Complex Numbers; Division 01 (Rect')

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Dr Palat Meethale Ushasree

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3 years, 1 month ago

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Dr Palat Meethale Ushasree 3 years, 1 month ago

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Michael Proudman 4 years, 9 months ago

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Michael Proudman 4 years, 9 months ago

Created this as a copy of Michael's copy of Gapfills in fractions.

Name	Status	Author	Last Modified
Equivalent Fractions	Ready to use	Rob Cade	17/06/2016 12:18
Gapfills in fractions	draft	Christian Lawson-Perfect	05/05/2021 14:17
Fractions: equivalent	Ready to use	Ben Brawn	26/08/2024 05:43
Algebraic fractions: simplifying/cancelling	Ready to use	Ben Brawn	06/02/2025 00:42
Algebra V: algebraic fractions (reasoning)	Ready to use	Luke Park	17/08/2016 17:19
Francis 's copy of Algebra V: algebraic fractions (simplification)	draft	Francis Duah	12/08/2016 14:48
Algebra V: algebraic fractions (simplification)	Ready to use	Nasir Firoz Khan	17/08/2016 17:31
Anthony's copy of Equivalent Fractions	draft	Anthony Brown	14/09/2016 20:17
Anthony's copy of Gapfills in fractions	draft	Anthony Brown	16/09/2016 00:01
Add two fractions with the same denominator	draft	Anthony Brown	27/08/2021 12:36
Add two fractions with the same denominator (some negative numbers).	draft	Anthony Brown	05/07/2017 23:14
Subtract two fractions with the same denominator	draft	Anthony Brown	05/07/2017 23:22
Subtract two fractions with the same denominator (some negative numbers).	draft	Anthony Brown	05/07/2017 23:21
Add two fractions with different denominators	draft	Anthony Brown	05/07/2017 23:08
Add two fractions with different denominators (some negative numbers)	draft	Anthony Brown	05/07/2017 23:13
Subtract two fractions with different denominators	draft	Anthony Brown	05/07/2017 23:03
Subtract two fractions with different denominators (some negative numbers)	draft	Anthony Brown	05/07/2017 23:00
Multiply two fractions	draft	Anthony Brown	11/07/2017 19:13
Multiply two fractions (some negative numbers)	draft	Anthony Brown	11/07/2017 19:14
Divide two fractions	draft	Anthony Brown	05/07/2017 20:59
Divide two fractions (some negative numbers)	draft	Anthony Brown	05/07/2017 20:58
Calculating powers (a^(-1))	draft	Anthony Brown	05/07/2017 17:01
Calculating powers ((-a)^(-b))	draft	Anthony Brown	05/07/2017 17:04
Calculating powers (-a^(-b))	draft	Anthony Brown	05/07/2017 16:17
Calculating powers (1^(-b))	draft	Anthony Brown	05/07/2017 17:00
Calculating powers (-a^(-a))	draft	Anthony Brown	05/07/2017 17:00
Calculating powers (-a^(-1))	draft	Anthony Brown	05/07/2017 13:14
Calculating powers (a^(-1))	draft	Anthony Brown	05/07/2017 13:13
Algebra V: algebraic fractions (reasoning)	draft	steve kilgallon	09/11/2016 19:00
Vicky's copy of Gapfills in fractions	draft	Vicky Hall	14/06/2017 14:46
gap in breuk	draft	Johan Maertens	10/07/2017 13:34
Calculating powers (a^(-b))	draft	Anthony Brown	11/07/2017 16:29
Add two fractions with the same denominator	draft	Anthony Brown	12/08/2017 13:03
Add two fractions with the same denominator (some negative numbers).	draft	Anthony Brown	12/08/2017 13:04
Subtract two fractions with the same denominator	draft	Anthony Brown	12/08/2017 13:05
Subtract two fractions with the same denominator (some negative numbers).	draft	Anthony Brown	12/08/2017 13:05
Add two fractions with different denominators	draft	Anthony Brown	12/08/2017 13:06
Add two fractions with different denominators (some negative numbers)	draft	Anthony Brown	12/08/2017 13:07
Subtract two fractions with different denominators	draft	Anthony Brown	12/08/2017 13:07
Subtract two fractions with different denominators (some negative numbers)	draft	Anthony Brown	12/08/2017 13:08
Multiply two fractions	draft	Anthony Brown	12/08/2017 14:42
Multiply two fractions (some negative numbers)	draft	Anthony Brown	12/08/2017 14:43
Divide two fractions	draft	Anthony Brown	12/08/2017 14:44
Divide two fractions (some negative numbers)	draft	Anthony Brown	12/08/2017 14:45
Calculating powers (a^(-1))	draft	Anthony Brown	12/08/2017 19:18
Calculating powers (-a^(-1))	draft	Anthony Brown	12/08/2017 19:19
Calculating powers (a^(-b))	draft	Anthony Brown	12/08/2017 19:20
Calculating powers (-a^(-b))	draft	Anthony Brown	12/08/2017 19:20
Calculating powers (-a^(-a))	draft	Anthony Brown	12/08/2017 19:21
Calculating powers (1^(-b))	draft	Anthony Brown	12/08/2017 19:22
Calculating powers ((-a)^(-b))	draft	Anthony Brown	12/08/2017 19:22
copy of fractions question	draft	Lovkush Agarwal	08/09/2017 14:33
Hannah's copy of Equivalent Fractions	draft	Hannah Bartholomew	14/09/2017 12:02
Johan's copy of Gapfills in fractions	draft	Johan Maertens	23/01/2018 16:01
Ida's copy of Algebraic fractions: simplifying/cancelling	draft	Ida Landgärds	13/04/2018 09:16
Algebra V: algebraic fractions (reasoning)	draft	heike hoffmann	14/11/2018 20:43
Algebraic fractions: simplifying/cancelling	Ready to use	heike hoffmann	14/11/2018 20:53
Terry's copy of Algebraic fractions: simplifying/cancelling	draft	Terry Young	03/04/2019 17:35
Complex Numbers; Division 01 (Rect')	Ready to use	Michael Proudman	08/02/2022 15:21
Merryn's copy of Fractions: equivalent	Ready to use	Merryn Horrocks	07/03/2024 01:07

There are 115 other versions that do you not have access to.

Question statement

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

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

Division is carried out by rationalising.

Multiply both "top and bottom" of the fraction by the complex conjugate of the bottom (the denominator).

$\Large \frac{z}{ w} = \frac{(a + bi)}{(c + di)}$

$\Large \frac{z}{ w} = \frac{(a + bi)}{(c + di)}\times\frac{(c - di)}{(c - di)}$

Give any introductory information the student needs.

			Name	Type	Generated Value

r1	integer	6
i1	integer	4
z1	number	6 + 4i

			Name	Type	Generated Value

r2	integer	3
i2	integer	2
z2	number	3 + 2i
z2C	number	3 - 2i

			Name	Type	Generated Value

Ans1	number	26
Ans2	number	13
Rans1	number	26
Ians1	number	0

			Name	Type	Generated Value

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(2..9)

JME function reference

Description

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

Generated value: `integer`

→ Used by:

Parts

Gap-fill
1. Gap Gap 0. Mathematical expression
  Add
2. Gap Gap 1. Mathematical expression
  Add
3. Gap Gap 2. Mathematical expression
  Add
4. Gap Gap 3. Mathematical expression
  Add
5. Gap Gap 4. Mathematical expression
  Add
6. Gap Gap 5. Mathematical expression
  Add
7. Gap Gap 6. Mathematical expression
  Add
8. Gap Gap 7. Mathematical expression
  Add
9. Gap Gap 8. Mathematical expression
  Add

Gap-fill

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

If:

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

Evaluate: $\large \frac{z_1}{z_2}$

Lay out the two "fractions":

$\Large \frac{z_1}{z_2}=$

Gap 0.Gap 1.

$\large \times$

Gap 2.Gap 3.

Multiply the two complex numbers on top, then those on the bottom:

$\Large \frac{z_1}{z_2}=$

Gap 4.Gap 5.

We can now write the answer as a single complex number with real and imaginary parts (they will most likely be fractions).

$\Large \frac{z_1}{z_2}=$ Gap 6. $+$ Gap 7. $i$

Now enter the complete answer as one entry, not forgetting to use fractions where needed:

$\Large \frac{z_1}{z_2}=$ Gap 8.

Advice

We are asked to carry out the division of two complex numbers by rationalising.

If:

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

Then:

$\frac{z_1}{z_2}=\frac{\var{z1}}{\var{z2}}$

We deal with this by multiplying both top and bottom by the complex conjugate of the denominator $z_2^*=\var{z2C}$

$\frac{z_1}{z_2}=\frac{\var{z1}}{\var{z2}} \times \frac{\var{z2C}}{\var{z2C}}$

We can do this as all we are really multiplying by is $1$ .

Now we multiply across the top, then across the bottom - just as we would do with ordinary fractions:

$\frac{z_1}{z_2}=\frac{\simplify{{z1}{z2C}}}{\simplify{{z2}{z2C}}}$

All that remains is to express this as a proper complex number, with separate real and imaginary parts:

$\frac{z_1}{z_2}= \frac{\var{Rans1}}{\var{Ans2}} + \frac{\var{Ians1}}{\var{Ans2}} i$

Give a worked solution to the whole question.

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Dr Palat Meethale Ushasree 3 years, 1 month ago

Michael Proudman 4 years, 9 months ago

Michael Proudman 4 years, 9 months ago

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

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

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

If:

Evaluate: $\large \frac{z_1}{z_2}$

We are asked to carry out the division of two complex numbers by rationalising.

If:

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Error

Complex Numbers; Division 01 (Rect')

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Dr Palat Meethale Ushasree 3 years, 1 month ago

Michael Proudman 4 years, 9 months ago

Michael Proudman 4 years, 9 months ago

Let zz and ww be any two complex numbers

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

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

If:

Evaluate: z1z2\large \frac{z_1}{z_2}

We are asked to carry out the division of two complex numbers by rationalising.

If:

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

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

Generated value: `integer`

Evaluate: $\large \frac{z_1}{z_2}$