How do IMAGINERY Number exist?

Question

If they are MAGINERY, like IMAGINERY friends,
then how can they exist?

SQR 1 = i?

Answer 1

Existence is a fuzzy word, but its meaning is most clear when referring to tangibles like dogs, cats, and Democrats. In this sense, then, no numbers "exist". There are just ideas expressed on paper to keep them organized. But the scrawlings themselves only represent these ideas; numbers are not made of graphite. They're defined in terms of how they relate to each other through operations like addition and multiplication. Any relationship to the real world falls into the realm of physical theory, not mathematics itself. 1 and i are both perfectly good numbers simply because the are well defined operationally. The rules they follow in accordance with their *definitions* include 1 * 1 = 1 and i * i = -1.

Answer 2

imaginary numbers (or complex numbers) are nothing like imaginary friends. Complex numbers are quite real. If you extend the definition of square root, it is any 1 value that when multiplied with itself gives the sqrt(x). For x = -1 it just so happens there is a value, but it is not scalar. If we embed this problem in the complex domain and use a 2x2 matrix, then:

sqrt(-1) =

[ 0, 1]
[ -1, 0]

If we square this matrix and bring it back down to 1 dimension, we do indeed get -1.

Complex numbers are often used in signal processing, radio transmission, fourier analysis... the topics are innumerable.

Answer 3

You mean imaginary number, it exists in your imagination. It seems that in mathematics they deal with everyting including the imaginary things. Squareroot of negative 1? Whats the use of that?