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.
Operators: You may think of -2 as being a different number to -+2 and you wouldn’t be alone. There are numerous references to negative numbers on the internet. But in reality 2 is the number and the + and – signs, like multiply and divide symbols tell us something about the number or how to deal with them. They are not a part of the number.
Plus (+) and minus (-) operators tell us the scale we are on or tell us the direction we are to move on that scale. A positive can tell us we are in credit at the bank and a minus that we are in debt. The number tells us our position on the credit or debit scale. When money is paid into the bank your position on the scale moves in a positive direction and when you make payments from your bank account your position on the scale moves in a negative direction.
A positive operation is a move in one direction along a scale whilst a minus operation moves us in the opposite direction. The latter operation may be described using words like reverse, reduction or backwards.
Multiplication and Operators: Some find it puzzling that when multiplying two minus numbers together we get a positive number. But, as explained above, there are no minus numbers. 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.
Square Roots: 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 of a negative operation or a positive of a positive one.
Complex Numbers: You may have encountered complex numbers, part real, part unreal or imaginary. The only complexity I see is in how we describe and explain them.
Numbers start at zero and rise to infinity. There are no unreal, imaginary or even negative numbers. The negative linked to a number further describes it or requires a number operation. It is not part of a number. Supposedly unreal numbers are actually real numbers operated on by i, which like +, -, x and ÷ is an operator, not a number or part of a number.
The operator i is unfortunately defined as an imaginary unit where i2 = -1 and unsurprisingly students and their parents conclude that i = √-1 and say WHAT!! – there isn’t a number which multiplied by itself makes the number – 1 and they are right.
They are right because -1 is not a number and the definition is wrong. It should not say that i2 = -1. It should say that i2 = – (the minus operator). Meaning operation i on operation i is a reversal or backward move on a number scale.
Operation i is a half turn toward a reversal. 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 operators normally go before numbers but also because behind the number i adds to confusion by looking 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 at number 6 on our scale turn our ruler through 90 degrees and then through 90 degrees again and measure out 3 so as to 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.