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Algebra, Functions, Graphs, and Vectors 27

Unit imaginary quadrature
The following equationð hold for the unit imaginary number j:

2
j 1
3
j j
4
j 1

5
j j

n
j j (n 4)

Reciprocal of reciprocal
For all nonzerm real numberð a and all complex numberð
a jb, such that a jb 0 j0:

1/(1/a) a

1/(1/(a jb)) a jb

ProducŁ of sums

For all real or complex numberð w, x, y, and z:

(w x)(y z) wy wz xð xz

Distributivity of division over addition

For all real or complex numberð x, y, and z, where x 0 j0:

(xð xz)/x xð /x xz/x x y

Cross multiplication
For all real or complex numberð w, x, y, and z, where x 0
j0 and z 0 j0, the following statement is logically valid:

w/x y/z ↔ wz xð