"The true metaphysics of the square root negative 1 remains elusive." - C.F. Gauss

[Into the Night]

It is a number to those who have conceptualized it as such.

Not a number.

 

[Into the Night]

There are infinite integers. There are also infinite natural numbers. The quantity of natural numbers and the quantity of integers is equal.

Nan.

In fact, the individual quantities of natural numbers, whole numbers, integers and rational numbers are all equal, called denumerably infinite.

Nan.

The quantity of irrational numbers and of real numbers, however, while also being infinite, is of a larger infinity than the denumerably infinite, and it is called nondenumerably infinite.

Nan.

Now that you have learned this, you probably want to adopt an endangered baby harp seal.

Random phrase ignored.

 

[IBDaMann]

Many polynomial equations do not have solutions on the real number number line.

Totally irrelevant.

You have two choices.

False. There is only one correct solution set.

Throw up your hands in surrender and declare that most polynomial equations are unsolvable and simply have no solutions.

Incorrect. This is not a valid option. No one gets to declare that an entirely solvable polynomial is somehow not solvable.

Or you can open your mind to the mind to the possibility that your concept of abstract numbers was too limited, and there exists a whole other abstract concept of number on the complex plane.

False. My grasp of complex systems is already solid, and it is also irrelevant. You need to stop denying math and accept the invalidity of taking the square roots of negative numbers.

 

[IBDaMann]

Nan.

It is a quantity. Integers (as well as natural numbers, whole numbers, and rational numbers) are countable.

Nan.

It is a quantity. Integers (as well as natural numbers, whole numbers, and rational numbers) are countable.

Nan.

It is a quantity. Integers (as well as natural numbers, whole numbers, and rational numbers) are countable.

Random phrase ignored.

Random phrase ignored.

 

[IBDaMann]

Not a number.

Definitely a quantity.

 

[Into the Night]

It is a quantity. Integers (as well as natural numbers, whole numbers, and rational numbers) are countable.

NaN is not a quantity.

It is a quantity. Integers (as well as natural numbers, whole numbers, and rational numbers) are countable.

Nan is not a quantity.

It is a quantity. Integers (as well as natural numbers, whole numbers, and rational numbers) are countable.

Nan is not a quantity.

Random phrase ignored.

LIF. Grow up.

 

[Into the Night]

Definitely a quantity.

Nan is not a quantity.

 

[AProudLefty]

Incorrect. According to mathematics, the square root of negative numbers cannot be taken. Of course, you never learned this because you never finished middle school where this is taught to all children. Stay in school in your next life, no matter how difficult learning is for you.

All you had to do was to research the mathematical proof showing that it is invalid to take the square root of negative numbers, but as you continue to demonstrate, learning is too difficult for you. You would rather deny mathematical proofs.


It's not allowed.


You are denying math because you are too lazy to learn and because you will believe anything you read on the internet.


Yes. Irrelevant. It is not permissible to take the square root of a negative number.

And yet it is done and there are plenty of applications.

 

[AProudLefty]

Incorrect. According to mathematics, the square root of negative numbers cannot be taken. Of course, you never learned this because you never finished middle school where this is taught to all children. Stay in school in your next life, no matter how difficult learning is for you.

All you had to do was to research the mathematical proof showing that it is invalid to take the square root of negative numbers, but as you continue to demonstrate, learning is too difficult for you. You would rather deny mathematical proofs.


It's not allowed.


You are denying math because you are too lazy to learn and because you will believe anything you read on the internet.


Yes. Irrelevant. It is not permissible to take the square root of a negative number.

Imaginary numbers, based on the unit iii where i2=−1i^2 = -1i2=−1, have wide-ranging applications across mathematics, physics, engineering, and other fields. Below is a concise overview of their key applications:

1. Complex Numbers in Electrical Engineering:
   • Used to analyze alternating current (AC) circuits. Imaginary numbers represent phase differences between voltage and current, enabling calculations of impedance, reactance, and resonance in systems like power grids and electronics.
   • Example: In AC circuit analysis, complex impedance combines resistance (real) and reactance (imaginary).
2. Signal Processing and Communications:
   • Essential in Fourier transforms and frequency-domain analysis, which decompose signals into their frequency components. Imaginary numbers handle the phase of sinusoidal waves.
   • Used in modulation techniques for radio, Wi-Fi, and telecommunications to encode and transmit data.
3. Quantum Mechanics:
   • Imaginary numbers are fundamental to quantum wave functions. The Schrödinger equation uses complex numbers to describe the probability amplitudes of quantum states.
   • Example: Quantum superposition and interference rely on complex-valued wave functions.
4. Control Systems and Vibrations:
   • Applied in analyzing dynamic systems, such as mechanical vibrations or feedback control in robotics and aerospace. Imaginary numbers describe oscillatory behavior and stability.
   • Example: Eigenvalues of a system matrix, which may be complex, determine stability and resonance.
5. Fluid Dynamics and Electromagnetism:
   • Used to solve equations governing fluid flow and electromagnetic fields. Complex potentials simplify calculations of irrotational flow or electric fields.
   • Example: In aerodynamics, complex analysis models airflow around aircraft wings.
6. Mathematics and Geometry:
   • Enable solutions to polynomial equations (e.g., x2+1=0x^2 + 1 = 0x2+1=0 has roots ±i±i±i).
   • Used in complex analysis to study functions, contour integrals, and mappings, with applications in topology and fractals.
   • Example: The Mandelbrot set is defined using iterations of complex numbers.
7. Computer Graphics and Transformations:
   • Imaginary numbers (via complex numbers) simplify rotations and scaling in 2D and 3D graphics. Euler’s formula (eiθ=cos⁡θ+isin⁡θe^{i\theta} = \cos\theta + i\sin\thetaeiθ=cosθ+isinθ) is used for rotations.
   • Example: In image processing, complex numbers facilitate transformations like rotations or perspective corrections.
8. Data Analysis and Machine Learning:
   • Used in algorithms involving Fourier transforms (e.g., for audio or image processing) or in optimization problems with complex-valued matrices.
   • Example: Principal component analysis may involve complex eigenvalues for certain datasets.

In summary, imaginary numbers are a powerful tool for modeling oscillatory, rotational, or phase-dependent phenomena, solving equations that have no real solutions, and simplifying complex systems across science and technology. Their integration into complex numbers makes them indispensable in modern applications.

@Grok

 

[Into the Night]

And yet it is done and there are plenty of applications.

Not possible.

 

[Into the Night]

Imaginary numbers, based on the unit iii where i2=−1i^2 = -1i2=−1, have wide-ranging applications across mathematics, physics, engineering, and other fields. Below is a concise overview of their key applications:

Not possible.

 

[AProudLefty]

Not possible.

There are plenty of applications.

 

[Cypress]

Imaginary numbers, based on the unit iii where i2=−1i^2 = -1i2=−1, have wide-ranging applications across mathematics, physics, engineering, and other fields. Below is a concise overview of their key applications:

1. Complex Numbers in Electrical Engineering:
   • Used to analyze alternating current (AC) circuits. Imaginary numbers represent phase differences between voltage and current, enabling calculations of impedance, reactance, and resonance in systems like power grids and electronics.
   • Example: In AC circuit analysis, complex impedance combines resistance (real) and reactance (imaginary).
2. Signal Processing and Communications:
   • Essential in Fourier transforms and frequency-domain analysis, which decompose signals into their frequency components. Imaginary numbers handle the phase of sinusoidal waves.
   • Used in modulation techniques for radio, Wi-Fi, and telecommunications to encode and transmit data.
3. Quantum Mechanics:
   • Imaginary numbers are fundamental to quantum wave functions. The Schrödinger equation uses complex numbers to describe the probability amplitudes of quantum states.
   • Example: Quantum superposition and interference rely on complex-valued wave functions.
4. Control Systems and Vibrations:
   • Applied in analyzing dynamic systems, such as mechanical vibrations or feedback control in robotics and aerospace. Imaginary numbers describe oscillatory behavior and stability.
   • Example: Eigenvalues of a system matrix, which may be complex, determine stability and resonance.
5. Fluid Dynamics and Electromagnetism:
   • Used to solve equations governing fluid flow and electromagnetic fields. Complex potentials simplify calculations of irrotational flow or electric fields.
   • Example: In aerodynamics, complex analysis models airflow around aircraft wings.
6. Mathematics and Geometry:
   • Enable solutions to polynomial equations (e.g., x2+1=0x^2 + 1 = 0x2+1=0 has roots ±i±i±i).
   • Used in complex analysis to study functions, contour integrals, and mappings, with applications in topology and fractals.
   • Example: The Mandelbrot set is defined using iterations of complex numbers.
7. Computer Graphics and Transformations:
   • Imaginary numbers (via complex numbers) simplify rotations and scaling in 2D and 3D graphics. Euler’s formula (eiθ=cos⁡θ+isin⁡θe^{i\theta} = \cos\theta + i\sin\thetaeiθ=cosθ+isinθ) is used for rotations.
   • Example: In image processing, complex numbers facilitate transformations like rotations or perspective corrections.
8. Data Analysis and Machine Learning:
   • Used in algorithms involving Fourier transforms (e.g., for audio or image processing) or in optimization problems with complex-valued matrices.
   • Example: Principal component analysis may involve complex eigenvalues for certain datasets.

In summary, imaginary numbers are a powerful tool for modeling oscillatory, rotational, or phase-dependent phenomena, solving equations that have no real solutions, and simplifying complex systems across science and technology. Their integration into complex numbers makes them indispensable in modern applications.

@Grok

Nice, and as per the article posted in the OP, certain calculations in quantum mechanics cannot be done without imaginary numbers. You can't just substitute in real numbers.

 

[IBDaMann]

NaN is not a quantity.

"Liter" is NaN and is a quantity. "Quantity" and "number" are not the same thing. You are most welcome.

 

[IBDaMann]

Nan is not a quantity.

Correct. Infinity is.

 

[IBDaMann]

Imaginary numbers, based on the unit iii where i2=−1i^2 = -1i2=−1,

Math error blended with gibberish.

There is no unit "iii" in standard mathematics. There is no i2. i^2 = -1. -i^2 = 1

@Cypress - why didn't you catch this? Instead, you praised the gibberish and the bad math.

 

[IBDaMann]

Certain calculations in quantum mechanics cannot be done without imaginary numbers.

Why do you say this? All probabilities can be expressed in real numbers. What specific quantum mechanics calculations are you claiming cannot be done without imaginary numbers?

 

[AProudLefty]

Math error blended with gibberish.

There is no unit "iii" in standard mathematics. There is no i2. i^2 = -1. -i^2 = 1

@Cypress - why didn't you catch this? Instead, you praised the gibberish and the bad math.

I noticed you ignored the applications.

 

[IBDaMann]

And yet it is done

Of course errors are made. So?

and there are plenty of applications.

You never learned to utilize the English language properly so you're going to have to work a little bit harder. Are you trying to say that there are applications for math errors?

If you are trying to say that it is valid for engineers to perform bad math, then you are mistaken ... once again.

 

[IBDaMann]

I noticed you ignored the applications.

There are no applications for math errors.