In the section on numbers we explained that cardinal numbers are counting numbers. They are only meaningful in the real world if they reference something. But in pure mathematics they can be manipulated without knowing what they refer to.
We may think of +2 as being a different number to -2 but in reality the 2 part is in each case the number and the positive and negative signs, like the multiply and divide symbols 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 scale we are on or tell us what to do with a number. Take for example a bank account where a positive operator tells us we are in credit with the bank and a minus operator tells us we are in debt. The + or – tell us which scale we are on. The number tells us our level of credit or debit on that scale.
We perhaps better know plus and minus operators as delivering changes in the position on the scale. When your wage or salary is paid into the bank your position on the scales moves in a positive direction and when you make payments from your account your position on the scales moves in a negative direction. We might think of a minus operation as producing a reverse in direction along a scale, a reduction in quantity or say as going backwards in time
Multiplication and division are both scaling operations. 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. Gambling on horses, etc will give you better odds but those odds are rarely in your favour.
I have always thought it better to replace “multipled by” and “times” with “of“. Six of nine is more descriptive than 6 multiplied by 9. When of is applied to fractions as with say 3/8 of 2/5 you immediately see that the answer must be a fraction of 2/5. We can view it as 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, as explained above, there is no minus number. We are in reality multiplying two numbers together and then combining two operations. In respect of the latter ask yourself what is a reverse of a reverse, a reduction of a reduction or a backwards of a backwards. Two minus operations equate to a plus operation.
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 operation in its 3 x 2 bit. We get 30 – 12 – 15 + 6 = 9 which is correct and demonstrates that a minus operation performed on a minus operation is a plus operation.
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 in terms of 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.
We are taught that the square root of say plus 9 can be either plus or minus 3. It leads us to believe that -3 is a different number to +3. But, I repeat + and – are operators, not numbers. They are not even part of a number. The root of the number 9 is always the number 3. A minus operation of a minus operation can result in 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 which again lead us into thinking that i is somehow a number
What no one seems to understand or teach is that i is not a number, imaginary or otherwise. It, like plus, minus, multiply and divide is an operator.
The 3i bit in our example comprises of a number 3 and an operator i that tells us what to do with that number. For consistency we should really have put the operator bit i in front of the number, making it look less like an algebraic unknown. But like so many historical decisions in maths and science we seem stuck with it.
So what does the operator i tell us? It turns our scale 90 degrees anti clockwise off our previous scale before applying the number. So the complex number 6 plus 3i means move 6 along a linear scale then turn the scale 90 degrees anti clockwise before moving 3 on a vertical scale. This 6 + 3i can describe a new point in the x-y plane or a vector with direction and length from the starting (in this case zero) point to the new point.
In 3d graphics, where objects are moved and rotated relative to one another, a single global coordinate system and multiple local coordinate systems are common. Those local coordinate systems can be translated (moved) and then rotated relative to the global one. If, having marked out 6 using a ruler on the above x axis, we lift the ruler and put its zero point at point 6 we are in effect created a local coordinate system at that point. The operator i tells us to turn the ruler through 90 degrees anti clockwise so that its positive scale is now vertically upward. We can think of this as rotating the coordinate system and we count 3 along the now vertical positive scale.
Complex numbers taught are generally confined to a single x, y plane but I hope you can see that a minus operator following the plus 6 would mean at the point 6 turn your ruler and therefore the coordinate system at point 6 through 180 degrees. Any counting we now do along the ruler is now in the negative global x direction.
i is an operator turning a positive scale through 90 degrees counter clockwise and i2 is an operator i acting on an operator i so as to turn the positive scale through 180 degrees and thereby equal to the minus operation. When we teach children i2 = -1 and i = √-1 they rightly say two numbers multiplied together cannot make -1.
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 the number 1. If we write i 1 = √-1 it makes sense because the root of the minus operation is the operation i and the root of 1 is 1. It surprises me that we are still teaching the same old real and imaginary/unreal ideas that I was taught 60 years ago as it doesn’t help 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 don’t think that helped in the student’s understanding. I hope you can now see that the vector had length only along a 90 degree rotated scale at the zero point.
Suppose in our graphical example above it had been 6 – 3i, better written as 6 – i 3. Now we have two operators (minus and i ) and 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.