Proving i=sqr(-1): Homework Equations & Solution

  • Thread starter Tusike
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In summary: So, our homework statement is that to prove that the complex numbers will only work so that they are an extension to real numbers if i=sqr(-1), we need to show that i^2= -1.
  • #1
Tusike
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Homework Statement


Our teacher said we can try to prove that the complex numbers will only work so that they are an extension to real numbers if i=sqr(-1).


Homework Equations


We know that |i|=1, and that for any complex numbers |a||b|=|ab|, and of course that the complex numbers are commutative, associative, and distributive.


The Attempt at a Solution



I didn't have much ideas of how to prove this, I figured I need to try proving i^2=-1, so denoting u for the real number 1 unit, I wrote |u+i||u-i|=|u^2-i^2|, knowing that |u|=|i|=1 we can write sqr(2)*sqr(2)=|u^2-i^2|, and so 2=|1-i^2|.

And from here, since |i|=1 and so |i*i|=1, the only way this can be is if i^2 = -1 and so i=sqr(-1). Is this an entirely precise and complete proof or did I miss something?
 
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  • #2
Just putting this out there, [itex]i^2 = 1[/itex] doesn't necessarily mean [itex]i = \sqrt{-1}[/itex]. It could be [itex]-\sqrt{-1}[/itex]...
 
  • #3
i = (0, 1)

To multiply complex numbers: (a, b)*(a', b')= (aa'-bb', ab'+a'b)

keep in mind (a, b) = a+bi where a and b are real numbers.

Can you now see why [itex] i^2= -1 [/itex] ?

This is also why one should take care to understand what the definition of i is. It can be... over-simplifying, to say [itex] i= \sqrt{-1} [/itex]
 
  • #4
@ArcanaNoir: in that multiplication rule you already used that i^2=-1.

Okay so then how do you prove whether i is + or -sqr(-1)?
 
  • #5
I didn't "use" i^2= -1

I showed how you can show it by computing (0,1)*(0,1)

As for i, i= (0,1), it's no more possible to show whether you use (0,1)*(0,1) or -(0,1)*-(0,1) than it is in real numbers.
 
  • #6
Defining "i" as the number whose square is -1 is ambiguous because there are two such numbers. one of them is i and the other is -i. We cannot say "the positive root" as we can in the real numbers because the complex numbers are not an "ordered" field. That is why, as ArcanaNoir says, it is better to define complex numbers as pairs of numbers with the defined multiplication.

That way, we define i to be the pair (0, 1), not (0, -1), but, other than that, there is no way to distinguish between "i" and "-i".
 
  • #7
Oh, I see it now. Thanks!
 

FAQ: Proving i=sqr(-1): Homework Equations & Solution

What is the significance of proving i=sqrt(-1)?

The number i, also known as the imaginary unit, is a crucial concept in mathematics and science. It allows us to work with and solve complex numbers, which have important applications in fields such as engineering, physics, and economics. Proving that i=sqrt(-1) helps solidify the foundations of complex number theory and allows us to confidently use it in various calculations and equations.

What are the steps to proving i=sqrt(-1)?

The proof of i=sqrt(-1) involves using the fundamental properties of complex numbers and basic algebraic manipulations. First, we define the imaginary unit i as the square root of -1. Then, we can use this definition to show that i squared is equal to -1. Finally, we can verify that taking the square root of -1 indeed gives us i. The detailed steps can be found in most complex analysis textbooks or online resources.

3. Can you explain the intuition behind the proof of i=sqrt(-1)?

The proof of i=sqrt(-1) is based on the idea of extending the real number system to include imaginary numbers. This means that we introduce a new number, i, that behaves like the square root of -1. By manipulating the properties of complex numbers, we can show that this new number satisfies the definition of the imaginary unit and is equal to sqrt(-1).

4. What are some real-world applications of the proof of i=sqrt(-1)?

The proof of i=sqrt(-1) has many real-world applications, particularly in fields that involve complex numbers such as electrical engineering, signal processing, and quantum mechanics. For example, engineers use complex numbers to analyze and design circuits, while physicists use them to describe the wave-like behavior of particles. The proof also allows us to use complex numbers in calculations and equations with confidence, knowing that the foundations of the theory are solid.

5. Can you provide a simple example of using i=sqrt(-1) in a calculation?

Sure, let's say we want to solve for x in the equation x^2+4=0. We can rewrite this equation as x^2=-4, and since the square root of -1 is i, we can take the square root of both sides to get x=i*sqrt(4). Simplifying further, we get x=2i or x=-2i. This is just one example, but there are many other ways in which the proof of i=sqrt(-1) allows us to manipulate and solve equations involving complex numbers.

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