A standard sequence to count with

The idea

A number system is a standard sequence of objects, names or written symbols with a fixed order; a collection is counted by a one-to-one mapping with it, and ideally the sequence is unending and easy to count with.

Counting without number names

A farmer wants to count a herd, but there are no number names. Pair each cow with the next letter of the alphabet, in the fixed order A, B, C, …

How many cows can the farmer count with the 26 letters of the English alphabet?

• Any number of cows
• Only up to 26 cows
• Only up to 10 cows

Letters or sticks?

Press + to make the collection bigger. Count it with the letters, then with the sticks. Watch what happens after 26.

A standard sequence

A number system is a standard sequence of objects, names or written symbols with a fixed order. A collection is counted by a one-to-one mapping with it, and ideally the sequence is unending and easy to count with.

Sticks: unending, but we need as many sticks as there are objects. Letters: easy to count with, but they stop at 26.

Each of these has its own problem. A good number system is unending and easy to count with.

Notes

A number system is a standard sequence of objects, names or written symbols with a fixed order. A collection is counted by a one-to-one mapping with it, and ideally the sequence is unending and easy to count with.

Check yourself

Why are the 26 letters not a complete number system?

A farmer counts a herd with the letters A, B, C, … The last cow gets the letter J. How many cows are in the herd?

Answer: 10

A B C D E F G H I J are 10 letters, so the herd has 10 cows.

How many sticks are needed to count 35 objects with sticks?

Answer: 35

One stick for each object: 35 objects need 35 sticks.

Which sequence is unending, but not convenient for large collections?

• They are not in a fixed order. They are in a fixed order: A, B, C, … Z.
• They run out after 26, so they are not unending — correct. Yes! Collections bigger than 26 cannot be counted with them.
• They cannot be paired with objects one-to-one. For up to 26 objects, each object can get its own letter.
• The letters A to Z. The letters stop at Z. They are not unending.
• The fingers of one hand. There are only 5 fingers, so this stops too.
• Sticks — correct. Yes! We can always add one more stick, but we need as many sticks as objects.