Landmark numbers
Give a few numbers their own symbols and write far less.
A tally of 2367
Writing 2367 as a tally needs one mark for every one. That is a very long row of marks. The Romans gave new symbols to some bigger numbers: X for 10, C for 100, M for 1000 and more.
With those new symbols, about how many marks do you think 2367 needs?
What this lesson covers
The idea
Numbers that are given a new basic symbol in a number system are landmark numbers; grouping by a sequence of landmark numbers, rather than one group size, makes number representation far more efficient.
A tally of 2367
Writing 2367 as a tally needs one mark for every one. That is a very long row of marks. The Romans gave new symbols to some bigger numbers: X for 10, C for 100, M for 1000 and more.
With those new symbols, about how many marks do you think 2367 needs?
- Fewer than 20
- About 200
- Still 2367
Switch on the landmarks
Only I (1) is on. The toy groups 2367 using the landmark numbers that are on, largest first, and counts the marks. Switch more on.
Landmark numbers
Numbers that are given a new basic symbol are landmark numbers. Grouping by a sequence of landmark numbers, rather than one group size, makes number representation far more efficient.
In the Roman system the landmark numbers include I (1), V (5), X (10), L (50), C (100), D (500) and M (1000).
Notes
Numbers that are given a new basic symbol are landmark numbers. Grouping by a sequence of landmark numbers, rather than one group size, makes number representation far more efficient.
Check yourself
Which of these is a landmark number of the Roman system?
With only I, X and C switched on, how many marks does 2367 need?
Answer: 36
2367 = 23 × 100 + 6 × 10 + 7 × 1, so 23 + 6 + 7 = 36 marks.
With all the Roman landmark numbers, how many marks does 2367 need?
Answer: 10
MMCCCLXVII has 10 marks: M M C C C L X V I I. Far fewer than the 2367 marks of a tally.
Why does using landmark numbers make writing numbers more efficient?
- 50, which has its own symbol L — correct. Yes! L is a basic symbol, so 50 is a landmark number.
- 40, because it is written XL. 40 has no symbol of its own. It is written using X and L.
- 60, because it is written LX. 60 has no symbol of its own. It is written using L and X.
- 25, because it is written XXV. 25 has no symbol of its own. It is written using X and V.
- Each landmark symbol stands for a big group, so fewer marks are needed — correct. Yes! Grouping by a sequence of group sizes needs far fewer marks.
- It makes the number smaller. The number stays the same. Only the way it is written is shorter.
- It uses only one group size. The Roman system uses a whole sequence of group sizes, not just one.