**Chapter 5**

"In the first place,” said Soth, after he had made himself more comfortable and taken a small bite of the kerbango Jonnie had produced, “there are an awful lot of different kinds of mathematics– different races, you know. I sort of kept my interest in life by collecting them.

“There have been existing systems for lots of different whole numbers.

There's the 'binary system' like the Chatovarians use: it has only two numerals in it, one and zero; that's so they can use them in computers, in which the electric current pulse, or the direction of magnetization of an element, has one of two values. One value corresponds to the numeral zero; the other to numeral one. Any number in any system can be translated to the binary system using only zero and one. Unwieldy for beings, but understandable for the computers.

“Then there's a system based on the integer three, an entirely different one based on four, another on five, another on six, another on seven, still another on eight, yet another on nine and so on. There's even been one on twenty and one on sixty.

“For paper computation the best system is called the 'decimal' system, based on ten.” (Jonnie knew about this from man-books.) "Psychlo mathematics are based on eleven; some people call this the 'undenary' system. It 's difficult so I won't try to teach you that.”

“Oh, I would love to know about the 'undenary' system!” said Jonnie. His use of the words 'love to know' gave him a twinge of conscience. He hated this confused mess!

“I can teach you the 'decimal' system much more easily,” said Soth. “Whenever they discover it on some planet they engrave the discoverer's name among the heroes.” He saw Jonnie wouldn't buy it and sighed. “All right,” he said. He got a sheet of somewhat crumpled paper. “I will write down the 'undenary' numerals for you.”

Jonnie said he already knew the Psychlo numerals but Soth shook his head.

“No, no,” said Soth. “I doubt very much if you do. To really understand a symbol, you have to know what it came from. Now all numerals as symbols were originally either the first letter of the word that spelled them or a number of dots or lines. Or they were pictographs that then became stylized until they were only a part of the original picture or a shorthand version of it.

“Now Psychlo numerals were originally pictographs. And then as time went on, they were written in a more and more simplified form until they now are what you see as the eleven separate Psychlo numerals. They were once called 'the road to happiness.'

Jonnie had not known that. He saw these numbers, these symbols, every time he flew. He began to get interested.

Soth was writing the numerals as pictographs, little pictures. “Zero is an empty mouth; see the teeth? One is a claw; just one talon. Two is a being and a pick. Three is a being, a shovel, and a rock. Four is a mine cart; see the four corners? Five is what we call the 'off’ paw, the one with five claws. Six is what we call the 'on' paw, the one with six claws. Seven is an ore chute. Eight is a pot smelter; see the smokestack and the smoke? Nine is a pile of metal ingots made like a pyramid; nine of them originally but now just the pyramid. Ten is a lightning bolt; symbol of power, now just a slash. Eleven is two paws clasped; that represents contentment.

“It’s a little moral lesson, you see. If you dig and smelt ore, it lifts you from starvation to power and contentment.” He laughed. “Very few people know this. All they know is what time and haste boiled them down to.” And above his pictures, he rapidly wrote the eleven Psychlo numeral symbols as they commonly appeared. They still bore the traces of the pictographs.

"I’m very glad to know that,” said Jonnie. And he was amused by it. The Psychlos had been miners from the start! “I can do a little arithmetic in this system.” He decided to really push it. “Where I get hung up is the Psychlo force equations.” And that was no lie. They gave him headaches.

Nothing ever balanced.

Soth was looking at him very keenly. “I think you are digging for the teleportation formulas.”

Jonnie shrugged. “We have a rig running. We are making rigs.”

“Yes, I heard that,” said Soth. “That's where all the new breathe-gas and goo-food came from. I heard there was a planet, Fobia, nobody could live on.” He was plainly puzzled. “Ah!” he exclaimed. “One of your scientists reevolved it in some other mathematics and you are trying to verify it against Psychlo equations.” He laughed and laughed.

Jonnie gave him another bite of kerbango.

“Ah, well. Not that it will do you much good. But it's small wonder you can't work it out. He laughed again. “You'd have to be a native of Psychlo!"[^{4}]

He laughed so hard he had to wipe his eyes.

“Oh, well,” he said at last, “you already have teleportation so what's the difference.” He took another big piece of paper and drew a huge circle on it. Then he had second thoughts and sat back and looked at Jonnie. "If I give you this,” said Soth, “what's it worth?”

“Money?” said Jonnie.

“A separate dome, access to compound libraries, and tools to experiment with computers. And not to be cross-fired elsewhere!”

“All right,” said Jonnie.

Soth made a fast list of what he had said. Then he added, "Breathe-gas and proper food for the rest of my life. I’m sorry to have to add this. But

I’ve only ten years or so to go so it's only ten years' worth. I won't add anything else.”

Jonnie signed it. He even put a paw print on it, using his fingernails. Soth looked like he had lost ten years of age.

With a flourish, Soth pulled the circle to him. Then he put another piece of paper on top of it. “Do you know anything about codes and ciphers?

Cryptography? Well anyway, here is the Psychlo alphabet.” And he wrote it out. “And here are the Psychlo numerals.” And he wrote them under the letters and then started the numerals over again until there was one written under each letter. “Do you see, here, that each letter has a number value?”

Jonnie said he did. Soth laid aside the top sheet and again addressed the big circle.

“This,” said Soth impressively, indicating the circle, “is the perimeter of the imperial Palace of Psychlo." He made a series of small slashes around the circle. “These are the eleven gates. A lot of people even on Psychlo never knew they had names. But they do:

“Going counterclockwise, the names of these gates are: 'Angel's Gate,' 'Betrayer's Gate,' 'Devil's Gate,' 'God's Gate,' 'Heaven's Gate,' 'Infernal Gate,'

'Monster's Gate,' 'Nightmare Gate,' 'Quarrel's Gate,' 'Regal Gate,' and 'Traitor's Gate.' Eleven gates, each with a name.”

He took a book, “Force Equations” off his shelf. “It doesn't matter which types of equations in Psychlo higher math. They're all the same. You mentioned 'force equations' so we'll use those. No difference.”

With a dig of his claw, he opened the book to the point where all the equations were summarized and pointed to the top one. “Now you see this 'B'? You might think it is a symbol for something in Psychlo mathematics. But there is no 'B' that represents anything mathematical except 'Betrayer's.'

He pulled the first paper back. “So where that 'B' occurs, we see that the letter 'B' has a number value of two. So we just have to add or subtract or whatever it says to do to 'B,' the number two.

“When we get to the second stage of the equation, there is no letter but a Psychlo mathematician knows you must take the second letter of 'Betrayer's,' which is 'E' and then look up the number value of 'E,' which is five, and factor the second stage of the equation with five. Now you get the same equation to its third stage and a mathematician knows he has to factor it with the number value of 'T' which is twenty. And so on.

"If the letter in the original equation were 'l,' then we would use its number value and follow right on down with the number values of the letters for 'Infernal.'

“You always have one of these letters in the first equation, so you always have the gate name. And you have to use it. When they put the equations together, they constructed them backward from the answer so a gate name would fit. Got it?”

Jonnie got it. A code and cipher mathematics!

No wonder nothing ever seemed to balance. This made even the original equations rigged.

And add to that all the complexity of a base-eleven math and you had what would appear to any outsider to be an utter mess.

He was glad he had the recorder running under his lapel. Completely aside from his being no native of Psychlo, the gate names themselves were weird.

“I have to be honest with you,” said Soth. “I don't know where I am getting all this impulse to be honest. But all this will be of limited use to you.”