Can someone give a physical meaning for imaginary numbers? The imaginary numbers, in my opinion, are truly imaginary. What do they even represent? Irrational numbers are, well, preposterous but I can accept them. √2, π and φ have some tangible meaning, but √(-1)? What does it mean? A solution of x^2+1=0? But that equation itself is artificial, representing nothing physical, at least as of now. Complex numbers represent vectors and are useful as phasors in electrical engineering, electromagnetism and other fields. But that is all they are – a tool, a short hand notation to ease the mathematical calculations – and not really real. Phasors allay the complexity of calculations but even without them we could still do all the calculations, albeit in a convoluted way. So do these imaginary numbers mean something in reality? Can someone give me an example? The closest real world counterparts of complex number I can think of are the probability amplitudes of quantum physics.
The meaning depends on what you measure. The most commonly used measurements that I'm aware of that give complex numbers are that of electrical impedance, and various ways to measure waves. Occasionally it's useful to measure position with complex numbers, such as in 2-D fluid flow problems.
Think of i as having the same value as "1" but in a different axis of measurement. If you find it helpful, imagine a number line; all expressible real numbers fall on this line. However, a number that departs from this line might have the same "real" component as another number but include an imaginary component. This could be the differnence between "5" and "5+2i." Like Hurkyl mentioned, how that is interpreted in real-world applications could differ fundamentally. In electrical engineering, we use j instead of i and we use it to connote a complex effective impedance of a load.
Well - so what is the physical meaning of π ? I may agree that natural numbers have direct meaning (no rabbits, one rabbit, two rabbits, three rabbits, plenty of rabbits). Maybe positive rationals could also be accepted (the rabbit weights 2.5 pound). But all the rest? What is the "physical meaning" of -3 ? Of 3/4? Or of π ? All numbers (except of naturals) are only abstracts used in abstract equations. If your answer to the meaning of -3 is that I may walk 3 steps forward, then 3 steps back (-3 steps) - then you should accept that walking 3 teps to the left may be noted as 3i steps, and 3 steps right as -3i steps.
Of course, as you suggest... mathematics [specifically, complex numbers] are tools of science. There is a way to deal with only real numbers if you are willing to use matrices. Let Z=aI+bJ, where a and b are real and I is the 2x2 identity matrix and J is the matrix [itex]\left(\begin{matrix} 0 & 1 \\ -1 & 0 \end{matrix} \right) [/itex]. You can find the analogue of complex arithmetic operations in terms of matrix operations. Do you have a problem with matrices? In the grand scheme of things... this is probably an indication that physical quantities are not just "the counting numbers"... not just scalars... but more complicated objects reflecting symmetries or other structure. Thus, we find it convenient to use vectors, matrices, spinors, complex numbers, quaternions, .... There are actually more irrational numbers than rational numbers.
π is the ratio of the circumference to the diameter of a circle that I can draw on sand. 3/4 is the weight of iron (in kilogram) that is 0.75kg. -3: cant think of physical meaning, but maybe there is one. If you interpret -3 as going backwards 3 steps, then -3 becomes only an aid to do calculations. This is the problem I am having with i. Is i simply a tool, or is it real?
I don't actually see the issue. Mathematics in general is simply a tool to help people understand reality, and in some sense the only mathematical reality is integers, things you can count - the rest is tools. Imaginary numbers are another tool, and can be critically important in some circumstances - roots of equations for example can be imaginary, and those roots can have real-number physical consequences even if you wish to scoff at the reality of the square root of negative one.
False. If you draw it on sand and measure it with a stick the ratio is probably 30:10. Maybe 31:10... But definitely not π. Your measurement has limited precission. So the measured ratio must be a rational number. π is an idealisation you use, as you were smart enough to understand Euclid's abstract view of geometry. Try to find intuitive one - better than counting steps back as negative. It is definitely a good tool - like all other kinds of numbers and the whole mathematics. And it is not real - it is imaginary But it may be used to describe real processes - and it is almost equally applicable for that purpose as real numbers.
See here: http://en.wikipedia.org/wiki/Imaginary_numbers under "geometric interpretations " and "applications". imaginary numbers are widely used in electrical phase measurements, feedback and signal processing analysis.
How does that differ in any substantive way from real numbers? The real numbers are a set of mathematical abstractions that follow certain rules and transformations. These rules and transformations are useful tools for calculating the predicted results of certain physical experiments. Similarly with complex numbers. The fact that they are called "imaginary" is merely a naming convention and not a reflection of their ontological status.
If you have limited precision, you don't have enough precision to measure a rational number. :tongue: Measuring devices measure what they measure -- and interpolation is one of the ways a ruler is used, and I can certainly interpolate to [itex]\pi[/itex] if I wanted to. I could even get a ruler that has a marking of [itex]\pi[/itex] centimeters. (I have seen clocks with such markings) It's good to remind people that their measurements have limited precision. It's bad to invoke limited precision to justify sacrificing accuracy to replace a measurement with a rational approximation, or to pedagogically cripple yourself into avoiding irrational numbers.
List of bad naming choices, that will continue to stir up misguided philosophical debates for ages to come: - real vs. imaginary numbers - real vs. fictitious forces - (intrinsic) curvature Feel free to extend
Multiplying by the number i rotates the plane pi/2 radians (90 degrees) counterclockwise. If you start at (1,0) in the plane and multiply by i then multiply by i again, where to you end up? At (-1, 0). So i^2 = -1. It's really that simple.
Ruler with pi mark? I envy you... I only have slide rule with pi, e, and roots of small integers on its logarithmic scale. Anyway - I insist that we may only measure natural numbers. Even rational measures are secondary to them and conventional (I may say that I am 1.83m tall, but it is derived from 183 cm). Egyptians used measuring rod and rope with equidistant knots, then they had to count how many rods (not: how much!) they had to mark along the measured distance. More precise measurement could be done with smaller (12 times shorter) rod - but then we again had integer number of 'short rods'. Real numbers are only 18th century (pre-atomic, pre-quantum, pre-information-theory) idealisation of continuous behaviour of the Nature and are not measureable by any means. In modern times of digital apparata such 'natural number' measurement is even more apparent.
I don't see how that's really different from using irrational numbers. To mirror your reasoning, I may say the ratio of a circle's circumference to its diameter is 3.1415..., but perhaps what I mean is, I made a measurement using a ruler with an infinite number of infinitesimal marks, and counted 3.1415...x10^inf of them. Similarly, I might have a ruler with every ith unit marked off, and thereby measure something to be 2+3i.