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Tyrion101
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Ok, so i by itself is -1, but i^2 is also -1, and I don't really understand why both are, and why we need i if it is just -1? Am I missing part of the definition?
Tyrion101 said:Ok, so i by itself is -1, but i^2 is also -1, and I don't really understand why both are, and why we need i if it is just -1? Am I missing part of the definition?
Tyrion101 said:Ok, that makes more sense than what I understood I was being told earlier.
jedishrfu said:No i by itself is i. Its defined as i=sqrt(-1) and is known as the imaginary number. So id say you are missing part of the definition.
for more info:
http://en.wikipedia.org/wiki/Imaginary_number
FeDeX_LaTeX said:I think it is usually defined as ##i^{2} = -1##.
jtbell said:You have to be aware that ##\sqrt{-1}## can be either +i or -i, just as ##\sqrt{1}## can be either +1 or -1. Usually we're OK if we take the positive root, but sometimes this gets us into trouble.
jtbell said:You have to be aware that ##\sqrt{-1}## can be either +i or -i, just as ##\sqrt{1}## can be either +1 or -1. Usually we're OK if we take the positive root, but sometimes this gets us into trouble.
jtbell said:You have to be aware that ##\sqrt{-1}## can be either +i or -i, just as ##\sqrt{1}## can be either +1 or -1. Usually we're OK if we take the positive root, but sometimes this gets us into trouble.
gopher_p said:It is a widely accepted convention among math educators and mathematicians that ##\sqrt{a}##, when ##a## is a positive real number, denotes the positive root; i.e the unique number ##b## such that ##b>0## and ##b^2=a##. Hence ##\sqrt{1}=1##. Full stop. Math notation only gets you into trouble when you abuse it.
Mark44 said:Furthermore, by the same convention, ##\sqrt{-1} = i##.
Mark44 said:Furthermore, by the same convention, ##\sqrt{-1} = i##.
By "same convention" I meant that we get only one value out of a square root, not that what we get is necessarily positive. What gopher_p was saying was restricted to square roots of nonnegative numbers.Tobias Funke said:By saying it's the same convention as gopher_p mentioned, you're saying that [itex]i>0[/itex].
Tobias Funke said:If we use the more general convention of choosing the smallest nonegative angle in polar form, then of course it's the same convention.
That would put the branch cut along the positive real axis. The standard convention for the principal square root is to put the branch cut along the negative real axis, with points a point x on the negative real axis mapping to i√|x|.Tobias Funke said:f we use the more general convention of choosing the smallest nonegative angle in polar form, then of course it's the same convention.
Mark44 said:Furthermore, by the same convention, ##\sqrt{-1} = i##.
HallsofIvy said:I don't think that objecting to saying that "i" is "the square root of -1" and "-i" is "the other one" in defining "i" is "pedantic".
You are sweeping a very important point, the distinction between i and -i under the rug!
Neither "[itex]i^2= -1[/itex]" nor "[itex]i= \sqrt{-1}[/itex]" works as a rigorous definition of i.
Much better, and what any really strict textbook will do, is to define the complex numbers as pairs of real numbers with addition defined by (a, b)+ (c, d)= (a+c, b+ d) and multiplication by (a, b)(c, d)= (ac- bd, ad+ bc). We can then show all of the usual properties of addition and multiplication (commutativity, associativity, distributive law), then show that the assignment x-> (x, 0) embeds the natural numbers in the complex numbers while defining i= (0, 1) gives [itex]i^2= (0(0)- (1)(1), (0)(1)+ (1)(0))= (-1, 0)= -1[/itex].
Also note that (b, 0)(0, 1)= (b(0)- (0)(1), 0(0)+ b(1))= (0, b)= bi using the definition "i= (0, 1)".
Using the notation, derived from the embedding of the real numbers in the complex numbers and the definition of ix as (0, 1), (a,b)= (a, 0)+ (0, b)= a(1, 0)+ b(0, b)= a+ bi gives the usual notation for the complex numbers.
gopher_p said:It's my understanding that there is no way to distinguish the two complex roots of the polynomial ##x^2+1## aside from arbitrary assignments of names. There are two roots. We'd like to give them names, so we pick one and call it ##i##. The other happens to be the additive inverse of ##i##, so we're kinda forced to call it ##-i## (poor bastard). Just by virtue of the order in which they get named, ##i## is "the" square root of ##-1## and ##-i## is "the other".
Or perhaps there are no roots of ##x^2+1##, and so we invent one and call it ##i##. And because ##i^2+1=0\Rightarrow i^2=-1##, it only makes sense to use (abuse) familiar terminology and say that ##i## is the/a square root of ##-1##. But wait! If I look at this other new thing, ##-i##, it turns out that it satisfies ##(-i)^2=-1## as well! So it's the other/another square root of ##-1##. Is there any way, other than their names, to tell these two roots apart? Not that I am aware of.
Also there is the completely philosophical (and likely off-limits) issue of whether this construction is the complex numbers or if it is merely isomorphic/homeomorphic to the complex numbers and whether there is even a difference.
In complex numbers, i is defined as the imaginary unit, which is equal to the square root of -1. Therefore, when we raise i to the power of 2, we are essentially squaring the square root of -1, which results in -1.
Yes, complex numbers can be visualized on a graph known as the complex plane. The real numbers are represented on the horizontal axis and the imaginary numbers are represented on the vertical axis.
Complex numbers are used in a variety of fields, including physics, engineering, and economics. They are used to represent and solve problems involving alternating currents, vibrations, and financial transactions, among others.
The imaginary unit i plays a crucial role in complex numbers as it allows for the representation of numbers that cannot be expressed with real numbers alone. It also allows for the solution of certain equations that would be impossible to solve without the use of complex numbers.
Yes, complex numbers can have a real part equal to 0, in which case they are purely imaginary, and an imaginary part equal to 0, in which case they are purely real. This is often seen in equations where the solution is a real number or a pure imaginary number.