+ y (Conjugate)
The complex conjugate (+z) of the numeric quantity: z
If z is real, then + behaves like Same (]) -- the identity function.
+ 7 0 _7 7 0 _7
Complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs.
J supports complex numbers and returns them as required. The way to write the scalar numeral with real part 3 and imaginary part 4i is: 3j4.
sqrt=: 3 : 'y^0.5' NB. (sqrt y) is y to the power of 0.5 sqrt=: ^&0.5 NB. (tacit alternative) sqrt 49 7 sqrt _1 0j1 + sqrt _1 0j_1 |sqrt _1 NB. The sq root of _1 has magnitude 1 1 | z=: 3j4 NB. vector repn of z is 3-4-5 triangle 5 NB. hence its magnitude is the hypotenuse + z 3j_4 | +z NB. Conjugate of z has same magnitude as z 5
Animation Illustrating Conjugate (+)
Current configuration does not allow embedding of the file http://www.youtube.com/v/8CWEDGY3sSM because of its mimetype application/x-shockwave-flash.: http://www.youtube.com/v/8CWEDGY3sSM
1. To test z is real not complex:
if. z=+z do. ... end.
2. Wherever the complex conjugate of a number is required.
Thus, to find the real part of z, you might think to use
z=: 3j4 -: z+ +z NB. (-:) is: Halve 3
though you'd normally separate a complex number into real and imaginary parts using Real/Imaginary (+.).
x + y (Plus)
The sum (x+y) of two numeric nouns: x and y .
Either or both of x, y can be scalar.
x + y [ x=: 5 [ y=: 2 3 4 7 8 9 x + y [ y=: 5 [ x=: 2 3 4 7 8 9
If both x and y are vectors, they must have the same number of entries (along the leading axis/axes) or length error results.
x + y [ x=: 100 200 [ y=: 2 3$i.6 100 101 102 203 204 205 x + y [ x=: 100 200 300 [ y=: 2 3$i.6 |length error | x +y[x=:100 200 300[y=:2 3$i.6
Note however the use of Rank (") to match-up x and y along axis 1 instead of (default) axis 0 --
x +"1 y 100 201 302 103 204 305
Animation Illustrating Plus (+)
Current configuration does not allow embedding of the file http://www.youtube.com/v/PIfB8aJ94vU because of its mimetype application/x-shockwave-flash.: http://www.youtube.com/v/PIfB8aJ94vU
To sum two numeric quantities
100 + 0 1 2 100 101 102
To sum the entries of a numeric vector
+/0 1 2 3 6
See: ../plus/Animations for additional trial animations to represent Conjugate and Plus.