A Story of Numbers · Lesson 4 of 8
The Idea of a Base
“Build the idea of base systems from Egyptian powers of 10 and a base-5 model.”
• Explain how Egyptian landmark numbers are formed from powers of 10. • Represent numbers using powers-of-10 landmark groupings. • Build and interpret a base-5 landmark system. • Define a base-n number system. • Generalise the landmark numbers of a base-n system as powers of n.
The Idea of a Base
Roman numerals use several landmark numbers, but their landmark sequence does not follow one simple multiplication rule. A major improvement appears when every next landmark is obtained by multiplying the previous landmark by the same number.
I. The Egyptian Number System
Start with the landmark 1. Group ten 1s to make 10. Group ten 10s to make 100. Group ten 100s to make 1000. The same rule repeats.
Because each step multiplies by 10, the landmark sequence is 10⁰, 10¹, 10², 10³, ... . A number is represented by using as many copies of each landmark value as needed.
Problem
Express 324 using powers-of-10 landmark values.
- 1.The largest landmark below 324 is 100.
- 2.Use three 100s: 300.
- 3.The remaining 24 is two 10s and four 1s.
- 4.So 324 = 3×100 + 2×10 + 4×1.
Problem
Express 2660 using landmark values.
- 1.2660 = 2×1000 + 600 + 60.
- 2.600 = 6×100.
- 3.60 = 6×10.
- 4.So 2660 = 2×1000 + 6×100 + 6×10.
II. Variations on the Egyptian System and the Notion of Base
What if we group in fives instead of tens? Start with 1. Five 1s make 5. Five 5s make 25. Five 25s make 125.
Now every landmark is a power of 5. The symbols used to draw the landmarks may vary; the mathematical structure is the fixed multiplication by 5.
Problem
Break 143 into powers-of-5 landmark values.
- 1.The largest landmark below 143 is 125.
- 2.143 − 125 = 18.
- 3.18 = 5 + 5 + 5 + 1 + 1 + 1.
- 4.So 143 = 1×125 + 3×5 + 3×1.
A number system in which the first landmark number is 1 and each next landmark is obtained by multiplying the current landmark by the same fixed positive integer n.
Problem
List the first four landmark numbers of a base-7 system.
- 1.7⁰ = 1.
- 2.7¹ = 7.
- 3.7² = 49.
- 4.7³ = 343.
- 5.So the sequence begins 1, 7, 49, 343, ...
The base tells how each landmark grows from the previous one. The drawing symbols are a separate choice.
Quiz
What are the first three Egyptian landmark numbers described here?
Which sequence belongs to a base-5 system?
What is the defining rule of a base-n system?
What is the next landmark after 49 in base 7?
Which decomposition matches 143 in the chapter's base-5 landmark system?
Practice Problems
- Express 784 as a sum of powers-of-10 landmark values.
- List the first six landmark numbers of a base-5 system.
- List the first five landmark numbers of a base-4 system.
- Break 137 into powers-of-5 landmark values using as few landmark groups as possible.
- Explain why powers of one fixed number create a more regular landmark sequence than the Roman landmark sequence.
Key Takeaways
• A base system uses a regular landmark sequence. • In the Egyptian system described here, the landmark numbers are powers of 10. • A base-5 system uses 1, 5, 25, 125, 625, ... as landmark values. • In a base-n system, the landmark numbers are n⁰, n¹, n², n³, ... . • Each landmark is obtained by multiplying the previous one by the base. • A regular power-based landmark sequence prepares the way for easier arithmetic.