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Is it possible to solve a radical equation where the root is a negative integer?

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Is it possible to solve a radical equation where the root is a negative integer?

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AKG

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[tex]x = \sqrt{-1}[/tex]

then:

[tex]x = i[/tex]

Where [itex]i[/itex] is the imaginary number defined as the square root of -1.

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Can you explain to me the logic behind imaginary numbers which apparently result when the radicand is negative and the root is an even integer?

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arildno

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(a,b)+(c,d)=(a+c,b+d)

and the multiplication:

(a,b)*(c,d)=(ac-bd,ad+bc)

Note that any complex number on the form (a,0) fulfills every property that a real number has!

An imaginary number is on the form (0,b).

If we make the multiplication (0,1)*(0,1) we get:

(0,1)*(0,1)=(0*0-1*1,0*1+0*1)=(-1,0)

This is the meaning of (0,1) as the "root" -1.

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AKG

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No. By convention I guess, [itex]\sqrt{4}[/itex] and [itex]4^{1/2}[/itex] are [itex]+2[/itex]. If you had a question like [itex]x^2 = 4[/itex], then the answer would be [itex]x=\pm 2[/itex]. Similarly, whereas [itex]\sqrt{4} = 2[/itex], [itex]\pm \sqrt{4} = \pm 2[/itex].Can "two" be negative?

An imaginary number is simply any real number times [itex]i[/itex]. So, if [itex]\sqrt{5.5225} = 2.35[/itex], then [itex]\sqrt{-5.5225} = \sqrt{5.5225} \times \sqrt{-1} = 2.35i[/itex]. You may want to check out the Mathworld.com article on Complex Numbers.

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HallsofIvy

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There is a very elegant branch of mathematics known as complex variable theory which deals in detail with the properties of complex numbers (a number with a real and imaginary part). This theory can be used to greatly simplify the solution of some problems that would be just about impossible to solve otherwise. As a simple example, try to prove the common trig identity Cos(A + B) = CosACosB - SinASinB using standard methods. Using Euler's Formula from Complex Variable Theory, it becomes almost trivial.

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arildno

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While sharing your dislike of that term (for the same reason), I dislike the term "irrational numbers" even more..geometer said:I dislike the term "imaginary numbers." These numbers exist as much as any other number exists.

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arildno

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Not really, the original word "ratio" means reason, and the term "irrational numbers" was explicitly formed to mean numbers that were "unreasonable"..geometer said:

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What are you going on about? Sources? The reason I ask is because this contradicts reality. The word "ratio" is from the Latin to calculate or reckon. Just because something sounds "cool" doesn't mean its right. Obviously, the common usage of rational follows from this definition.arildno said:Not really, the original word "ratio" means reason, and the term "irrational numbers" was explicitly formed to mean numbers that were "unreasonable"..

*Nico

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arildno

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It sort of fitted the documented opposition in history from the evolution of new numbers (as in the Pythagoreans' ambivalence to the discoveries of irrational numbers, European mathematicians opposition to negative numbers in the 16'th century, and, about the same time, or a bit later, the coining of term "imaginary" in opposition to "real").

It might well be a myth in the case of irrational numbers, so, assuming that, thx.

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jcsd

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arildno

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Now that's REALLY COMPLEX, jcsd

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jcsd

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Few numbers are PERFECT thougharildno said:Now that's REALLY COMPLEX, jcsd

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arildno

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But most of them are FRIENDLY

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jcsd

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Yes I've ceratinly found some numbers are very AMENABLE to my needs *groan*

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arildno

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I think this thread has degenerated into a PRIME example of stupid math jokes..

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