In the section on numbers we explained that cardinal numbers are counting numbers. Whilst they are only meaningful in the real world if they reference something, in the subject of pure mathematics they can be manipulated without knowing what they refer to.
Positive, negative, multiply and divide symbols are not a part of a number; they tell us something about the number or how to deal with it. They are operators.
Operators like plus (+) or minus (-) can convey to us the nature of the scale we are on, whether it be a positive or negative scale An example of this is our bank account where those on the positive scale are in credit and those on the negative scale are in debt. But note the + and – in this case tell us which scale you are on but nothing about the position and numerical level of your credit or debit on these scales. Likewise the numerical value alone tells us nothing of which scale you are on.
We perhaps better know plus and minus operators as delivering changes in the position on the scale as when your wage or salary is paid into the bank or when you make payments from your account. As an operation we can view a minus as things like a reverse of direction, a reduction in quantity or say a backwards in time
Multiplication is a scaling operation. If you have a bank account that pays interest, and it is in credit, the bank will periodically scale your account upward, though I don’t know of any bank that doubles or trebles your money. If you are in debit to the bank they will scale your debit upward and no doubt by a larger multiplying operator.
I have always thought it better to replace “multipled by” and “times” with “of“. Six of nine items is more descriptive and when applied to fractions as with 3/8 of 2/5 you immediately see that the answer must be a fraction of 2/5. It is 3 lots of 1/8 of 2/5.
Some find it puzzling that when multiplying two minus numbers together we get a positive number. But what we are really doing is multiplying two numbers together and then combining two operations. Ask yourself what is a reverse of a reverse, a reduction of a reduction or a backwards of a backwards. They are all + operations.
My granddaughter’s algebra text book teaches the FOIL (first, outer, inner, last) method for multiplying double brackets. Let’s apply this method to some simple numbers that involve a minus of a minus, say (6 – 3)(5 -2) which involves a minus of a minus in its 3 x 2 bit. We get 30 – 12 – 15 + 6 = 9 which is only correct if a minus operation performed on a minus operation is a plus operation.
Simple dividing (÷) or sharing out is best thought of as how many of this are there in that? Dividing is like multiplication a scaling operator and one can always be expressed as the other. Multiplying by 1/3 is the same as dividing by 3 for example and dividing by 1/8 the same as multiplying by 8. Think about that and you see why the divide by a fraction rule of invert and multiply works.
In dealing with square roots we are taught that the square root of say plus 9 can be either plus or minus 3. We are led along the road of believing that -3 is a different number to +3. But 3 is the number and + and – are operators, in this case telling us whether the number is on a rising or descending scale. The root of the number 9 is always 3. We explained above that a minus operation of a minus operation is a plus operation and so the root of a positive operation can be a negative operation as well as a positive one
In later school mathematics we may encounter complex numbers like 6 +3i. We are told they have a real number part and an unreal or imaginary number part identified by the letter i (engineers use the letter j). We learn that i2 = -1 and that i = √-1. What no one seems to understand or teach is that i is not a number, imaginary or otherwise, it is not even part of a number, it is an operator. The 3i bit in our example is very much a real number 3 and an operator i that tells us what to do with it.
So what does the operator i tell us? It says do a 90 degree anti clockwise off your linear scale before applying the number. So the complex number 6 plus 3i means move 6 along a linear scale then do a turn 90 degrees anti clockwise before moving 3 on a vertical scale as per the diagram.
Another way of looking at this, and particularly useful in preparing a student for 3d graphics and vectors, is that any point can have its own coordinate system. The point 6 on the x scale can have its own coordinate system. If our scale is a ruler we move our ruler so that its zero point is at point 6. The operator i tells us to turn the ruler along with the new (at point 6) coordinate system through 90 degrees anti clockwise so that its positive scale is now vertically upward. We count 3 along it.
Using the same sort of thinking a minus operator following the plus 6 would mean at the point 6 turn your ruler and the coordinate system at point 6 through 180 degrees. Any counting we now do on the scale rule is in the negative direction of the global coordinate system we started with.
In 3d graphics we are often keeping track of local coordinate systems as to their position and rotation relative to a global system.
We teach i2 = -1 and i = √-1 and whilst these are true children are clever enough to say two numbers multiplied together cannot make -1 and they are right. Like all other operators i is meaningless if not accompanied by a number. So when we see i on its own we really mean i acting on 1. We should really have put the operator bit i in front of the number so that it doesn’t look like an algebraic unknown.
i2 is really an operator i acting on an operator i acting on a number 1. i2 means turn the scale 90 degrees then turn the scale again 90 degrees giving us a negative scale. That is why i2 = -1. We can now also see that √-1 is actually the root of a minus operation and the root of 1. Well we have just seen that the i operation is the root of the minus operation and of course the root of 1 is 1. So we can write i = √-1 but always remember that i and the minus sign are not of themselves numbers or parts of numbers. It surprises me that we are still teaching the same old real and imaginary/unreal ideas that I was taught 60 years ago. The words don’t add to our understanding.
I read somewhere a response to a student internet question The answer started “lets take a purely imaginary vector”. The student asked what’s a “purely imaginary vector” and the answer was “one with no real component”. I hope you can now see that the no real component bit was a zero move along the plus scale and the purely imaginary bit was some vector length along a 90 degree rotated scale at that zero point.
Of course in our graphical example above it might have been – 3i, which I said above is better written as -i 3. Now with the two operators together we see we are required to make a turn at our + 6 position on the scale through 180 degrees and then through 90 degrees before moving 3. We move 3 down.
Complex numbers along with matrices and quaternions are much used in generating computer and film graphics and in electrical engineering where alternating currents in circuits that have inductance and/or capacitance are out of phase with voltages. They can also describe vectors and 2d translations and when used with an angular measure describe complex rotations.