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What are imaginary numbers? - Mathematics Stack Exchange
You ask why imaginary numbers are useful. As with most extensions of number systems, historically such generalizations were invented because they help to simplify certain phenomena in existing number systems. For example, negative numbers and fractions permit one to state in a single general form the quadratic equation and its solution (older solutions bifurcated into many cases, avoiding ...
complex numbers - What is $\sqrt {i}$? - Mathematics Stack Exchange
The square root of i is (1 + i)/sqrt (2). [Try it out my multiplying it by itself.] It has no special notation beyond other complex numbers; in my discipline, at least, it comes up about half as often as the square root of 2 does --- that is, it isn't rare, but it arises only because of our prejudice for things which can be expressed using small integers.
Why don't we define "imaginary" numbers for every "impossibility"?
Particularly for the imaginary numbers you mentioned, the square root of −1 1 was contemplated because it simplified manipulations on polynomials when looking for their roots.
Historical Motivation for the Creation of Complex/Imaginary Numbers?
Does anyone know from a historical perspective what kinds of problems/situations resulted in the need and creation of complex/imaginary numbers? Recently, my friend was asking about why we need complex numbers and what are there applications.
Why were Imaginary numbers invented - Physics Forums
Imaginary numbers, specifically the unit i, were invented to solve the equation x² + 1 = 0, which has no solution in the real number system. The development of number systems progressed from counting numbers to whole numbers, integers, rationals, and finally to real numbers, each time addressing limitations in solving equations.
Rules of imaginary numbers - Mathematics Stack Exchange
Rules of imaginary numbers Ask Question Asked 10 years, 10 months ago Modified 2 months ago
"Where" exactly are complex numbers used "in the real world"?
There are plenty of applications of complex numbers, but from what I have seen they are typically used to simplify solving a math equation, and the end result is still a real number. Or in some cases (lile quantum) a 2d vectors are represented with complex numbers, but could be represented with 2d vectors.
imaginary numbers - how can I understand them - especially as they ...
13 In another question here, about roots of equations being imaginary, the accepted answer said something interesting about "imaginary" (as a technical word in math) not meaning "not real". I understand that imaginary numbers are "manipulable and usable to do all kinds of things". And thus, "they exist" at least, in the realm of mathematics.
Why do complex numbers lend themselves to rotation?
Why do complex numbers love doing this so much? I can understand why these theorems work; however, aside from basic knowledge of polar coordinates, I do not intuitively understand what property of complex numbers make rotation and angle such a common/convenient idea.
Are there imaginary numbers other than i? • Physics Forums
The discussion centers on the existence of "imaginary" numbers beyond the standard unit i in complex analysis. Participants explore extensions of complex numbers, including quaternions and octonions, which introduce additional dimensions and properties such as non-commutativity. The hyperreals and dual numbers are also mentioned as alternative number systems that expand upon traditional real ...
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