In a treatise with a single point of view are considered basic concepts of both discrete approximations and transforms, - general, such as polynomial, difference, Fourier, Chebyshev, and more special, such as orthogonal and biorthogonal discrete cosine transforms (DCT), Haar's, Walsh's and Karhunen Loeve. There besides are considered certain basic approaches to obtaining the transforms with needed properties - algebraic, geometrical and probabilistic. Also a general condition for equivalence of discrete approximations had been established and some relationships describing an effect of quantization on errors of a discrete approximation or transform obtained and analyzed in application to orthogonal transforms. In addition there considered some basic matrix algorithms and properties of biorthogonal matrices.