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Holocene
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Actually, can you take a root of any negative number?
Holocene said:Actually, can you take a root of any negative number?
ehj said:how do you know that these complex number exists :P?
ehj said:how do you know that these complex number exists :P?
ehj said:I just thought there was a "proof" of its existence? Like nearly everything else in mathematics, shouldn't there be an argument for it? Or maybe that's quite advanced?
There are technical difficulties with that. Since complex numbers, just like real numbers, have two square roots, defining i as [itex]\sqrt{-1}[/itex], or as "the number such that i2= -1", is ambiguous.Tricore said:All complex numbers can be written on the form [tex]a + ib[/tex] where [tex]a[/tex] and [tex]b[/tex] are real numbers, and [tex]i[/tex] is the imaginary unit, DEFINED as [tex]i=\sqrt{-1}[/tex]. This is just definitions.
Are you asserting that real number "exist in a physical sense"? Can you draw a line of length [itex]\pi[/itex]? For that matter, can you draw a line of length 1?So complex numbers exist because we created them. They don't exist in a physical sense - you can't draw a line on paper with a complex/imaginary length, but perfect circles don't exist in a physical sense either.
I'm not complaining (too much). I agree with most of what you say.Becuase of that, it doesn't make sense to ask for a proof that they exist. What we CAN prove however, is that there exists a solution to [tex] \sqrt[n]{-x} [/tex] within the complex numbers, and also that there is no solution within the real numbers.
HallsofIvy said:There are technical difficulties with that. Since complex numbers, just like real numbers, have two square roots, defining i as [itex]\sqrt{-1}[/itex], or as "the number such that i2= -1", is ambiguous.
What we can do is define the complex numbers as pairs of real number, (a, b), with addtion and multiplication defined by (a, b)+ (c, d)= (a+ c, b+ d), (a, b)*(c, d)= (ac- bd, ad+ bc). Of course one must then prove that this is "well defined" and forms a field (but it does NOT form an "ordered" field).
HallsofIvy said:Are you asserting that real number "exist in a physical sense"? Can you draw a line of length [itex]\pi[/itex]? For that matter, can you draw a line of length 1?
Can't I say that a line, of length [itex]\sqrt{5}[/itex], parallel to the imaginary (i.e. y-axis in an xy-coordinate system) is such a thing?Tricore said:Thanks for correcting me
No, I'm not.
What I was trying to say is that you can draw a line that's (almost) say 5cm, but it doesn't make sense to draw a line that's almost [tex]\sqrt{-5}[/tex]cm.
Yes, it is possible to take an even root of a negative number. However, the result will be a complex number rather than a real number. This is because the even root of a negative number involves taking the imaginary unit, i, as a factor.
An even root is a mathematical operation that involves finding a number that, when multiplied by itself a certain number of times, will result in a given value. For example, the square root is an even root because it involves finding a number that, when multiplied by itself, will result in the given value.
The difference between an even root and an odd root is the number of times the number is multiplied by itself to get the given value. An even root involves an even number of multiplications, while an odd root involves an odd number of multiplications.
No, it is not possible to simplify an even root of a negative number. This is because the result will always involve the imaginary unit, i, which cannot be simplified further.
Yes, there are several real-life applications of taking an even root of a negative number. For example, in engineering and physics, complex numbers are often used to represent alternating current in electrical circuits. In mathematics, complex numbers are used in fields such as calculus and differential equations.