I understand how to solve problems dealing with imaginary numbers, but I don't understand the reason why they exist and what they really do. Could somebody please explain to me what the point of them is? What I don't understand is that wouldn't multiplying by negative one just do the same thing?
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1You are mistaken when you say "it takes out the negative". – Paul Sundheim Nov 09 '14 at 00:04
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@PaulSundheim What would it do then? – IHeartBunnies Nov 09 '14 at 00:04
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@IHeartBunnies They give an answer to "$\sqrt{-1}={}?$". Multiplying $-1$ by negative one wouldn't give you the right answer to your question. (Also, it turns out that imaginary and complex numbers are useful in other branches of math, like trigonometry. It can be shown that:$$(\cos(A)+i\sin(A))(\cos(B)+i\sin(B))=(\cos(A+B)+i\sin(A+B)),$$a fact that is very useful.) – Akiva Weinberger Nov 09 '14 at 00:19
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Ok, thanks. That makes a little more sense now. – IHeartBunnies Nov 09 '14 at 00:21
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Why does 14 exist? – Mariano Suárez-Álvarez Nov 09 '14 at 02:53
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@MarianoSuárez-Alvarez What do you mean by this question? – IHeartBunnies Nov 10 '14 at 01:55
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My point is that the ontological status (that is, why it exists, in what sense does it exists, etc) of the number 1+i is not at all different from that of the number 14. The only difference is that you are more familiarized with the latter so you just don't even think to ask those questions about it. – Mariano Suárez-Álvarez Nov 10 '14 at 02:03
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@MarianoSuárez-Alvarez I understand what you are saying, but I mean really what is the point of them? – IHeartBunnies Nov 10 '14 at 02:12
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The point of $1+i$ is exactly the same as the point of $14$. – Mariano Suárez-Álvarez Nov 10 '14 at 02:17
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@MarianoSuárez-Alvarez I understand why you are saying. I don't know exactly how to word this, but what exactly do they represent? If they don't really exist, what is the reasoning behind having them? – IHeartBunnies Nov 10 '14 at 02:25
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What does 14 represent? – Mariano Suárez-Álvarez Nov 10 '14 at 02:27
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Let us continue this discussion in chat. – IHeartBunnies Nov 10 '14 at 02:28
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When you are asking about why they exist, I take it you mean why they were developed? Because if you're really asking about whether numbers exist, that becomes a philosophical and rather complicated question about our ontological commitments to mathematical entities.
They were first noticed possibly when mathematicians were solving quadratic polynomials, i.e. $ax^2+bx+c=0$. You'll quickly notice that sometimes we get solutions involving taking the square root of negative value. Mathematicians dismissed this as being absurd until they began to work on finding a formula for the roots of the general cubic polynomial, i.e. $ax^3+bx^2+cx+d=0$.
As for what they do, they have a lot of applications within and outside of mathematics. We're able to solve a lot of problems which appear to be firmly fixed in the real numbers using complex numbers. Within mathematics, this can be seen in geometry, calculus, etc. Outside of mathematics, it is extremely useful to physics and thus useful to engineering, particularly electrical engineering.
If our goal was to "get rid" of the negative, sure, multiplying by a negative number would get rid of it symbolically but then that changes our equation algebraically.
inkievoyd
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3Just to be clear: "absurd" results in the quadratic formula always indicate non-real solutions. On the other hand, Cardano's formula for solving the cubic sometimes involves a root of a negative, but if we pretend that $i$ exists and can be manipulated normally, the simplified form of the formula is real (imaginary parts in different terms cancel). – jxnh Nov 09 '14 at 00:19