Vocabulary/jdot
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j. y Imaginary
Rank 0 -- operates on individual atoms of y, producing a result of the same shape -- WHY IS THIS IMPORTANT?
Multiplies y by 0j1, which represents the imaginary unit i .
Equivalent to 0j1 * y
j. 5 NB. y real, resulting in the pure imaginary number 0+5i 0j5 5 * 0j1 0j5 j. 3j4 NB. y complex, result is the product (0+1i)*(3+4i) = 3i+4i^2 = -4+3i _4j3 j.(^:4) 3j4 NB. each step rotates the vector by π/2 3j4
x j. y Complex
Rank 0 0 -- operates on individual atoms of x and y, producing a result of the same shape -- WHY IS THIS IMPORTANT?
Combines x and y into a complex number having x as its real part and y as its imaginary part
Equivalent to x + 0j1 * y
3 j. 4 NB. arguments are separated from verb by spaces (x j. y) 3j4 ^ ^ 3j.4 NB. this is a different number, complex constant (3+0.4*i) 3j0.4 1 j. -:%:2 NB. composing complex number (1+sqrt(2)/2*i) 1j0.707107
Common uses
1. Work with complex arithmetic.
2. Some primitives use a complex argument as a way of putting two numbers into one atom
NB. (":) Format
o.>:i.3 NB. first three multiples of π
3.14159 6.28319 9.42478
(o.>:i.3) ,: (*: o.>:i.3) NB. multiples and their squares
3.14159 6.28319 9.42478
9.8696 39.4784 88.8264
13j9 ": (o.>:i.3) ,: (*: o.>:i.3) NB. x is (w j. d), producing formatted output
3.141592654 6.283185307 9.424777961 NB. with d decimal digits and a field width of w
9.869604401 39.478417604 88.826439610
NB. (#) Copy
2j1 # 2 3 5 7 NB. x is (n j. f), specifying n copies followed by f fills
2 2 0 3 3 0 5 5 0 7 7 0
0 2j1 1 3 # 2 3 5 7
3 3 0 5 7 7 7
NB. (i:) Steps
i: 2j3 NB. y is (a j. n), calling for n steps in intervall [-a,a]
_2 _0.666667 0.666667 2
3. Handle x/y screen coordinates as complex numbers instead of pairs of reals.
j4xy=: _2 j./\ ] NB. a list of xy-coords --> a list of complex nos xy4j=: [: , +. NB. a list of complex nos --> a list of xy-coords j4xy 3 4 _2 3 3j4 _2j3 xy4j 3j4 _2j3 3 4 _2 3
Related primitives
Real/Imag (+. y), Signum (Unit Circle) (* y), Length/Angle (*. y), Magnitude (| y), Circle Functions (x o. y), Angle * Polar (r.)