Last week, in Number Sense 040, Awkward Goat and Billy Goat Gruff discovered they had invented a clock, by walking around a tree. At least, they discovered a facet of clock arithmetic. Billy Goat Gruff, however, suspected he'd been snookered, somehow, so this week finds them back at the same tree.

“All right,” said Billy Goat Gruff. “So this imaginary fence/circular number line acts like a clock. But what if I'd supposed fewer fence posts?”

“I'm not sure what you're getting at,” replied Awkward Goat.

“Last week, you asked me to imagine a number of fence posts, and I said there were twelve. That worked out well, because there are twelve hours on a clock. But would it work if I'd picked a different number? What if I'd picked, say, seven.”

“Well, I guess that would depend on what you mean by, 'that worked out',” said Awkward Goat. “Seven might be days of the week. Is that what you mean?”

“Or eight. What about eight?”

“Hmmm. The direction you're facing, perhaps,” said Awkward Goat.

“But,” continued Awkward Goat, “as fun as coming up with real world applications is, it's more interesting to me to see if this clock arithmetic works mathematically.”

“What do YOU mean, then, by 'works mathematically?'” asked Billy Goat Gruff.

“We discovered that we could subtract, using our twelve post circle last week,” replied Awkward Goat, “and come up with a sensible answer.”

“It was only sensible because it was twelve, though, like twelve hours in a day,” said Billy Goat Gruff.

“Right. But to me, 'works mathematically' means that subtraction on a circular number line would have the same mathematical properties as subtraction on a straight number line. And clock addition would have the same properties as regular addition, and so on.”

“That does sound like something that would interest you,” observed Billy Goat Gruff.

“Let's start with addition,” began Awkward Goat. “And let's use a five post fence, because, off hoof, I can't think of any application for five.”

“How are we going to keep track, if there's no application? How will we know if our answers are right or wrong?” asked Billy Goat Gruff.

“Excellent question, Billy Goat Gruff,” replied Awkward Goat. “I knew you were going to be a great help to me in figuring this out.”

Awkward Goat scratched out a five numbered clock on the ground.

“We check our answers to regular addition problems,” said Awkward Goat, “by referring to an addition table.”

“True,” agreed Billy Goat Gruff.

“So, I suppose the first thing we should do is make an addition table,” continued Awkward Goat, “so we can check our work.”

“The first row is the same as the header row,” said Billy Goat Gruff, “because adding zero doesn't change the other number.”

“Oh, very good!” exclaimed Awkward Goat. “There's a property that's the same as normal addition!”

“What, already?” asked Billy Goat Gruff. “We've barely started. What property is that?”

“Clock addition has an identity element,” replied Awkward Goat. “There is a number we can add that doesn't change the original number. A number plus the identity element is the same number.”

“Identity element,” snorted Billy Goat Gruff. “Fancy name for zero.”

“It's only zero for addition,” replied Awkward Goat. “Zero isn't the identity element for multiplication.”

“Course it is,” argued Billy Goat Gruff, “zero times anything is zero. You get the same number you started with.”

“You've got your number and your identity element mixed up,” said Awkward Goat. “The way it works is, a number times the identity element is the same number. So, 'anything times zero is anything,' but that isn't true, because anything times zero is always zero.”

“I was hasty,” admitted Billy Goat Gruff. “Let's see, hmmm, …, 3 times what is 3? Oh, it's one. You multiply by one to get the same number.”

“So one is the identity element for multiplication, and zero is the identity element for addition,” summarized Awkward Goat. “They're not the same number.”

“I GOT that, Awkward Goat,” said Billy Goat Gruff. “You don't have to rub it in.”

Have fun in the comments.

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