## + 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.
Complex_conjugation

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

### Common uses

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

### Common uses

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```