# multiplicative inverse

Today’s note is a guest tasting by Daniel Ginsberg, @NemaVeze.

Mathematics seems peculiarly prone to confusions between the symbols through which ideas are communicated and the ideas themselves. —David Pimm

What do you remember from middle school math? Do you remember how to divide fractions? For example, if I ask you to divide two thirds by five sixths, what do you do?

Well, you take the second one and flip it. Then you multiply.

You flip it, huh? What does that mean? Five sixths is a concept, a philosophical proposition. It’s a portion of an abstract unit which is divided into six segments but you only take five. How do you flip that?

Don’t be obtuse. Five sixths is a five, and then a line under it, and a six on the bottom. You flip it. You write the six on top and the five on the bottom. Then you multiply.

Right, but when you say “flip it,” that’s just a trick of notation. It’s like saying that multiplying something ten times is the same as writing a zero on the end. It’s how you write down what you’re doing, and it’s a kind of shorthand for what you’re doing, but it’s not what you’re really doing.

Okay, smarty pants, what am I really doing?

You’re taking advantage of the fact that multiplication and division are really the same thing, a concept that’s obscured by integers and then brought back into clarity with fractions. The rational numbers are a field; they can be added, subtracted, multiplied, divided. To divide by a number is the same as multiplying by its reciprocal. “Flipping” is the notation; what you’re really doing is taking the multiplicative inverse.

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Multiplicative inverse begins staccato in a telegraph clatter of plosives, to tail off through nasals and fricatives of lessening intensity. Like the function one over x, it traces a swooping descent toward zero.

Okay, not like the function. Like the graph of the function.

How precise is the pronunciation of multiplicative inverse. More than technical, it sounds technological. It has a feel of moving parts that fit together like a watch, or like the Platonic ideal of a watch that makes the sound of an unlocking iPhone. The sound of precision machinery for a precise piece of terminology.

But what is it? What layers of abstraction are buried here? Multi-plic-at-ive from plectere, folding. Cousin to the plectrum that picks out the notes on your guitar (and music as you know is mathematics), to thesolar plexus (a kick in which some people would take if it would get them out of math class), to the multiplex where you go when your math homework is done. In-verse, like inversion, turning upside down, but in fact it’s more of a reverse, doing it backward. If multi-plic-ation is folding over and over, the inverse is unfolding.

With all this morphology, the words’ structure reflects the nesting of concepts by which mathematics proceeds. At the core is the idea that you can take a quantity (of what? Doesn’t matter), copy it repeatedly (how many times? Indefinitely), and sum up the results. Like folding a sheet of note paper along the vertical: if there are 30 lines then each fold makes 60, 90, 120 little boxes. But the process to multiply becomes an object called multiplication that imputes multiplicative properties to other objects. Multiplicative inverse, multiplicative identity. Mathematical objects that exist in our minds, if we train our minds to hold them. Math isn’t a science; it’s an appliable philosophy.

Flip it and multiply. Phooey, he says, shaking his head, pushing his glasses upward on his nose.