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Math and Arithmetic
Adjectives and Articles
What's the physical meaning of i aka the square root of -1 Does it have any physical existence and if not then why is it used to describe some real physical quantities like some terms in electricity?
December 14, 2009 10:30AM
"i" is actually a special kind of number called an "operator". But what is an operator?
OK, here goes:
The symbols "+" and "-" can be used to show if the value of a number is either positive or negative but, as as well as performing that "sign" function, the symbols "+" and "-" can also be used in a different way: they can be used as "operators" to show what has to be done to perform a calculation.
In these examples the + and - symbols are being used as operators:
1 + 2 = 3
3 - 2 = 1
But in the next example the minus symbol in (-1) is being used to show the sign of the value 1 and the plus symbol is being used as an operator:
(-1) + 4 = 3
Hopefully you may now understand the statement that "i is an operator". But what does the operator "i" actually do? OK, it does something very special: if you plot a value of a number on an (x, y) graph, the operator "i" moves that value round the origin of the graph (x=0, y=0) by 90 degrees.
So if you start with a point on the graph representing a value of, say plus 1 (x=1, y=0), then applying the operator i to that value will change it to be represented on the graph by a new point (x=0, y=1).
OK, but why would anyone bother to do that? Well, one way to explain it is if you consider this: if you apply the operator i to that new point what do you get? You get another new point on the graph which represents the value -1, having graph points (x=-1, y=0). So applying the operator i twice changes a value of plus 1 into a value of minus 1. That total operation, of applying the operator i twice, can be thought of as (i times i) or i2. (We read that out loud as "i squared".)
The end result of all this is that applying the operator i2 to a value is exactly the same as if you multiply that value by (-1).
So it turns out that, in mathematics, thinking of i as being the same as the square root of -1 makes good sense! It may be worth mentioning here that the use of the operator i when working with numbers is known in math as "working with imaginary numbers" or "working with complex numbers". A complex number is a very interesting thing because it includes both a "real" and an "imaginary" part.
Using the operator i to work with imaginary and complex numbers makes it very easy to calculate several things which only work because of a delay or an advance in time for one value (such as, say, current) compared to the normal time of another value (such as, say, voltage). An example is the current flowing in an alternating circuit which has some reactance in it such as inductance and/or capacitance.
Several things such as radio tuning circuits, electric motors, fluorescent lights, etc., must use some reactance. Including some reactance is the only way to get them to work. Radio tuning circuits, electric motors, fluorescent lights, etc., have to be designed by electrical or electronics engineers. Use of complex numbers helps the design engineer because the current in such a circuit lags or leads the voltage by a portion of a wavelength. All the points making up a complete cycle (i.e. one wave) can be represented within a full circle (360 degrees) drawn on a graph, so the operator i can be used to calculate precisely the number of degrees of lead or lag the current is flowing at compared to the voltage, depending on the reactance in the circuit.
At a low frequency, such as the mains supply, a circuit which only has resistance and practically no reactance will take a current "in phase" with the applied voltage. (That means there is no lead or lag, even though the applied voltage is alternating.) Imaginary numbers would not be needed to calculate stuff for a purely resistive circuit running at a low frequency such as the mains supply because everything at that frequency can be calculated accurately enough just using real numbers.
However, even with the same type of wiring, if it is to be used to carry high frequencies - such as radio waves or broadband internet waves - complex numbers would have to be used to work out the actual currents flowing and their phasing compared to the voltage. This would be necessary because any reactance present in a circuit will change proportionally to the frequency. The higher the frequency of the supply, the higher the circuit's reactance will grow...
If you want to study stuff like how to design radio circuits, electronics or electrical engineering projects, or if you want to work in any similar areas of applied science, you would have to learn all about i including the reason it was invented and how it can be used to work with real, imaginary and complex numbers.
'i' has never been defined as the square root of -1. The postulate of i says a number called i exists, such that i x i = -1.
A square root is only applicable to positive numbers. Therefore what is interesting about i is that its sign is undefined, a bit as if one who would be considering a circular trajectory would not mind going around by the left or by the right. That's what the number i can mean if it is used for the analysis of mechanical or electrical oscillations for example.
As i x i = -1 is a postulate it does not require any physical existence because a postulate, by definition, does not require demonstration.
As such the postulate itself is not important, it is what it allows us to describe that is. Whether i exists or not does not matter: when used it is an efficient way allowing us to build mathematical tools to describe many phenomenons.
It might be a bit the same for life: does it really matter if God exists as long as you do what Jesus would have done?
At the end of the day the bet of the existence of 'i' is worth it because it works. So maybe the bet of the existence of God is also worth it, as long as it drives us to behave like the idealistic picture we all have in our mind of his supposed son.
Other mathematical postulates are for example:
1 - A point has no volume and no surface, which is an aberration as a form or a volume is made of points and is possible to measure. In some situations summing points of value 0 can lead to a non-zero value. Conclusion: maybe the whole concept of measurement is based on a nonsense assumption. Yeah fine. But it works!
2 - "Past one point in a straight line passes one and one only parallel line."
This is one of the assumptions of Cartesian flat geometry. If you change that assumption to this other one: "Past one point in a straight line pass an infinite number of parallel lines" you will end up with spherical geometry.
Just works Dude!
It is not because the fundamental assumption seems wrong that the whole model is wrong, for example the assumption that the mass does not vary with the speed perfectly suits the needs of common mechanics but does not fit the needs for studying high speed particles for which the mass variation must be considered.
Try thinking this way. You may be able to visualize the Real Line - axis representing real numbers in sequence. The real line is not actually a line. It is an intercept of a plane in real world. Complex numbers are all those numbers in that plane. We dont feel the number - but we feel the effect of that number, little technically, whenever a representation on real line tends to change direction.
After all, do we visualize negative number. Negative numbers are measures of the absent. When we say 75% is present, it is physical. But we might also say 25% is absent. We could equivalently say -25% is present. Absence makes an effect on the presence. Complex numbers are also such tools, that exist (taking a privilege to say so) to describe effect of mathematical measurements that do not fall exactly on the real line.