Rive

FACTORS AND PRIMES — a factor is a number that divides another into equal whole groups with nothing left over; twelve rives cleanly into 2×6, 3×4, and 4×3. A prime number cannot be riven into equal rows at all except one-times-itself; primes are the numbers all others are built from.

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01 Opening

Rive grew up in a quarry, splitting stone, and he learned about numbers from the way the stone chose to break.

His family cut paving-blocks from the grey hillside above the town of Scarp. A rough block came off the face, and Rive's job — from the time he could swing a small maul — was to rive it: to split it, along the grain, into equal smaller blocks that a mason could lay in tidy rows. A block that made twelve paving-stones could be riven into two rows of six, or three rows of four, or four rows of three, or six rows of two. Every one of those splits was even — no scrap, no odd corner, every stone the same size.

But some blocks would not do it. A block that made seven stones could not be riven into equal rows at all. Two rows left one stone over. Three rows left one over. You could only lay it as one long row of seven, or seven single stones — one-times-itself, and nothing else. The stone simply refused.

"Some numbers split," Rive's aunt told him, sighting along a chisel. "Some don't. The ones that split, you can lay in rows. The ones that won't — you keep them whole. They're the stubborn ones. They're also," she added, "the ones everything else is built from, though it took me years to see it."

02 Rive

Rive's given name was Bram, but he had been called Rive since he was twelve, when a mason asked him for a block that would tile a courtyard exactly, and Bram walked the whole quarry, tapping blocks, until he found one that would rive into the mason's precise number of rows with nothing wasted. "That boy can rive a number to order," the mason said, and the name held.

He came to think of every number as a block waiting to be tested. To find a number's factors — the numbers it splits into equally — you simply tried to rive it: does it break into rows of two? rows of three? rows of four? Each clean split named a pair of factors. Twelve broke at two (giving 2 and 6), at three (giving 3 and 4), and there was no need to look past its own square-root, because after that the pairs only repeat in the other order.

And the stubborn blocks — the ones that broke at nothing but one-times-themselves — those he learned to call primes. Two, three, five, seven, eleven, thirteen: the un-riveable ones. Not broken. Whole.

03 Rive

An academy quarrier came for building-stone and found Rive sorting a season's blocks into "splits cleanly" and "won't split," and he stopped, because that was exactly the lesson he could never get children to feel.

"You are factoring," he said. "And your stubborn stones are the prime numbers. We teach that every number is either built from primes or is a prime itself, and children think it is a rule from nowhere. You have it in your hands."

Rive left the quarry the next spring. He has taught factors and primes for eight years, and he keeps, on his classroom floor, a set of small wooden cubes for riving.

04 Rive

Rive begins each year with twelve cubes. "Rive them into equal rows," he says, and the children discover, delighted, that twelve is generous — two sixes, three fours, four threes, six twos. He writes each split as a pair: 2×6, 3×4, and shows that these pairs are the factors of twelve: 1, 2, 3, 4, 6, 12. "A factor," he says, "is any number that rives it cleanly. Twelve has six of them. Twelve is an easy stone."

Then he hands them seven cubes and says, "Now rive these." The children try. Two rows leave one over. Three rows leave one over. They cannot do it. And Rive smiles and says, "You're not failing. Seven is a prime. It won't rive into equal rows except one row of seven. Its only factors are one and itself. It is not a broken number — it is a building number." Then he shows them the point: he rives thirty into 2×15, then rives the 15 into 3×5, and holds up the primes at the bottom — 2, 3, 5 — and says, "Every number, if you keep riving, comes down to a handful of primes. The primes are the blocks. All the others are walls built from them."

05 Closing

A girl named Odile stayed behind. She said she had always thought the numbers that wouldn't split were the worst ones — the failures, the ones that didn't work — and it turned the world upside-down to hear that the stubborn primes were the most important numbers of all.

Rive sat on an upturned cube beside her. He said the quarry had taught him the same lesson the same way. "For years I thought a block that wouldn't rive was a bad block," he said. "Then my aunt showed me that those were the true stones — the ones every wall is finally made of. The number that resists being broken is the number everything else is built from." Odile turned a single un-riveable cube over in her hands — the stubborn seven — and felt something rearrange quietly inside her, the feeling of learning that a thing you had filed under problem had been the point the whole time.

Rive is patient, tactile, and cannot look at a number without silently testing whether it splits. He still goes up to Scarp at the year's turning to help with the season's cutting, and he still tests each block by hand. And he still, he admits, feels a quiet tenderness toward the stubborn ones — a warmth that surprises him every time — because he remembers being a boy who thought the unbreakable blocks were failures, and he loves that they turned out to be the very stones the world is built on.

The Numberverse ensemble

Rive is part of Numberverse's distributed-narrative cast. Each character embodies a different curricular primitive; together they teach the full subject.