Products of landmark numbers in a base
In a base, a landmark times a landmark is a landmark.
Landmark times landmark
To multiply two big numbers we multiply their landmark numbers. In base 10, for example, 10 × 100 = 1000, which is another landmark number.
In the Roman system, will the product of two landmark numbers (1, 5, 10, 50, 100, …) always be another landmark number?
What this lesson covers
The idea
In a system with a base, the product of any two landmark numbers is another landmark number, which makes multiplication simple; Roman landmark numbers lack this, so multiplying with them is difficult.
Landmark times landmark
To multiply two big numbers we multiply their landmark numbers. In base 10, for example, 10 × 100 = 1000, which is another landmark number.
In the Roman system, will the product of two landmark numbers (1, 5, 10, 50, 100, …) always be another landmark number?
- Yes, always
- No, not always
- Never
Look at the products
Tap a landmark on the left to open its row. A product that is a landmark number turns green; one that is not turns red. Look at both tables.
Powers multiply into powers
In a system with a base, the product of any two landmark numbers is another landmark number, which makes multiplication simple; Roman landmark numbers lack this, so multiplying with them is difficult.
Landmarks of base n are powers of n, and nᵃ × nᵇ = nᵃ⁺ᵇ. Base 10: 100 × 1000 = 10² × 10³ = 10⁵. Roman: 5 × 5 = 25 has no symbol.
Notes
In a system with a base, the product of any two landmark numbers is another landmark number, which makes multiplication simple; Roman landmark numbers lack this, so multiplying with them is difficult.
Check yourself
In base 10, what is 100 × 1000?
Answer: 100000
100 × 1000 = 10² × 10³ = 10⁵ = 100000. The product is another landmark number.
In base 5, 25 × 125 = 5² × 5³ is a landmark number. It is 5 to the power of what?
Answer: 5
5² × 5³ = 5²⁺³ = 5⁵ = 3125, the next landmark number after the ones we listed.
Which product is not a landmark number of the Roman system (I, V, X, L, C, D, M)?
Why is multiplying with Roman numerals difficult?
- 5 × 5 = 25 — correct. Yes! 25 has no symbol of its own. It must be written XXV.
- 5 × 10 = 50. 50 is a landmark number: L.
- 10 × 10 = 100. 100 is a landmark number: C.
- 10 × 100 = 1000. 1000 is a landmark number: M.
- A product of landmark numbers is not always a landmark number — correct. Yes! A product like 5 × 5 = 25 has no symbol, so we need extra work to write it.
- Roman landmark numbers are too big. Size is not the problem. The problem is that the products do not always land on another landmark number.
- Roman numerals have no symbol for 1. I is the symbol for 1. The problem is that products like 5 × 5 = 25 are not landmark numbers.