What do they mean by imaginary numbers?

Question

i saw it on a book on differrent theorys about the universe

Answer 1

Imaginary numbers are a very useful mathematical idea, and they do have a connection with reality, despite what everyone else seems to think.

Consider an ordinary number line. If we are standing on 0, we see -1, -2, -3... in one direction, and +1, +2, +3... in the other direction.

If we multiply by -1, it simply reverses our direction. If we were pointing in the positive direction, multiplying by -1 points us in the negative direction (180 degrees from our initial orientation).

If we multiply by -1 twice, we change the direction we face twice, doing a 360-degree turn and ending up back where we started. Hence (-1)^2 = 1.

But consider a number that will only take us 180 degrees around when we multiply by it TWICE. For instance, if we are facing in the positive direction, and we multiply by that number twice, we will end up pointing in the negative direction. That number is "i", which equals (-1)^(1/2), or the square root of -1.

So what is i? It's simply a 90-degree rotation. Multiplying by i sets you perpendicular to your original orientation. Multiplying by i twice turns you 180 degrees. Hence i^2 = sqrt(-1)^2 = -1. Finally, multiplying by i four times leaves you back where you started. Hence i^4 = (i^2)^2 = (-1)^2 = 1.

Answer 2

An imaginary number is the square root of a negative number.

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In real numbers, there are roots that cannot be solved (or quadratic that cannot be solved, which is a different way of saying the same thing).

For example (x^2 + 1) cannot be factored and x^2 + 1 = 0 cannot be solved. This is the same problem stated two different ways.

So in come complex numbers.

In complex numbers, we add a "number" normally called i (sometimes j if i is already used for something else, like in electrical engineering).

i is a number such that i^2 = -1.

In popular speech, we say that i is the square root of minus one; in maths, we still don't like to admit that negative numbers can have square roots, so we say it as above.
"i is a number such that i squared equals minus one"

So, if we allow x = i, then x^2 + 1 becomes
i^2 + 1 = -1 + 1 = 0
and we have solved the quadratic x^2 + 1 = 0.

A complex number (usually called z instead of x) is written in the form
z = a + bi
where a and b are real numbers.

The "a" part is the "real" part of the complex number and the "bi" part is called the imaginary part.

The number 1 + i is a complex number where a = 1 and b = 1.
If we square it (1 + i)^2 = (1+i)(1+i) = 1 + i + i + i^2, we get:
1 + 2 i + i^2 = 1 + 2i - 1 = 2i.
If we square it again:
(i + i)^4 = [(1+i)^2][(1+i)^2] = [2i][2i] = 4 i^2 = -4.

So (1+i) is the fourth root of -4

(1+i) is a complex number. When we squared it we got a purely imaginary number 2i (no real part -- in other words, z = 0 + 2i, where a = 0) and when we squared it again, we got a real number -4 (z = -4 + 0i, where b = 0)

Answer 3

The concept of imaginary numbers was invented to answer the question - What is the square root of a negative number. Since the square of every number, whether positive or negative is a positive number, there are nor real numbers which are the square root of a negative number. so, by definition, the imaginary number "i" is the square root of -1. In other words, i^2 = -1.

Answer 4

It is basicly casued by taking the square root of a negitive number
Sqrt(-1)=i
Sqrt(-25)=5i

Answer 5

Complex numbers. They come in the role of a +bi

where a is a real number and b is a constant and i is an imagery number.

The most basic application is that i = Sqrt of negative one.

Answer 6

Numbers such as the square root of negative two...try calculating that.

Answer 7

Now, you're sure that this book is real? And not imaginary? And exactly who are "they"?

Answer 8

they are just square roots of negative numbers. there really is no such thing...but are necessary for completion of upper level math equations.

Answer 9

i have no idea