Imaginary Numbers question

What exactly is 2i, 3i,4i etc. and how can I understand what that actually means? How can I picture 2i so that it makes sense to a person who must see concrete things to make any meaning of them? I am in the process of reading a book called Quantum Mind and it is drawing analogies about mathematics and the mind and at the point I am in the book it is discussing imaginary numbers but I am not able to comprehend or picture these numbers. Any help would be appreciated.

i is defined as the square root of -1. (2i)^2 would be -4. Obviously, imaginary (or complex) numbers are called so because the square root of a negative number cannot exist in our number system.

No help with visualizing them though… sorry.

See complex numbers as couples of numbers where there is
a real part (re) and an imaginary part (im). so 5 + 3i is (5,3) and 2i is (0,2). (0, 1) * (0, 1) = -1 Couples of numbers can be plotted in the x and y axis. The plane is called the complex plane and the diagram is called an argand diagram.

I see that Zweistein has already mentioned “argand diagram”. Look here for a picture:

It is very useful to see multiplication of a complex number by i as rotating that complex number by 90º in the plane, so:

(a + bi) x i = -b + ai

Think about how that looks on the plane.
Then consider what happens if you muliply by i[sup]2[/sup]… we rotate it twice by 90º. Of course, we also know that by i[sup]2[/sup] = -1, and we can confirm that multiplication by -1 rotates a number by 180º in the plane.

Multiplication by i[sup]3[/sup] rotates a point 270º
Multiplication by i[sup]4[/sup] rotates a point 360º
See if you can follow this logic through and ask, what is the square-root of i in the complex plane? (Hint, square-root of i == i[sup]½[/sup])

If you’re used to picturing real numbers as points on the real number line (with positive numbers to the right of 0 and negative numbers to the left), then you can imagine imaginary numbers as living above (or below) the real number line. (i is one unit above 0, 2i is 2 units above 0, -2i is 2 units below 0, etc.) And then, as Zweistein has already said, you can picture complex numbers (with a real and an imaginary part) as corresponding to points (and/or vectors) in the 2-dimensional complex plane.

insider, I presume that the book you’re reading is this one:

Have you considered the possibility that this book is just a bunch of crank theories? If so, there’s no reason that learning about imaginary numbers will tell you anything interesting about “quantum mind,” whatever that term means. I suspect that the author of the book doesn’t really know much about imaginary numbers and anything you learn about them won’t be any help in understanding the book.

What you have to understand is that imaginary numbers, like all mathematical concepts, are entirely abstract. Concepts like addition are easy to relate to reality, by imagining two things being combined. Imaginary numbers don’t directly relate to anything like that so it’s not useful or necessary to try. Just remember that i[sup]2[/sup]=-1 and remember that there a few basic rules for operating on complex numbers, which are often the same as those for real numbers (e.g. 2i+3i=5i). Don’t try to convince yourself that the number i actually exists. True, it doesn’t, but technically, neither does the number 1. Both are just abstract concepts, which are sometimes useful for solving problems. Hope this helps.

I would love hear Homer Simpson’s take on "imaginary’ numbers:)

One way to look at imaginary numbers is to think of them as an operator, that is, they are tools to perform a specific task.

Think of + as being one task - as another and i as yet another.

When looked upon this way on an Argand diagram, powers of i can be used to rotate a vector, or when drawn using differant axis they can be used to represent relative displacements in time of regularly varying quantities.

i is the solution to x[sup]2[/sup] = -1 and (so is -i).

Imaginary numbers are just as ‘real’ as real numbers, many physical problems require the use of imaginary numbers (or complex numbers), It’s just that we don’t genrally experince them in everyday life.

Both real numbers and imaginary numbers are subsets of the complex field.

I think you’re attributing an unwarranted significance to the “complex field”. It’s just 2-space. Complex numbers are simply coordinate pairs. Complex arithmetic is simply a useful way of dealing with equations in 2-space.

The Heaviside/Maxwell/Gibb scalar/vector/tensor formulation is a way of extending the arithmetic into n-space.

In 2-space in complex notation, i denotes a direction orthogonal to the default direction. In the Heaviside notation, there is no default direction. Each one is denoted by a symbol i, j, k, etc.

The complex field is just the set of all real, imaginary and other complex numbers which also happens to form a field.

Complex quantites also turn up in physics for example in wave equations. In classical wave equations only Re(z) is given physical signifcance, in quantum mechanical wave equations the whole of z has a ‘physical’ signifance as it is the probailty amplitude.

What I should say is thta the complex field is just a more genralized system of numbers than the real numbers.

The complex field is the smallest algebraically complete field containing the reals, and may be the smallest algebraically complete field (in the sense that every algebraically complete field contains a field isomorphic to C). A field is algebraically complete iff it contains the roots to every polynomial with coefficients from that field.