# Why are imaginary numbers called "imaginary"? If they really exist

If Imaginary numbers do exist and have real applications, then why do we call imaginary numbers "imaginary numbers"? . They exist. They're used all the time. What makes them "imaginary"?

etotheipi
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You're right! You can look at how Spivak defines complex numbers, as a pair ##(a,b)## along with the definitions ##(a,b) + (a,d) = (a+c, b+d)## and ##(a,b) \cdot (c,d) = (a\cdot c - b\cdot d, a\cdot d + b \cdot c)##. The ##x+iy## notation is recovered by setting ##i = (0,1)##, and ##(a,0) = a##. There's nothing very strange about this at all. And imaginary numbers are just a subset of ##\mathbb{C}##, of the form ##(0,b)##.

Historically, mathematicians have a tendency to be skeptical about new developments, hence why "imaginary" was introduced as a bit of a diss.

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PeroK
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If Imaginary numbers do exist and have real applications, then why do we call imaginary numbers "imaginary numbers"? . They exist. They're used all the time. What makes them "imaginary"?
The name originally was due to skepticism and has stuck. There is, however, an important distinction between real and complex/imaginary numbers in physics. Complex numbers tend to be part of the mathematical machinery, but measurements are always real numbers. You can measure an angle ##\theta##, but you can't do a measurement and get an imaginary angle ##i\theta##.

This is hard-wired into Quantum Mechanics, where measurements are the eigenvalues of Hermitian operators - and these operators always have real eigenvalues. There are other important operators in QM that have complex or imaginary eigenvalues, but these explicitly are not measurable quantities.

etotheipi
fresh_42
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If Imaginary numbers do exist and have real applications, then why do we call imaginary numbers "imaginary numbers"? . They exist. They're used all the time. What makes them "imaginary"?
Not what, whom! It was René Descartes 1637. The reason was very likely, that the complex zeros of polynomials, for which the imaginary numbers were first needed, couldn't be seen in their graphs.

FactChecker, berkeman and aheight
What i have understood is that imaginary numbers are called imaginary because Mathematicians thought up an answer to the square root of -1, and they simply "imagined" a solution. Later when it was found that Imaginary numbers really exist,The name was not changed beause the named sound impressive as by "IMAGINARY NUMBERS" it was left to us to imagine a solution to the square root of negative numbers aince we are imagining the number so it got its name "Imaginary Numbers".

Here comes an important point- IMAGINARY NUMBERS are Imaginary but their existance is not Imaginary they really exist. it was imaginary in the sense as it was left to the people's imagination to imagine a solution to the square root of negative numbers and use the letter i this was fancy and impressive.
So,IMAGINARY NUMBERS are Imaginary but their existance is not Imaginary.they are "imaginary" but not in the sense of fantasy or fiction.It is just the same word being used differently.....

Please tell me have I understood it correctly.....

hutchphd
Homework Helper
You will need to explain what you mean when you say a "real" number "exists". I think this is fraught.
I prefer a more teleological approach: is the concept useful? Of course then one needs to define "useful"

fresh_42
Mentor
Please tell me have I understood it correctly.....
You did not. The correct version is in my post #4. Descartes used it 1637 for the first time. Mathematicians at this point in time were still busy to solve polynomial equations. Imaginary numbers occurred in the formulas of Italian mathematicians like Cardano, Tartaglia or Ferrari a century ago. Descartes, however, coined the name imaginary. If you want to know his thoughts, you have to dig up the original article, likely a letter.

It is not helpful to fantasize some speculative explanations if history is as clear as in this case.

Descartes, however, coined the name imaginary.
My question is not how , its why did he coined the name Imaginary if these numbers really exists.Help me.

I prefer a more teleological approach: is the concept useful? Of course then one needs to define "useful

fresh_42
Mentor
My question is not how , its why did he coined the name Imaginary if these numbers really exists.Help me.
Look up Helmuth Gericke: Geschichte des Zahlbegriffs. Bibliographisches Institut, Mannheim 1970, p. 66. and the references therein.

Again, and for the third time: If you want to know why Descartes named it so, you will have to read the original occurrence.

The question has been answered. Speculations are meaningless, so this thread will be closed.

https://en.wikipedia.org/wiki/Discourse_on_the_Method
https://gallica.bnf.fr/ark:/12148/btv1b86069594/f5.image

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robphy