A pebble for every cow
Natural numbers
No words for numbers
Long ago, a herder lived by the Saraswati river. Every morning her cows went to the forest. Every evening they came home. She had no words for numbers and could not write.
How could she make sure that no cow was lost?
What this lesson covers
The idea
Counting by one-to-one correspondence, matching one object to another, gives the natural numbers ℕ = {1, 2, 3, 4, …}, the counting numbers.
No words for numbers
Long ago, a herder lived by the Saraswati river. Every morning her cows went to the forest. Every evening they came home. She had no words for numbers and could not write.
How could she make sure that no cow was lost?
- Remember every cow's face
- Put one pebble in a pot for each cow that leaves
- Hope for the best
The pebble pot
Send the cows out. Put one pebble in the pot for each. In the evening, take one pebble out for each cow that comes home.
Cows, pebbles, numbers
Matching one object to one other object is called one-to-one correspondence. Counting this way gives the natural numbers ℕ = {1, 2, 3, 4, …}.
Every cow has its own pebble, and every pebble has its own number: 1, 2, 3, 4, 5, 6. The last number tells how many cows there are.
Notes
Matching one object to one other object gives the natural numbers ℕ = {1, 2, 3, 4, …}.
Check yourself
In the evening the pot still holds 2 pebbles. How many cows have not come home?
Answer: 2
Each pebble stands for one cow that went out. A pebble that is left stands for a cow that is still away. 2 pebbles, so 2 cows.
Which of these is a natural number?
Why does the herder put one pebble for each cow?
The natural numbers are {1, 2, 3, 4, …}. What does the “…” tell us?
- −3. You never count cows or pebbles with a negative number.
- 1/2. You never reach one half by counting 1, 2, 3, …
- 7 — correct. Yes. 7 is one of the counting numbers 1, 2, 3, …
- So that every cow is counted exactly once — correct. Yes. One pebble for one cow: nothing is missed and nothing is counted twice.
- So that the pot looks full. The point is to match, not to fill the pot.
- Because pebbles are lucky. It is not luck. The matching is what tells her whether a cow is missing.
- The list stops at 10. There is no last natural number. The dots mean the list keeps going.
- The list goes on and on — correct. Yes. After every natural number there is a next one.
- The list stops at the number of cows. The numbers are not tied to the cows. They keep going for any count.