Complex Numbers; Multiplication 01 (Rect')

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Michael Proudman 4 years, 10 months ago

Michael Proudman 4 years, 10 months ago

Let $z$ and $w$ be any two complex numbers

$z=a+bi$ and $w=c+di$

Then

$z w = (a + bi)(c + di)$

$z w = ac-bd+(ad+bc)i$

If:

Evaluate:

If:

Evaluate:

If:

Evaluate:

If:

Evaluate:

If:

Evaluate:

We are asked to evaluate the following multiplications of complex numbers.

Given:

Evaluate:

Given:

Evaluate:

Given:

Evaluate:

Given:

Evaluate:

Given:

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Checkpoint description

Michael Proudman 4 years, 10 months ago

Michael Proudman 4 years, 10 months ago

Let zz and ww be any two complex numbers

z=a+bi z=a+bi and w=c+di w=c+di

Then

zw=(a+bi)(c+di)z w = (a + bi)(c + di)

zw=ac−bd+(ad+bc)iz w = ac-bd+(ad+bc)i

Error in variable testing condition

Error

Warning

If:

Evaluate:

If:

Evaluate:

If:

Evaluate:

If:

Evaluate:

If:

Evaluate:

We are asked to evaluate the following multiplications of complex numbers.

Given:

Evaluate:

Given:

Evaluate:

Given:

Evaluate:

Given:

Evaluate:

Given:

Let $z$ and $w$ be any two complex numbers

$z=a+bi$ and $w=c+di$

$z w = (a + bi)(c + di)$

$z w = ac-bd+(ad+bc)i$

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Michael Proudman

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Published this.

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Created this as a copy of Complex Numbers; Addition & Subtraction 01.

Name	Status	Author	Last Modified
Complex Numbers; Basics 02	Ready to use	Michael Proudman	08/02/2022 12:44
Complex Numbers; Addition & Subtraction 01	draft	Michael Proudman	11/01/2023 16:23
Complex Numbers; Multiplication 01 (Rect')	draft	Michael Proudman	07/12/2021 10:31

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Question statement

Give any introductory information the student needs.

			Name	Type	Generated Value

			Name	Type	Generated Value

			Name	Type	Generated Value

			Name	Type	Generated Value

			Name	Type	Generated Value

r5	integer	9
i5	integer	-3
z5		Error evaluating variable z5: Function `i` is not defined. Is `i` a variable, and did you mean `i*(...)`?
cc5		Error evaluating variable cc5: Variable `z5` is not defined.

			Name	Type	Generated Value

Definition of variable i is empty.

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Value

xxxxxxxxxx
 
random(2..25)

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Advice

Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

$\large z_1=\var{z1}$ and $\large z_2=\var{z2}$

$\large z_1z_2=$ Gap 0.

$\large z_3=\var{z3}$ and $\large z_4=\var{z4}$

$\large z_3z_4=$ Gap 1.

$\large z_5=\var{z5}$ , $\large z_6=\var{cc1}$

$\large z_5z_6=$ Gap 2.

$\large z_1=\var{z1}$ and $\large z_1^*=\var{cc1}$

$\large z_1z_1^*=$ Gap 3.

$\large z_2=\var{z2}$ and $\large z_2^*=\var{cc2}$

$\large z_1z_1^*=$ Gap 4.

Advice

The technique should be very familiar. It is exactly the same as multiplying out two brackets.

The workings below probably include more steps than really neccesary once you get used to doing it.

$\large z_1=\var{z1}$ and $\large z_2=\var{z2}$

$\large z_1z_2=(\var{z1})(\var{z2})$

$\large z_1z_2=(\var{r1})(\var{r2})+(\var{r1})(\var{i2})i+(\var{i1})(\var{r2})i+(\var{i1})(\var{i2})i^2$

$\large z_1z_2=\simplify{{r1}{r2}}+\simplify{{r1}{i2}}i+\simplify{{i1}{r2}}i+\simplify{{i1}{i2}}i^2$

$\large z_1z_2=\simplify{{r1}{r2}}+\simplify{{r1}{i2}+{i1}{r2}}i+\simplify{{i1}{i2}}i^2$

Now here is the extra "twist" that comes from doing this with complex numbers. Remember that $i=\sqrt{-1}$ so $i^2=-1$ :

$\large z_1z_2=\simplify{{r1}{r2}}+\simplify{{r1}{i2}+{i1}{r2}}i+\simplify{{i1}{i2}i^2}$

$\large z_1z_2=\simplify{{z1}*{z2}}$

$\large z_3=\var{z3}$ and $\large z_4=\var{z4}$

$\large z_3z_4=(\var{z3})(\var{z4})$

$\large z_3z_4=(\var{r3})(\var{r4})+(\var{r3})(\var{i4})i+(\var{i3})(\var{r4})i+(\var{i3})(\var{i4})i^2$

$\large z_3z_4=\simplify{{r3}{r4}}+\simplify{{r3}{i4}}i+\simplify{{i3}{r4}}i+\simplify{{i3}{i4}}i^2$

$\large z_3z_4=\simplify{{r3}{r4}}+\simplify{{r3}{i4}+{i3}{r4}}i+\simplify{{i3}{i4}}i^2$

Remember that $i=\sqrt{-1}$ so $i^2=-1$ :

$\large z_3z_4=\simplify{{r3}{r4}}+\simplify{{r3}{i4}+{i3}{r4}}i+\simplify{{i3}{i4}i^2}$

$\large z_1z_2=\simplify{{z1}*{z2}}$

$\large z_5=\var{z5}$ and $\large z_6=\var{cc1}$

$\large z_5z_6=(\var{z5})(\var{cc1})$

$\large z_5z_6=(\var{r5})(\var{rcc1})+(\var{r5})(\var{icc1})i+(\var{i5})(\var{rcc1})i+(\var{i5})(\var{icc1})i^2$

$\large z_5z_6=\simplify{{r5}{rcc1}}+\simplify{{r5}{icc1}}i+\simplify{{i5}{rcc1}}i+\simplify{{i5}{icc1}}i^2$

$\large z_5z_6=\simplify{{r5}{rcc1}}+\simplify{{r5}{icc1}+{i5}{rcc1}}i+\simplify{{i5}{icc1}}i^2$

Remember that $i=\sqrt{-1}$ so $i^2=-1$ :

$\large z_5z_6=\simplify{{r5}{rcc1}}+\simplify{{r5}{icc1}+{i5}{rcc1}}i+\simplify{{i5}{icc1}i^2}$

$\large z_5z_6=\simplify{{z5}*{cc1}}$

$\large z_1=\var{z1}$ and $\large z_1^*=\var{cc1}$

$\large z_1z_1^*=(\var{z1})(\var{cc1})$

$\large z_1z_1^*=(\var{r1})(\var{rcc1})+(\var{r1})(\var{icc1})i+(\var{i1})(\var{rcc1})i+(\var{i1})(\var{icc1})i^2$

$\large z_1z_1^*=\simplify{{r1}{rcc1}}+\simplify{{r1}{icc1}}i+\simplify{{i1}{rcc1}}i+\simplify{{i1}{icc1}}i^2$

$\large z_1z_1^*=\simplify{{r1}{rcc1}}+\simplify{{r1}{icc1}+{i1}{rcc1}}i+\simplify{{i1}{icc1}}i^2$

Remember that $i=\sqrt{-1}$ so $i^2=-1$ :

$\large z_1z_1^*=\simplify{{r1}{rcc1}}+\simplify{{r1}{icc1}+{i1}{rcc1}}i+\simplify{{i1}{icc1}i^2}$

$\large z_1z_1^*=\simplify{{z1}*{cc1}}$

$\large z_2=\var{z1}$ and $\large z_2^*=\var{cc2}$

Evaluate:
$\large z_2z_2^*=(\var{z2})(\var{cc2})$

$\large z_2z_2^*=(\var{r2})(\var{rcc2})+(\var{r2})(\var{icc2})i+(\var{i2})(\var{rcc2})i+(\var{i2})(\var{icc2})i^2$

$\large z_2z_2^*=\simplify{{r2}{rcc2}}+\simplify{{r2}{icc2}}i+\simplify{{i2}{rcc2}}i+\simplify{{i2}{icc2}}i^2$

$\large z_2z_2^*=\simplify{{r2}{rcc2}}+\simplify{{r2}{icc2}+{i2}{rcc2}}i+\simplify{{i2}{icc2}}i^2$

Remember that $i=\sqrt{-1}$ so $i^2=-1$ :

$\large z_2z_2^*=\simplify{{r2}{rcc2}}+\simplify{{r2}{icc2}+{i2}{rcc2}}i+\simplify{{i2}{icc2}i^2}$

$\large z_2z_2^*=\simplify{{z2}*{cc2}}$

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Error evaluating variable z1: Function i is not defined. Is i a variable, and did you mean i*(...)?

Error evaluating variable cc1: Variable z1 is not defined.

Error evaluating variable rcc1: Variable cc1 is not defined.

Error evaluating variable icc1: Variable cc1 is not defined.