Eng:How to Use Complex Numbers in Scratch

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This tutorial shows how to calculate complex numbers in Scratch. A complex number is a number with a real part and an "imaginary" part, which is equal to the square root of -1.

Representing a complex number on a plane

Preparing

Because complex numbers have both a real and an imaginary part, six variables are needed - two for each input and two more for the answer.

(a :: variables) // The 1st real part
(b :: variables) // The 1st imaginary part
(c :: variables) // The 1st real part
(d :: variables) // The 2nd imaginary part
(e::variables) // The real part of the answer
(f::variables) // The imaginary part of the answer

Addition

$(a+bi)+(c+di)=(a+c)+(b+d)i$

set [e v] to ((a) + (c))
set [f v] to ((b) + (d))

Subtraction

$(a+bi)+(c+di)=(a+c)+(b+d)i$

set [e v] to ((a) - (c))
set [f v] to ((b) - (d))

Multiplication

$(a+bi)(c+di)=ac+adi+bci-bd$

set [e v] to (((a) * (c)) - ((b) * (d)))
set [f v] to (((a) * (d)) + ((b) * (c)))

Division

${\frac {a+bi}{c+di}}={\frac {ac-adi+bci+bd}{c^{2}+d^{2}}}$

set [e v] to ((((a) * (c)) + ((b) * (d))) / (((c) * (c)) + ((d) * (d))))
set [f v] to ((((b) * (d)) - ((a) * (c))) / (((c) * (c)) + ((d) * (d))))

Exponents

Square

$(a+bi)^{2}=a^{2}+2abi-b^{2}$

set [e v] to (((a) * (a)) - ((b) * (b)))
set [f v] to ((2) * ((a) * (b)))

Cube

$(a+bi)^{3}=a^{3}+3a^{2}bi-3ab^{2}-b^{3}i$

set [e v] to (((a) * ((a) * (a))) - ((3) * ((a) * ((b) * (b)))))
set [f v] to (((3) * ((a) * ((a) * (b)))) - ((b) * ((b) * (b))))

Inverse

$(a+bi)^{-1}={\frac {a-bi}{a^{2}+b^{2}}}$

set [e v] to ((a) / (((a) * (a)) + ((b) * (b)))
set [f v] to ((0) - ((b) / (((a) * (a)) + ((b) * (b)))

Absolute value

The absolute value of a complex number measures its distance from 0 using the Pythagorean theorem. $|a+bi|={\sqrt {(a^{2}+b^{2})}}$
([sqrt v] of (((a) * (a)) + ((b) * (b))))

Displaying values

The scripts listed in this tutorial all store the answer as two separate values. To display the answer to the user, the following script may be used:

if <(b) \< (0)> then
    set [answer v] to (join(e::variables)(join[ - ](join((0) - (f))[i])
else
    set [answer v] to (join(e::variables)(join[ + ](join(f)[i])

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