Base-n number system
Each landmark is the same number of times the one before.
A new ladder of landmarks
The Egyptians grouped 10 collections to make the next landmark: 1, 10, 100, 1000, … Suppose we group 5 collections instead, again and again, starting from 1.
What do you think the first four landmark numbers are?
What this lesson covers
The idea
In a base-n number system the first landmark number is 1 and each next one is the previous one times a fixed number n, so the landmarks are n⁰ = 1, n, n², n³, …; base 10 is decimal.
A new ladder of landmarks
The Egyptians grouped 10 collections to make the next landmark: 1, 10, 100, 1000, … Suppose we group 5 collections instead, again and again, starting from 1.
What do you think the first four landmark numbers are?
- 1, 5, 25, 125
- 1, 5, 10, 15
- 1, 5, 6, 7
Group again and again
Start with 1. Each tap makes 5 groups of the last landmark. That is the next landmark.
Powers of n
In a base-n number system the first landmark number is 1 and each next one is the previous one times a fixed number n, so the landmarks are n⁰ = 1, n, n², n³, …; base 10 is decimal.
Base 10: 1, 10, 100, 1000, … Base 5: 1, 5, 25, 125, 625, …
143 = 125 + 5 + 5 + 5 + 1 + 1 + 1 in base 5: take as many of the largest landmark as possible.
Notes
In a base-n number system the first landmark number is 1 and each next one is the previous one times a fixed number n, so the landmarks are n⁰ = 1, n, n², n³, …; base 10 is decimal.
Check yourself
In base 5, which landmark number comes after 25?
Answer: 125
25 × 5 = 125. The landmarks of base 5 are 1, 5, 25, 125, 625, …
In base 3, the landmarks start 1, 3, 9, … What is the fourth landmark?
Answer: 27
1, 3, 9, 27: each one is 3 times the one before. The fourth landmark is 27 = 3³.
Which list is the landmark numbers of a base-n system?
Write 143 in base 5 landmarks, taking as many of the largest as possible. How many 5s are used?
Answer: 3
143 = 125 + 5 + 5 + 5 + 1 + 1 + 1. After 125, 18 is left: three 5s and three 1s.
- 1, 4, 16, 64, … — correct. Yes! First 1, then each is 4 times the one before: base 4.
- 1, 4, 8, 12, …. This adds 4 each time. In a base system each landmark is multiplied by the same number.
- 2, 4, 8, 16, …. The first landmark of a base system must be 1.
- 1, 2, 4, 6, …. The first jump is × 2 but the next ones add 2. We need to multiply by the same number every time.