{ } = 0
{*} = 1
{**} = 2
{***} = 3
{****} = 4
{*****} = 5
{******} = 6
{*******} = 7
{********} = 8
{*********} = 9
"See how clever they are?" said Sarah. "The brackets let us tell at a glance that there’s nothing in the first set. And see what they’re doing? Establishing digits for the numbers zero through nine — the aliens are using base ten, which may mean they’ve got the same number of fingers we have, or it might just mean that they’ve decoded some of our TV, and have seen that that’s how many fingers we’ve got. Oh, and notice that this chart gives us their equals sign, too."
He got up and helped himself to another slice; when you skipped the crust, you went through pizza awfully quickly.
"Anyway," she continued, "they immediately give us the basic mathematical operators. Again, I’ve rendered them in familiar notation." She rotated the wheel on her mouse, and this scrolled into view:
[Question] 2+3
[Answer] 5
[Question] 2-3
[Answer] -1
[Question] 2*3
[Answer] 6
[Question] 2/3
[Answer] 0.6
"See what they’ve done here? They’ve established a symbol for ‘question,’ and another for ‘answer.’ And they’ve also established a symbol for a decimal place, and a symbol for repeating indefinitely, which I’ve shown as that ‘and’ thingy."
"Ampersand," said Don, helpfully.
She gave him an I-knew-that scowl, and went on. "Next up, they give us a symbol for ‘the relationship between,’ which I’ve shown as a colon, and that lets us get a bunch of other concepts." She made this appear:
[Question] 2/3 : 0.6
[Answer] =
[Question] 5 : 3
[Answer] ›
[Question] 9 : 1
[Answer] ››
[Question] 3 : 5
[Answer] ‹
[Question] 1 : 9
[Answer] ‹‹
[Question] 1 : -1
[Answer] [opposite]
"See?" she said. "We’re getting into judgment calls. Nine is judged to be not just greater than one but much greater than one, and one, in turn, is much less than nine.
Next they give us their symbols for correct and incorrect." This appeared on screen:
[Question] 2+5
[Answer] 7 [correct]
[Question] 3*3
[Answer] 9 [correct]
[Question] 8-3
[Answer] 6 [incorrect]
"And then," said Sarah, "things get really exciting."
"I can hardly contain myself," Don said.
She whapped him lightly on the arm, and nibbled at her own piece of pizza before changing the screen. "This came later in the message. Look."
[Question] 8/12
[Answer 1] 4/7 [incorrect]
[Answer 2] 4/6 [correct] [alpha]
[Answer 3] 2/3 [correct] [beta]
"See what they’re saying there? I’ve assigned Greek letters to the two new symbols they’re establishing. Can you puzzle out what alpha and beta mean?"
To his credit, he stopped shoveling cheese and pepperoni into his mouth and studied the screen carefully.
"Welllll," he said at last, "both answer two and answer three are correct, but, um, well, answer three is more correct, right? ’Cause, I mean, they’ve reduced the fraction."
"Bravo! That’s exactly right! Now, think about that: they’ve just given us a way to express some very powerful concepts." She touched a key, and the terms alpha and beta were replaced with words:
[Question] 8/12
[Answer 1] 4/7 [incorrect]
[Answer 2] 4/6 [correct] [bad]
[Answer 3] 2/3 [correct] [good]
"That is, they’ve given us a term for distinguishing between an answer that, while technically correct, isn’t preferable from one that is preferable — distinguishing a bad answer from a good one. And, just to drive home the point that they are making that distinction — that these terms should be translated as polar opposites — they give us this."
[Question] [bad] : [good]
[Answer] [opposite]
Sarah translated. "What is the relationship between ‘bad’ and ‘good’? Why, they’re opposites, just like one and negative one, as we saw before. They’re saying these terms should be treated as actual opposites, in a way that ‘right’ and ‘more right,’ which would have been the other possible way of translating alpha and beta, aren’t."
"Fascinating," he said.
She touched her mouse, and a new display appeared. "Now, what about things that aren’t clear-cut? Well, try this. What does gamma mean?"
{3 5 7 11 13 } = [gamma]
"Odd numbers?" he said. "Every other number?"
"Look again. There’s no nine."
"Oh, right. Oh, and, um, hey, there’s that ‘and’ thingy again."
"Ampersand," said Sarah, imitating Don’s helpful tone from earlier. He grinned.
"Right," she said, "but I’ll give you a hint — something I gleaned from other examples. When the ampersand is right up against another digit, it means that digit is repeated forever. But if there’s a space before it — a little gap in the transmission, as there is here — I think it means that this sequence goes on forever."
"Three, five, seven, eleven, thirteen…"
"I’ll give you another hint. The next number in the sequence would be seventeen."
"Um, ah…"
"They’re primes," she said. " Gamma is their symbol for prime numbers."
"Ah. But why start with three?"
She was grinning broadly now. "You’ll see. This is the beauty part." She darted her mouse around. "There’s a little more set theory, which I won’t bore you with, that establishes a symbol for ‘belongs to this set,’ and then we get this…"
[Question] 5 [belongs to] [prime numbers]
[Answer] [correct]
"Does five belong to the set of prime numbers — or, more colloquially, the question is ‘Is five a prime number?’ And the answer is yes; indeed, five was one of the sample numbers we used in naming the set ‘prime numbers.’ "
She made another similar Q A pair appear:
[Question] 4 [belongs to] [prime numbers]
[Answer] [incorrect]
"Is four a prime number?" said Sarah, interpreting. "No." She rotated her mouse’s wheel again:
[Question] 3 [belongs to] [prime numbers]
[Answer] [correct]
"Is three prime? Yup, sure is. And what about two? Ah, well, let’s have a look."
More mouse movements, and this appeared:
[Question] 2 [belongs to] [prime numbers]
[Answer 1] [correct] [good]
[Answer 2] [incorrect] [good]
[Answer 3] [delta]
"Huh?"
"My precise reaction," said Sarah, smiling.
"So what’s delta?" Don said.
"See if you can figure it out. Look at answer one and answer two for a moment."
He frowned. "Hey, wait. They can’t both be good answers. I mean, two is a prime number, so saying that it isn’t can’t be a good answer."
She smiled cryptically. "They give exactly the same three answers for the number one," she said, scrolling the screen.