A narrow view of numbers
Negative numbers are a source of confusion for many who are getting their feet wet in mathematics. The problem comes with having too narrow an idea of number, i.e. restricting the notion to embody only quantity.
We are introduced to numbers, even negative numbers, through quantity. If I have 5 apples and eat three of them, then I have only 2 left. If I loan 3 of them to you and you eat them, then you have negative 3 apples and I will send Bubba after you with a tire iron to get them back if I have to. It doesn’t take much imagination to figure out that this can be tied to money and once things get tied to money they’ve got a way of capturing our attention. We like to keep track of our money. One device to keep track of money is the number line.
I’ve indicated it above with dots draw on it to indicate the integers, a zero to give us a starting point, and an arrow to indicate the positive direction. We can use it to keep track of our money. If I have $5, then I can indicate that by marking it as below.
However, if I buy something that costs $7, then I go $2 in the hole and move the marker backwards, in a negative direction.
Thus, the line is a device to keep track of personal wealth, with numbers to the right of 0 denoting what I have and those to the left what I owe. Of course, as I’ve set this up, it is particularly egocentric. The person I borrowed from still counts himself as having that $2 and would remind himself of that by noting it on his number line.
By bringing in the number line, I’ve introduced geometry to the situation. Zero is my starting point and the (+) and (-) signs indicate the direction which I am going away from zero. I can think of the line as being First Street, which runs in front of my house and zero as being the corner of First Street and Catalpa, which I can see from the window of my home-office. By setting East as the positive direction, I can then tell people I am going 5 blocks and they would be able to figure out that my destination is the corner of First and Walnut, while if I told them I was going negative 2 blocks they would be able to discern I was headed to First and Georgia.
This works fine for addition. As for multiplication, if I am multiplying a positive number by a positive number, I can think of that as repeated addition; if I am multiplying negative by positive, I can think of it as repeated subtraction; and while I am convinced I could do something similar for negative times a negative, thinking about it makes my head swim. Fortunately, I don’t have to because we’ve introduced geometry to the problem.
Let us go back to borrowing versus lending. When Bubba borrows from Billy Bob, Bubba owes, but Billy Bob is owed. The same transaction is positive from one point of view, but negative from the other. Given Bubba’s predilection for non-repayment, the negative end is often hard to discern, but I digress. The point is the positivity or negativity is simply one of point of view. We could view multiplication has having two effects: one on size and one on direction. Multiplication by a positive number might change size but it preserves direction while multiplication by a negative might preserve size, but it changes direction. The negative may be viewed as flipping a number over to the other side. Hence, a negative times a negative is a positive.
Numbers as two pieces
When we multiply numbers, we can think of them as having two pieces. The first of these is their absolute value, i.e. their length, and the other is their sign. Effectively, this is what we do: (5)(-3)=(+5)(-3)=(+)(-)(5)(3)=(-)(15)=-15 or (-2)(-4)=(-)(-)(2)(4)=+8=8.
In the movie Paint Your Wagon, the innocent farm boy utters the line, “But I’ve never lied to my parents before.” To this Ben Rumson answers, “Well when you do it opens up a whole new world!”
When geometry is introduced to arithmetic, a whole new world is opened up. For example, I can expand my explorations beyond First Street by introducing complex numbers. I can take the complex number x+yi and take it to mean go x units in the east-west direction and go y units in the north-south direction. Recall that i is the square root of -1. I can add x+yi to u+vi by taking the sum of the two to be (x+u)+(y+v)i. This works out well geometrically because if I first go x units east-west and y units north-south and follow this by u units east-west and v units north-south this gets me to the same place as going x+u units east-west and y+v units north-south. Go for a walk and see for yourself.
Thus again we are called to expand our notion of number as being simply quantity. On the plane, as we are, we can think of the complex number x+yi as being the directed line segment from the origin of our axes to the number x+yi, as below.
Complex numbers mimic real numbers in that they are best thought of as multiplying in two pieces as well. They have their length, the distance from the origin to the point (x,y), and their direction, which is the angle from the positive real axis to the directed line segment which is associated to x+yi. This is labeled by the Greek letter alpha below.
When two complex numbers are multiplied together, the length of the product is the product of the lengths of the two numbers involved and the angle of the product is the sum of the angles of the two numbers involved. So, for example, the imaginary number i has a length 1 and angle of 90 degrees. So i times i has a length of 1 and an angle of 180. Notice that the number -1 fits this description exactly. Therefore i times i is equal to -1 just like it is supposed to.
This is how the formal multiplication works out. To prove this is a beautiful exercise in trigonometry, but it would put some folks off. I am willing to show you, but only if you beg me.