Imaginary numbers are numbers that exist out of the scope of real numbers.
√(-1) does not exist, but this entire field assumes that this has an associated constant that is called i.
Edit (by Haris):
sqrt(-1) doesn't exist within the real numbers.
If you imagine the number as a 2D plane with the Real Numbers on the horizontal (x) axis, then the imaginary numbers would be on the vertical (y) axis.
Imaginary is really a poor term that comes from the poor understanding of the numbers from early mathematicians. They are just as real as the "Real Numbers", but they occupy a different space than the Real Numbers.
From a purely mathematical point of view, there are two sets of numbers.
One set, dear to us in our daily lives... Real .
the other ... Irreal.
Both have identical subsets within their own respective sets.
One set... Rational.
The other ... Irrational.
The rational set contains booth.. integers and fractions.
The Irrational set contains numbers that can not be represented by fractions.
A fraction can be represented by a quotient with finite numbers of digits in its numerator and denominator.
More could be explained about these and other sets...
Imaginary numbers are those that exist such that when you multiply two of these they result in a negative real number as their product...
i3 x i12 = -36
-12/i3 = i4
1 + i1 = i1 + 1
(1 + i1) * (1 + i1)
= (1 + i1) * 1
+ (1 + i1) * i1
= (1+ i) + (i1-1 )
And these numbers are not that Irreal or Imaginary.
Engineers take them in consideration in the construction of bridges; so in electronics..
Consider the polynomial function p(x) = x² + 1. If you drew it on paper it is a parabola sitting on 1 which open upwards. It doesn't intersect with the x-axis - there are no real roots. But we can ask: Is there a field in which p(x) splits into linear factors? The answer is positive. In fact, Kronecker could show some 130 years ago that if K is a field and P(x) a non-constant polynomial with coefficients in K, then there is a field extension L which contains K such that P(x) has a zero in L. This is also known as "Fundamental Theorem of Field Theory".
So far it has been stated that there is a number "i" such that i² = -1, but no proof of that fact was given. Fortunate for us as programmers, the proof can be constructive - which will give us and idea of how to implement complex numbers in programming (if the type is not naturally supported as e.g. in C99/C++ or Fortran).
In your last question we have seen how fractions are constructed from pairs of integers. Again, we shall look at pairs of real numbers. Naturally, addition is as we would expect it, componentwise: (a, b) + (c, d) = (a+c, b+d)
To turn it into a field, we need to furnish those pairs with a multiplication which is invertible for all but (0, 0). We can do that as follows: (a, b) * (c, d) = (ac - bd, ad + bc). This multiplication has to satisfy a number of field axioms, it has to be associative, commutative and distributive. All of which to show is easy and just a bunch of algebra which I leave out. I want to emphasise some points though:
1) (1,0) * (a,b) = (a,b)
so (1, 0) is the multiplicative unit
2) (a,0) * (b,0) = (ab,0)
multiplication acts on the subset of pairs with second entry 0 just like multiplication of real numbers.
3) (0,1) * (0,1) = (-1,0) = -(1,0)
(0,1) satisfies x²+1=0.
In light of (2), the first component is called "real part", the second component "imaginary part". Traditionally, (0,1) is written "i", or "j" in electrical engineering where "i" already means "electric current".
The field constructed (and thus it exists) is called the field of complex numbers C, and it extends the field of real numbers since the real numbers are canonically embedded through x -> (x,0). It is constructed as adjuction of the root "i" of x²+1 to the set of reals: denoted R[i].
As pairs of real numbers it is a vector space of dimension 2 over the reals. As such, any complex number can be expanded in terms of the base vectors (1,0) and (0,1). Since (1,0) is the real 1 and (0,1) is called "i", we usually do not write the base vectors as pairs but expand as (a,b) -> a + bi
Numbers with real part zero are called "imaginary numbers".
The constructive proof shows how we can implement complex numbers in programming: as structures with two real values, "real" and "imag" where the addition and multiplication is just as defined above.
Complex numbers certainly play an important role in mathematics, but especially in physics and engineering. For programming, digital signal processing, data encoding make use of complex numbers. More areas may likely apply.
As interesting a topic the topic of field extensions is, the complex numbers are already algebraically closed. Since the dimension of the vector space [L:K] is the degree of the irreducible polynomial whose roots are adjoined, we immediately see that any polynomial over the reals splits into factors of degree at most 2. As we have seen from the field of fractions, as pairs of integers, that set Q is countable. The real numbers are not countable - so we can expect many field extensions to reside between Q and R. Most important for programming are field extensions over the finite field of binary numbers, GF(2).
The theory of field extensions intertwines with group theory, as there is a Galois connection between the subfields of a field extension and the subgroups of the K-invariant automorphism group of the extension [L:K]. A fascinating theory that touches various fields of algebra. But I am rambling ...
Okay, it seems my last answer didn't convince anyone (I thought it was a great answer, it not only constructed the complex numbers, it showed how to implement them including prospects of uses in mathematics and programming).
I have another (constructive) one. It is a little more sophisticated and does not immidiately lend itself to implementation in programming, until we realise that they are described by pairs of real numbers.
In the same motivation as above, we are looking for a field extending the real numbers R such that the polynomial p(x)=x²+1 has a root.
Let R[X] denote the polynomial ring over the reals. As we have seen, p(X) is irreducible in that ring. Since R[X] is a unique factorisation domain, p(X) is prime. And since R[X] is a principal ideal ring, the prime ideal I=(X²+1), generated by p(X), then is maximal. It is a known result from undergraduate algebra that the factor ring R[X]/I then is a field.
The claim is now that the residue class Y of X modulo I satisfies Y²+1 congruent 0 modulo I. We see that by squaring Y = X + I:
(X + I)²
= X² + I
= X² - (X²+1) + I
= -1 + I
The residue class Y is a root of p(X)=X²+1 in the field R[X]/I. It is usually denoted "i" and called "the imaginary unit". In the end we are dealing with polynomials with degree at most 1 since, as we have seen Y² is in the same class as -1. As such, the field of complex numbers, C, is isomorphic to R² as vector space. So any complex number can be written as pair (a, b) = a + bi. Complex numbers with real component identical zero are called "imaginary numbers".
real number = number * cos (0) = number
imaginary = number * cos (90) = 0
means if a number is real its not sean by the imaginary axis and vic virsa.
A practical number in coding or real life, has to represent something that is build from various functions or values. if thies valuse are independent from each other you can describe or code it as each one to be imaginary from the other values; where the effect never cross between each other, in math you assign them perpendicular to each other at 90° ~ where you can measure the other value effect on the value that is being calculated by cos (angle deference)