Complex numbers

Our understanding is that the square of a number can never be negative. In that case, how do we solve x2 = -9? Sensibly, in math we have the concept of i, as a solution, that is, x = 3i. Numbers such as i, -i, 3i, and 2.27i are called imaginary numbers. "A real number" + "an imaginary number" forms a "complex number".

Complex number = (real part) + (imaginary part) I

The following examples show complex number representation using the Breeze library for Mathematics:

import breeze.linalg.DenseVector 
import breeze.math.Complex 
val i = Complex.i 

// add 
println((1 + 2 * i) + (2 + 3 * i)) 

// sub 
println((1 + 2 * i) - (2 + 3 * i)) 

// divide 
println((5 + 10 * i) / (3 - 4 * i)) 

// mul 
println((1 + 2 * i) * (-3 + 6 * i)) 
println((1 + 5 * i) * (-3 + 2 * i)) 

// neg 
println(-(1 + 2 * i)) 

// sum of complex numbers 
val x = List((5 + 7 * i), (1 + 3 * i), (13 + 17 * i)) 
println(x.sum) 
// product of complex numbers 
val x1 = List((5 + 7 * i), (1 + 3 * i), (13 + 17 * i)) 
println(x1.product) 
// sort list of complex numbers 
val x2 = List((5 + 7 * i), (1 + 3 * i), (13 + 17 * i)) 
println(x2.sorted) 

This preceding code gives us the following result:

3.0 + 5.0i
-1.0 + -1.0i
-1.0 + 2.0i
-15.0 + 0.0i
-13.0 + -13.0i
-1.0 + -2.0i
19.0 + 27.0i
-582.0 + 14.0i
List(1.0 + 3.0i, 5.0 + 7.0i, 13.0 + 17.0i)