Last week, in Number Sense 041, Awkward Goat and Billy Goat Gruff found that clock arithmetic has an identity element for addition: zero. This week they keep looking for more properties in common with ordinary arithmetic.
“Now we've finished filling in the addition table, it seems odd,” said Billy Goat Gruff, “to have two numbers adding up to zero.”
“It does seem unusual” replied Awkward Goat. “But I imagine if we'd learned clock arithmetic first, ordinary arithmetic would seem odd, just going on and on like it does. Think of asking someone the time and hearing, 'Oh, it's 17,345,297 o'clock!' as an answer. Most goats would have to look out the door to see whether it was day or night with an answer like that.”
“Hah! That's true” said Billy Goat Gruff. “So where do we go from here?”
“Well, another thing I notice about our new addition table, is that there are no extra numbers.”
“What do you mean by extra numbers?” asked Billy Goat Gruff.
“We have five numbers, 0, 1, 2, 3 and 4,” began Awkward Goat.
“That's because we only have five fence posts on our imaginary circular fence,” replied Billy Goat Gruff.
“I wasn't finished,” complained Awkward Goat. “If we add any two of those numbers, according to our addition chart, the sum is always one of those numbers. We don't get, say, six or nineteen or any numbers that aren't in the original five. There are no 'extra' numbers.”
“That seems obvious, now that you put it that way,” said Billy Goat Gruff. “We walk around the fence in a circle, so we only pass by fence posts that have one of those five numbers.”
“It's not just that we pass by them, Billy Goat Gruff,” said Awkward Goat. “Since we are adding whole numbers, we always end our walk at a numbered post. We never stop in between.”
“All right, then. No extra numbers,” agreed Billy Goat Gruff. “Is that a property?”
“It is,” said Awkward Goat. “It means addition is closed for our five fence posts arithmetic.”
“Closed?” asked Billy Goat Gruff. “That's an odd name for a property.”
“I think of it like this,” said Awkward Goat. “These five numbers are inside a fenced-in yard. And no matter which of them I add, the sum is also inside the same fenced-in yard. There's no escape. The yard is closed.”
“I,” said Billy Goat Gruff, “would just jump over that fence!”
Have fun in the comments.