The number parts of negative numbers and fractions are just rational whole numbers between zero and infinity. They are no different to the number parts of positive numbers. Positive, negative, multiply and divide symbols are not a part of those numbers; they are instructions that tell us how to position or deal with the number. They are operators.
Operators like plus (+) or minus (-) often convey to us a value on a scale that extends through zero from minus infinity to plus infinity. An example of this we all must encounter is with our bank account and although some will describe themselves as deeply in debt I doubt that they are anywhere near infinity on that scale.
If your account with the bank is in credit you reside on the positive side of a money scale and the bank is in possession of some of your money. If your account is in debit you have taken from the bank more than you had in it and you owe them money and your position may be viewed as on the negative side of the 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 a plus operation moves your money balance up the scale and when you make payments from your account your money balance moves in a reverse direction down that scale. The negative operator reverses the normal positive operator. I say normal because when we are given a number on its own we take it to be a positive one.
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. If you are in debit to the bank they will scale your debit upward and no doubt by a larger multiplying operator. When using the multiplying operator (x) I prefer to think “of”. Six of nine items is more descriptive than six multiplied by nine 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.
Some find it puzzling that when multiplying two minus numbers together we get a positive number. But if we separate the operations from the numbers and multiply them separately we get a minus operation of a minus operation. As each minus operation signifies a reverse or turn through 180 degrees the result is a positive operation that acts on the result of the multiplied number values.
My granddaughter’s algebra text book teaches the FOIL (first, outer, inner, last) method for multiplying double brackets. If we apply it to some simple numbers that involve a minus of a minus, say (6 – 3)(5 -2) we get 30 -12 -15 + 6 = 9 which demonstrates that minus of minus is a plus.
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 can always be expressed as a multiplication. Dividing by 3 for example is the same as multiplying by 1/3 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.
In later school mathematics we may encounter complex numbers like 5 +4i, that comprise of a real number portion and an imaginary number portion identified using the letter i. We learn that i2 = -1 so that i = √-1. This often leads to the common sense question how can we have a square root of -1 and the answer given should be that we cannot. The reason we have this difficulty is because we treat i as an algebraic unknown (4i in our example above). That is grossly misleading because i is not an algebraic unknown, i is not a number at all; i is an operator like those we have so far encountered and as is our practice with other operators it would be better if it preceded the number (instead of writing 6i we should write i 6)
So what does the operator i do? Well it does a 90 degree anti clockwise turn off our linear scale above 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 as illustrated in the diagram.
We are told i2 = -1 and i = √-1 and whilst this is true it helps in understanding why this is so. 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 1 and so i2 is i 1 x i 1. As i is a turn through 90 degrees i of i is a turn through 180 degrees and the same as the minus operator. The 1 of 1 is 1. Hence i2 = -1 We also see that √-1 is the root of the value 1 which is 1 and the root of the minus operator(-) which is i and that is why we can say i = √-1
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 with inductors or capacitors are out of phase with voltages. They can describe vectors and 2d translations and when used with an angular measure can describe rotations