What is Multiplication – part XII

In this post I sliced the various ways we’ve looked at multiplication into two basic types, which I called scalar multiplication and unit multiplication.  This is not the only useful way to slice and dice the various multiplicative structures we’ve collected in this series of posts.  This post addresses another one.

The picture below is taken from an earlier post, and it suggests multiplicative relationships that are thought of as a quantity going in and another quantity coming out.  What’s happening inside the box is that the quantity coming in is being multiplied by some amount that is specific to the box. I suggest the name unary multiplication for this.  There is single quantity going in, and another single quantity coming out.  The relationship is multiplicative – specifically, directly proportional – implying that if I double the quantity going in, the quantity coming out will double also.  If I double the number of kids, I will also double the number of thumbs; if I double the number of eyes, I will also double the number of noses; if I double the number of items in my basket, the price after the 35% discount will also double.

The Mentos people, who always put 14 candies in a roll, will think of the relationship between the quantity of rolls and the quantity of candies as a box with one quantity going in and another coming out.  The Brits before 1971 would always have 240 pennies in a pound.   (In British English, the plural of penny is pence, I’m using the American plural.)  After 1971, they always have 100 pennies in a pound.  Before 1971 I could go to a bank, give them an amount of pounds and get an amount of pennies in return.  I could represent the action as a box with one quantity going in, one quantity going out, in a multiplicative relationship.  After 1971, the same thing, though there obviously was a major break that happened in 1971.  The box that had the quantity of pounds going in and the quantity of pennies coming out would have had to be fixed in 1971.

All these notions of multiplication are embedded in a more general notion, one where there are two quantities going in, and one quantity coming out.  Sometimes there is a situation where it is natural to think of two quantities that can be independently varied.  A simple example of this is shown in the figure below: The figure shows different rectangles, with different widths and different heights.  Each rectangle covers a number of little squares.  The 1-by-1 covers a single square, the 2-by-10 covers 20 squares.  The number of squares covered can be found by multiplying the width by the height.  This is a common situation where I think of neither the width nor the height as fixed. If I think of the action as a box with stuff going in and stuff going out, it would have two inputs and one output.  This is called binary multiplication¹.

The distinction between unary multiplication and binary multiplication is very much in the eye of the beholder.  It is in the way you think about this thing called multiplication rather than anything “out in the real world.”  This doesn’t make it any less real or any less important.  It is ideas and notions about multiplicative relationships and multiplicative structures that we have been looking at.  In a sense, none of those exist in the real world.  That thumbs are double the number of faces lives in an observation, not in any act of multiplication that faces do or that thumbs do.  Even bunnies don’t really multiply, it’s just that after a while there are more of them.  We look at the world, and we appreciate multiplicative structures.  It’s in the eye of the beholder, in the form of various mathematical ideas that are learned and refined, added to and replaced.  A child who completely understands “double” may not yet understand binary multiplication.  It is a different mathematical idea.

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(Note 1: Often, multiplication of binary numbers is also abbreviated as ‘binary multiplication’.  I’m not talking about binary numbers here, but about operators, specifically multiplication.  For operators, the terminology unary operator, binary operator, ternary operator are common, and refer to the number of inputs.)

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