from a mythological starting point. One b’ak’tun = 20 k’atuns, one k’atun = 20 tuns, one tun = 18 winals, and one winal = 20 days. How many days are in one complete b’ak’tun?
    (You do not need to compute the final number.)
  5. Explain why the Maya needed both a 365-day calendar and a 260-day calendar. What different purposes did each serve?
  6. The Maya dot-and-bar numeral system uses dots (= 1) and bars (= 5). Which characteristic (additive, alphabetic, multiplicative, or positional) would you associate with this numeral system itself, and why?
  7. Name one feature that distinguishes Maya civilization as mathematically or scientifically sophisticated.

Part 2: Reflection Questions

The Incas

  1. Machu Picchu sits above 2,400 m, connected by 30,000+ km of roads. What mathematical and social organization does this require? What would have to be known, recorded, and communicated?

The Maya

  1. The Maya developed zero independently. Why is zero a mathematically significant idea? What becomes difficult or impossible without it?
  2. Reaching age 52 — one Calendar Round — was considered a full life cycle. What does this tell us about how mathematics and culture intertwine? Can you think of a parallel elsewhere?
  3. The Mayan calendar was developed independently of Europe, China, and the Islamic world, with comparable precision. Does this suggest mathematics is a universal human activity, or a cultural product?
  4. Based on what you saw in class — Maya art, architecture, writing — how does the reality of Maya civilization compare to common assumptions about “ancient” cultures?

Comparing Both Civilizations

  1. Name one specific mathematical idea from each civilization and explain what challenge or question motivated it.
  2. What surprised you most about either civilization? How has your impression shifted?

Quiz Problem Rubric

Points Criteria
3 Correct answer with reasoning/work shown
2 Partially correct with some reasoning shown
1 Correct answer without reasoning/work OR significant attempt with some understanding
0 Incorrect or blank

Notes