An imaginary number is a component of a complex number (satisfying i2 = - 1). In MRI, complex data are used for example in the Fourier transforms.

Out of phase component of the signal from a quadrature detector.

An imaginary number is a part of a complex number. Complex numbers are an extension of the real numbers. A complex number has a real and an imaginary part. The imaginary unit (i) is equal to the square root of -1. The complex conjugate is a pair of complex numbers with identical real parts and imaginary parts which differ only in sign (e.g.: 3 + 7i and 3 - 7i).

The coordinate system most frequently used to quantitatively describe a n-dimensional space.
In 2 dimensions, i.e. a plane, it describes any point as a function of 2 perpendicular unit vectors (1,0) and (0,1) and in 3 dimensions as a function of 3 perpendicular unit vectors (1,0,0), (0,1,0) and (0,0,1). Functions in 2 dimensions are often conveniently described using the so-called theory of functions. When using this type of mathematical description, the imaginary number
i = √(-1) is introduced to label the y-axis.
a + ib is then actually a 2 dimensional vector with a x-axis component of 'a' and a y-axis component of 'b'.
The 'a' is called the real part and the 'b' the imaginary part of the function, an expression that is frequently encountered in MRI, where the real image is a pixel-wise representation of 'a' and the imaginary image a pixel-wise representation of 'b', with 'a' and 'b' the components of the xy-magnetization along the x- and y-axis, respectively.
(Renatus Cartesius/Rene Descartes, 1596-1650, French philosopher and mathematician)

The symmetry in k-space is a fundamental property of Fourier transformations. For a two-dimensional example, let g(x,y) be a complex function, i.e. the value of g at any (x,y) is a complex number. If nothing is known about the function g, data throughout all of k-space is needed to fully characterize it.
If the function g is 'real', meaning that at every (x,y) the imaginary component of g(x,y) is zero, then you only need half as much data to characterize g. The result is redundancy between the data on one half of k-space and the other. Specifically, if G(kx,ky) is the Fourier transformation of g(x,y), and g(x,y) is real, then G(kx,ky)=G*(- kx,- ky), where * indicates a complex conjugate. The data in mirrored positions in k-space, i.e. (kx,ky) versus (- kx,- ky), are conjugates of each other.

See Imaginary Numbers and Complex Conjugate.