Separate symbols need endless new symbols
Every new landmark needs one more symbol, forever.
A symbol for every landmark
In the Egyptian system every landmark number has its own symbol: one for 1, another for 10, another for 100, and so on. It worked well for numbers up to a crore (10⁷).
If we want to write numbers as big as we like, how many different symbols will we need?
What this lesson covers
The idea
A base system that gives each landmark number its own symbol needs an unending sequence of new symbols to represent larger and larger numbers.
A symbol for every landmark
In the Egyptian system every landmark number has its own symbol: one for 1, another for 10, another for 100, and so on. It worked well for numbers up to a crore (10⁷).
If we want to write numbers as big as we like, how many different symbols will we need?
- Only a few more
- More and more, with no end
- Exactly 10
Keep making numbers bigger
Each tap makes the biggest landmark 10 times bigger. A landmark number needs a brand-new symbol. Watch how many symbols pile up.
An unending sequence of symbols
A base system that gives each landmark number its own symbol needs an unending sequence of new symbols to represent larger and larger numbers.
Number representation has the same challenge as before, in a different form: we cannot keep inventing symbols for ever. The next idea solves this.
Notes
A base system that gives each landmark number its own symbol needs an unending sequence of new symbols to represent larger and larger numbers.
Check yourself
A system has a separate symbol for each landmark 1, 10, 100, 1000, 10000. How many symbols is that?
Answer: 5
1, 10, 100, 1000, 10000 are five landmark numbers, so five symbols are needed.
The landmark numbers are 10⁰, 10¹, 10², …, 10⁸. How many different symbols are needed?
Answer: 9
The powers 0, 1, 2, 3, 4, 5, 6, 7, 8 are 9 landmark numbers, so 9 symbols.
Now we want to write a number 10 times bigger than our biggest landmark. What must we do?
What is the drawback of giving every landmark number its own symbol?
- Invent a new symbol for the next landmark — correct. Yes! Each new landmark needs a new symbol. That never ends.
- Write the biggest symbol 10 times. That goes back to marking again and again. We wanted one symbol for each landmark to avoid exactly this.
- Nothing, the old symbols can show any number. The old symbols only reach so far. Each bigger landmark needs a symbol of its own.
- Larger and larger numbers need an unending sequence of new symbols — correct. Yes! There is no end to the symbols we would have to invent and remember.
- Small numbers cannot be written. Small numbers are written easily. The trouble is with bigger and bigger numbers.
- It needs only 10 symbols. It needs more and more symbols, one for every landmark number.