# Why imaginary co-ordinates and complex numbers?

In summary: If the equation is nonlinear, the exponential will be inaccurate.In summary, modern physics uses imaginary co-ordinates to represent sinusoidal functions, and complex numbers to represent waves. These tools are useful for representing real phenomena.
Most of advance/modern physics has i(imaginary components like E and P are represented so ) in it..How does these imaginary co-ordinates or axes fit into application of physics which explains real world phenomenon..Hope my question sounds reasonable.?.THANK YOU IN ADAVANCE

i believe imaginary coordinates are just convenient ways to express sinusoidal functions
e^(ix)=cosx +I*sinx .
There's this thing call Fourier series that says anything in the real world can be described by adding many sines and cosines together, so that's why many physicists choose to describe the world to describe the world with sines and cosines, and use these imaginary number exponentials to express these sines and cosines...

Complex numbers are not removed from the physical world at all if you actually understand them. You can think of them as either points in a plane or amplitudes and phases of waves.

They're handy for representing rotations.

That and the maths depends on it.
No 'physical observable' takes a complex value in QM, even if their operators are complex.

Things like the probability of something happening or an expectation value or whatever are the magnitude squared of a complex wavefunction usually, the magnitude of a complex number is a real number and so you don't get complex results for real physical observables.

I used to think of complex numbers as merely a useful trick and shorthand for two real numbers, and complex numbers were "artificial" whereas real numbers really existed in nature. Well, I have since changed my stance, and now regard all numbers as mere tools for human models, and this puts complex numbers and real numbers on more equal footing. In modern algebra, various kinds of numbers are not somethings that are discovered from nature, but rather somethings that are defined by their operations.

If complex numbers are more concise than real numbers for representing some theory, they are useful. Of course, any theory with complex numbers can be rewritten in real numbers only. It's silly to consider which representation is more "correct" if they both give the same results.

BUT, there is a note of technical importance. Exponentials are useful in representing real wave-like solution mainly if those solutions are of a linear equation of motion.

## 1. Why do we need imaginary co-ordinates and complex numbers in science?

Imaginary co-ordinates and complex numbers play a crucial role in many areas of science, particularly in physics, engineering, and mathematics. They allow us to describe and analyze phenomena that cannot be represented by real numbers alone, such as electrical circuits, quantum mechanics, and waves. Without the use of these numbers, many scientific concepts and theories would be incomplete or inaccurate.

## 2. How do imaginary co-ordinates and complex numbers differ from real numbers?

Real numbers are those that can be represented on a traditional number line, including positive and negative whole numbers, fractions, and decimals. Imaginary co-ordinates and complex numbers, on the other hand, involve the use of "imaginary" units, denoted by the letter i, which represents the square root of -1. Complex numbers also include a real component and an imaginary component, expressed as a+bi, where a and b are real numbers.

## 3. Can imaginary co-ordinates and complex numbers be visualized?

While real numbers can be easily visualized on a number line, imaginary co-ordinates and complex numbers cannot be represented on a traditional number line. However, they can be represented on a complex plane, which is a two-dimensional graph where the real component is plotted on the x-axis and the imaginary component is plotted on the y-axis. This allows for a visual representation of complex numbers and their relationships.

## 4. What are some practical applications of imaginary co-ordinates and complex numbers?

Imaginary co-ordinates and complex numbers have numerous practical applications in various fields of science. For example, they are used in electrical engineering to analyze AC circuits, in quantum mechanics to describe the behavior of subatomic particles, and in signal processing to analyze and filter signals. They are also used in fields such as fluid dynamics, optics, and control systems.

## 5. Are complex numbers just a mathematical abstraction or do they have real-world significance?

While complex numbers may have been originally developed as a mathematical tool, they have proven to have immense real-world significance. They are used to model and solve a wide range of problems in science, engineering, and mathematics. They provide a more complete and accurate representation of many natural phenomena, and their applications continue to expand as our understanding of the physical world grows.

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