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Nuke
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I think you will find this very usefull https://www.youtube.com/watch?v=T647CGsuOVU&list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF watch the entire series, is really well done!

To the point.

The confusion begins by calling a certain set of numbers (those that belong to ℝ) as "real" numbers.

In reality what is "real" and what is not, is a philosophical question, and thus, it sits outside of the practice of mathematics.

"Real" numbers are not more "real" than any other number... Natural numbers are real numbers too. And of course, imaginary numbers (and complex numbers) are also... real numbers too.

Or in other words... All numbers are equally real as the "real" ones and equally imaginary as the "imaginary" ones.

Imaginary numbers are DEFINITELY as valid and as real and as usefull and as "physical" as any other number. They are NOT mathematical hacks, or "tools" that we use in a "shut up and calculate" fashion.

They make PERFECT sense in mathematical logic, and they are the DIRECT extension of the function of powers and square roots.

Imaginary numbers are basically numbers that sit at right angles from the "real" number line. So everytime we use a system of 2 axes, we knowingly or unknowingly use imaginary and complex numbers. They are hardwired into our physical reality and they are not at all weird or spooky.

The fact that $\sqrt{-1}$ exists, is not weird at all... We don't square a "real" number to get -1, we square a number that sits at right angles from the real number line, its not weird that when we do square it, it gives -1. Its just the next logical step after we have defined powers and square roots.

Its school that has made us all a great disservice, by not teaching us imaginary logic correctly, and made us all believe that somehow we can introduce all kinds of nonesense into math - we cant.

Imaginary numbers are not nonesense, they are valid numbers, as valid as any other number that you can think off.

Everytime you use a graph of 2 axes, I don't know if you use vectors or polar forms or whatever, you are still using imaginary numbernumbers even withought knowing it.

I think you will find this very usefull https://www.youtube.com/watch?v=T647CGsuOVU&list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF watch the entire series, is really well done!

To the point.

The confusion begins by calling a certain set of numbers (those that belong to ℝ) as "real" numbers.

In reality what is "real" and what is not, is a philosophical question, and thus, it sits outside of the practice of mathematics.

"Real" numbers are not more "real" than any other number... Natural numbers are real numbers too. And of course, imaginary numbers (and complex numbers) are also... real numbers too.

Or in other words... All numbers are equally real as the "real" ones and equally imaginary as the "imaginary" ones.

Imaginary numbers are DEFINITELY as valid and as real and as usefull and as "physical" as any other number. They are NOT mathematical hacks, or "tools" that we use in a "shut up and calculate" fashion.

They make PERFECT sense in mathematical logic, and they are the DIRECT extension of the function of powers and square roots.

Imaginary numbers are basically numbers that sit at right angles from the "real" number line. So everytime we use a system of 2 axes, we knowingly or unknowingly use imaginary and complex numbers. They are hardwired into our physical reality and they are not at all weird or spooky.

The fact that $\sqrt{-1}$ exists, is not weird at all... We don't square a "real" number to get -1, we square a number that sits at right angles from the real number line, its not weird that when we do square it, it gives -1. Its just the next logical step after we have defined powers and square roots.

Its school that has made us all a great disservice, by not teaching us imaginary logic correctly, and made us all believe that somehow we can introduce all kinds of nonesense into math - we cant.

Imaginary numbers are not nonesense, they are valid numbers, as valid as any other number that you can think off.

Everytime you use a graph of 2 axes, I don't know if you use vectors or polar forms or whatever, you are still using imaginary number even withought knowing it.

I think you will find this very usefull https://www.youtube.com/watch?v=T647CGsuOVU&list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF watch the entire series, is really well done!

To the point.

The confusion begins by calling a certain set of numbers (those that belong to ℝ) as "real" numbers.

In reality what is "real" and what is not, is a philosophical question, and thus, it sits outside of the practice of mathematics.

"Real" numbers are not more "real" than any other number... Natural numbers are real numbers too. And of course, imaginary numbers (and complex numbers) are also... real numbers too.

Or in other words... All numbers are equally real as the "real" ones and equally imaginary as the "imaginary" ones.

Imaginary numbers are DEFINITELY as valid and as real and as usefull and as "physical" as any other number. They are NOT mathematical hacks, or "tools" that we use in a "shut up and calculate" fashion.

They make PERFECT sense in mathematical logic, and they are the DIRECT extension of the function of powers and square roots.

Imaginary numbers are basically numbers that sit at right angles from the "real" number line. So everytime we use a system of 2 axes, we knowingly or unknowingly use imaginary and complex numbers. They are hardwired into our physical reality and they are not at all weird or spooky.

The fact that $\sqrt{-1}$ exists, is not weird at all... We don't square a "real" number to get -1, we square a number that sits at right angles from the real number line, its not weird that when we do square it, it gives -1. Its just the next logical step after we have defined powers and square roots.

Its school that has made us all a great disservice, by not teaching us imaginary logic correctly, and made us all believe that somehow we can introduce all kinds of nonesense into math - we cant.

Imaginary numbers are not nonesense, they are valid numbers, as valid as any other number that you can think off.

Everytime you use a graph of 2 axes, I don't know if you use vectors or polar forms or whatever, you are still using imaginary numbers even withought knowing it.

Source Link
Nuke
Nuke
  • 117
  • 1
  • 8

I think you will find this very usefull https://www.youtube.com/watch?v=T647CGsuOVU&list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF watch the entire series, is really well done!

To the point.

The confusion begins by calling a certain set of numbers (those that belong to ℝ) as "real" numbers.

In reality what is "real" and what is not, is a philosophical question, and thus, it sits outside of the practice of mathematics.

"Real" numbers are not more "real" than any other number... Natural numbers are real numbers too. And of course, imaginary numbers (and complex numbers) are also... real numbers too.

Or in other words... All numbers are equally real as the "real" ones and equally imaginary as the "imaginary" ones.

Imaginary numbers are DEFINITELY as valid and as real and as usefull and as "physical" as any other number. They are NOT mathematical hacks, or "tools" that we use in a "shut up and calculate" fashion.

They make PERFECT sense in mathematical logic, and they are the DIRECT extension of the function of powers and square roots.

Imaginary numbers are basically numbers that sit at right angles from the "real" number line. So everytime we use a system of 2 axes, we knowingly or unknowingly use imaginary and complex numbers. They are hardwired into our physical reality and they are not at all weird or spooky.

The fact that $\sqrt{-1}$ exists, is not weird at all... We don't square a "real" number to get -1, we square a number that sits at right angles from the real number line, its not weird that when we do square it, it gives -1. Its just the next logical step after we have defined powers and square roots.

Its school that has made us all a great disservice, by not teaching us imaginary logic correctly, and made us all believe that somehow we can introduce all kinds of nonesense into math - we cant.

Imaginary numbers are not nonesense, they are valid numbers, as valid as any other number that you can think off.

Everytime you use a graph of 2 axes, I don't know if you use vectors or polar forms or whatever, you are still using imaginary number even withought knowing it.