This is a follow up from yesterday’s post on what is a number. There I related how I was pleased with a class discussion I had with my 10th grade class on what a number is. The key I wanted us to agree on was that a number was both a geometric and algebraic object and that we could view addition and multiplication through both lenses. Geometrically numbers tell us both an amount and a direction. In my situation with my class I was especially interested in the geometric meaning of multiplication as I was introducing complex numbers, and the reason complex numbers are so awesome is that geometric meaning of multiplication. (Without that aspect, I see no particular advantage of thinking about a point in the plane as a complex number as opposed to a pair of real numbers).
With my class the 2nd day discussion–which by this point was more of me talking then I might have liked–focused on the idea that when we multiply by a number the length of the number (or its absolute value, or norm, or size, or magnitude, or modulus, or whatever else people like to call it) plays one role. That tells us how much to “stretch” the number by. The direction of the number (or its sign, or angle, or argument, or whatever else people like to call it) plays another role. It tells us how much to change the direction of the number. So -7 times 3 tells us both to stretch 3 by a factor of 7 (giving 21) and change the direction 180 degrees to -21. Likewise -7 times -3 should stretch -3 by a factor of 7 (giving -21) and change the direction 180 degrees to 21. This might not be concrete enough yet for a model of negative times a negative, but I feel it can and should be the basis for one.
The nice thing about this interpretation of multiplication is that it not only works well with negative numbers, but extends and motivates complex numbers. If we wanted to rotate a point/number by 90 degrees counterclockwise without any stretching, we should multiply by the point/number that has a length of 1 (ie it is 1 unit away from 0) and that is located in the direction that is 90 degrees counterclockwise from the positive direction. That number has been given the name of i. If we multiply by i again we still won’t stretch (since the length is 1) but we will rotate 180 degrees. So i times itself is -1. I like this definition of i in terms of where it is located in the number plane (we’ve now moved beyond the number line) as opposed to just saying that it is the square root of -1 for many reasons. First of all, it’s motivated. Second of all, it clearly exists–I can see that point on the plane, we’re just giving it a name now. Thirdly, it’s well-defined–i is not the only number that squares to give -1, but it is the only point located 1 unit away from 0 at a 90 degree counterclockwise angle to 1. The number 3i can now be thought of in several ways. Start with i and stretch it by a factor of 3. Start with 3 and rotate it 90 degrees counterclockwise. We could even think of it as 3 units in the “i direction” just as we may think of -3 as three units in the “negative” direction or 3 turned around or -1 stretched by 3.
Since last May when I had this discussion with my 10th graders I’ve come across this post by Ben Braun which has me thinking about the distinct models of multiplication we use in terms of whether we are multiplying by something with a unit, or a scalar that does not have a unit. This is something I’m going to think about some more and follow up on later, but any thoughts on it now are more than welcome.