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 in each case is the number and the positive and negative signs, like multiply and divide symbols tell us something about the number or how to deal with it. They are operators.
Operators like plus (+) and minus (-) tell us the scale we are or deliver changes to scale positions. In a bank account a positive tells us we are in credit whilst a minus tells us we are in debt. The + or – tell us which scale we are on. The number tells us our position on the credit or debit 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 positive operation as a movement in one direction along a scale and a minus operation as going in the opposite direction. The latter operation may be described using words like reverse, reduction and backwards. in time
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 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.
Applying the FOIL (first, outer, inner, last) method to multiplying out of the brackets in the simple (6 – 3)(5 -2) we get 30 – 12 – 15 + 6 = 9 which is correct. The + 6 is obtained by multiplying the two numbers 3 and 2 to get the number 6. The plus operator is the result of a negative operator acting on a negative operator.
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. 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” or “times” with “of“. Six of nine is more descriptive than 6 multiplied by 9. Applied to fractions as with say 3/8 of 2/5 you immediately see the answer is a fraction of 2/5.
Simple division (÷) is best thought of as how many of this are there in that? Division can always be expressed as a multiplication. Multiplying by 1/3 is the same as dividing by 3 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 6 is the real number part, +3i the unreal or imaginary number part as identified by the letter i (engineers use the letter j). We learn that i2 = -1 and that i = √-1 and unsurprisingly students and their parents say WHAT!! – there isn’t a number which multiplied by itself makes – 1.
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.
So what does the operator i do? It turns our scale 90 degrees anti clockwise before applying the number. So the complex number 6 plus 3 i means move 6 along a linear scale then turn our ruler through 90 degrees anti clockwise before marking 3 on our now vertical scale. Our 6 + 3i would be far better written as 6 + i 3 because we put operators before numbers but also because behind the number it confusingly looks like an algebraic unknown. We can think of our 6 + i 3 as describing either a new point in the x-y plane or a vector to the point 6,3 from point 0,0.
Suppose we change our 6 + i 3 to 6 + i2 3. What does that mean? It means turn our ruler through 90 degrees and then through 90 degrees again and measure out 3 so as tu finish up at point 3,0. I hope you can see that our i2 operation is the same as a – (minus) operation (there is no number involved in this operation equality)._
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. Surely we can teach the above and stop the WHAT!! response of students and their parents.