Добавить материал и получить бесплатное свидетельство о публикации в СМИ
Эл. №ФС77-60625 от 20.01.2015
Свидетельство о публикации

Автоматическая выдача свидетельства о публикации в официальном СМИ сразу после добавления материала на сайт - Бесплатно

Добавить свой материал

За каждый опубликованный материал Вы получите бесплатное свидетельство о публикации от проекта «Инфоурок»

(Свидетельство о регистрации СМИ: Эл №ФС77-60625 от 20.01.2015)

Инфоурок / Математика / Конспекты / Тематические материалы для подготовки к уроку математики на тему "Argand diagram"
ВНИМАНИЮ ВСЕХ УЧИТЕЛЕЙ: согласно Федеральному закону № 313-ФЗ все педагоги должны пройти обучение навыкам оказания первой помощи.

Дистанционный курс "Оказание первой помощи детям и взрослым" от проекта "Инфоурок" даёт Вам возможность привести свои знания в соответствие с требованиями закона и получить удостоверение о повышении квалификации установленного образца (180 часов). Начало обучения новой группы: 28 июня.

Подать заявку на курс
  • Математика

Тематические материалы для подготовки к уроку математики на тему "Argand diagram"

Выбранный для просмотра документ Маthematics Grade 12 Argand diagram Tasks Тоxаnоvа А.Zh..docx

библиотека
материалов

Argand diagram


hello_html_54202739.png

hello_html_3ce90fc0.png

hello_html_3ce90fc0.png



Answers:

hello_html_m29bb4657.gif


hello_html_m7dc2b615.gif

hello_html_1c862998.gif


hello_html_530257fc.gif

hello_html_242b6539.gif


hello_html_4edaea7b.gif


hello_html_34c60a47.gif


Выбранный для просмотра документ Маthematics Grade 12 Argand diagram Test Toxanova A.Zh..docx

библиотека
материалов

Argand diagram


Questions 1-3 refer to the Argand diagram below.

hello_html_m23e3d494.gif


1. In the Argand diagram, the point A represents the complex number

  1. –3 + 2i

  2. 3 – 2i

  3. 2 – 3i

  4. –2 + 3i


2. In the Argand diagram, the point B represents the complex number

  1. 1 + 4i

  2. –4 – i

  3. –1 – 4i

  4. 4 + i


3. In the Argand diagram, the point C represents the complex number

  1. 3 – i

  2. 1 – 3i

  3. –3 + i

  4. –1 + 3i


Questions 4-6 refer to the Argand diagram below. The point representing the complex number (z) is shown on the diagram.

hello_html_m7b5fc732.gif

4. The point which represents (z*) is

  1. V

  2. R

  3. Q

  4. T


5. The point which represents (iz) is

  1. U

  2. S

  3. P

  4. Q


6. The point which represents (–z) is

  1. V

  2. R

  3. S

  4. T


7. The point z on the Argand diagram below is:

hello_html_706a7d64.jpg



8. If the amplitude of a complex number is then the number is

  1. neither real nor imaginary

  2. purely imaginary

  3. purely real

  4. 0


9. If z represents a complex number then is


10. If a = 3 + i and z = 2 – 3i then the points on the Argand diagram representing az, 3az and –az are

  1. Vertices of a right angled triangle

  2. Vertices of an equilateral triangle

  3. Vertices of an isosceles triangle

  4. Collinear




Answers:


1

2

3

4

5

6

7

8

9

10

A

C

А

A

A

C

B

B

D

D


Выбранный для просмотра документ Маthematics Grade 12 Argand diagram Theory + examples Toxanova A.Zh..docx

библиотека
материалов

Argand diagram


SP reference

Learning Objective

ММ 12.1

locate complex numbers on the complex plane (using Argand’s diagram)

Skills

Success criteria

Learners achieve this learning objective, if

Knowledge and understanding

represent complex numbers correctly on the Argand diagram

state complex numbers indicated on the Argand diagram


It is very useful to have a graphical or pictorial representation of complex numbers.

For example, the complex number z = 3 + 4i is represented as a point in the xy plane with coordinates (3, 4) as shown in Figure 1. That is, the real part, 3, is plotted on the x axis, and the imaginary part, 4, is plotted on the y axis.


hello_html_m4887d349.gif

Figure 1. Argand diagram which represents the complex number 3 + 4i by the point P(3,4).


More generally, the complex number z = a + ib is plotted as a point with coordinates (a, b) as shown in Figure 2.


hello_html_73f13542.gif


Figure 2. Argand diagram which represents the complex number a + bi by the point P(a, b).


Because the real part of z is plotted on the horizontal axis we often refer to this as the real axis. The imaginary part of z is plotted on the vertical axis and so we refer to this as the imaginary axis. Such a diagram is called an Argand diagram. Engineers often refer to this diagram as the complex plane.


Examples

Plot the complex numbers z1 = 2 + 3i, z2 = −3 + 2i, z3 = −3 − 2i, z4 = 2 − 5i, z5 = 6, z6 = i on an Argand diagram.

Solution

The Argand diagram is shown in Figure 3.

hello_html_286d4ac4.gif

Figure 3. Argand diagram showing several complex numbers.


Note that purely real numbers lie on the real axis. Purely imaginary numbers lie on the imaginary axis.


Another observation is that complex conjugate pairs (such as −3 + 2i and −3 2i) lie symmetrically about the x axis.


Finally, because every real number, a say, can be written as a complex number, a + 0i, that is as a complex number with a zero imaginary part, it follows that all real numbers are also complex numbers. As such we see that complex numbers form an extension of the sets of numbers with which we were already familiar.



Выбранный для просмотра документ Маthematics Grade 12 Argand diagram Tasks Тоxаnоvа А.Zh..docx

библиотека
материалов

Argand diagram


hello_html_54202739.png

hello_html_3ce90fc0.png

hello_html_3ce90fc0.png



Answers:

hello_html_m29bb4657.gif


hello_html_m7dc2b615.gif

hello_html_1c862998.gif


hello_html_530257fc.gif

hello_html_242b6539.gif


hello_html_4edaea7b.gif


hello_html_34c60a47.gif


Выбранный для просмотра документ Маthematics Grade 12 Argand diagram Test Toxanova A.Zh..docx

библиотека
материалов

Argand diagram


Questions 1-3 refer to the Argand diagram below.

hello_html_m23e3d494.gif


1. In the Argand diagram, the point A represents the complex number

  1. –3 + 2i

  2. 3 – 2i

  3. 2 – 3i

  4. –2 + 3i


2. In the Argand diagram, the point B represents the complex number

  1. 1 + 4i

  2. –4 – i

  3. –1 – 4i

  4. 4 + i


3. In the Argand diagram, the point C represents the complex number

  1. 3 – i

  2. 1 – 3i

  3. –3 + i

  4. –1 + 3i


Questions 4-6 refer to the Argand diagram below. The point representing the complex number (z) is shown on the diagram.

hello_html_m7b5fc732.gif

4. The point which represents (z*) is

  1. V

  2. R

  3. Q

  4. T


5. The point which represents (iz) is

  1. U

  2. S

  3. P

  4. Q


6. The point which represents (–z) is

  1. V

  2. R

  3. S

  4. T


7. The point z on the Argand diagram below is:

hello_html_706a7d64.jpg



8. If the amplitude of a complex number is then the number is

  1. neither real nor imaginary

  2. purely imaginary

  3. purely real

  4. 0


9. If z represents a complex number then is


10. If a = 3 + i and z = 2 – 3i then the points on the Argand diagram representing az, 3az and –az are

  1. Vertices of a right angled triangle

  2. Vertices of an equilateral triangle

  3. Vertices of an isosceles triangle

  4. Collinear




Answers:


1

2

3

4

5

6

7

8

9

10

A

C

А

A

A

C

B

B

D

D


Выбранный для просмотра документ Маthematics Grade 12 Argand diagram Theory + examples Toxanova A.Zh..docx

библиотека
материалов

Argand diagram


SP reference

Learning Objective

ММ 12.1

locate complex numbers on the complex plane (using Argand’s diagram)

Skills

Success criteria

Learners achieve this learning objective, if

Knowledge and understanding

represent complex numbers correctly on the Argand diagram

state complex numbers indicated on the Argand diagram


It is very useful to have a graphical or pictorial representation of complex numbers.

For example, the complex number z = 3 + 4i is represented as a point in the xy plane with coordinates (3, 4) as shown in Figure 1. That is, the real part, 3, is plotted on the x axis, and the imaginary part, 4, is plotted on the y axis.


hello_html_m4887d349.gif

Figure 1. Argand diagram which represents the complex number 3 + 4i by the point P(3,4).


More generally, the complex number z = a + ib is plotted as a point with coordinates (a, b) as shown in Figure 2.


hello_html_73f13542.gif


Figure 2. Argand diagram which represents the complex number a + bi by the point P(a, b).


Because the real part of z is plotted on the horizontal axis we often refer to this as the real axis. The imaginary part of z is plotted on the vertical axis and so we refer to this as the imaginary axis. Such a diagram is called an Argand diagram. Engineers often refer to this diagram as the complex plane.


Examples

Plot the complex numbers z1 = 2 + 3i, z2 = −3 + 2i, z3 = −3 − 2i, z4 = 2 − 5i, z5 = 6, z6 = i on an Argand diagram.

Solution

The Argand diagram is shown in Figure 3.

hello_html_286d4ac4.gif

Figure 3. Argand diagram showing several complex numbers.


Note that purely real numbers lie on the real axis. Purely imaginary numbers lie on the imaginary axis.


Another observation is that complex conjugate pairs (such as −3 + 2i and −3 2i) lie symmetrically about the x axis.


Finally, because every real number, a say, can be written as a complex number, a + 0i, that is as a complex number with a zero imaginary part, it follows that all real numbers are also complex numbers. As such we see that complex numbers form an extension of the sets of numbers with which we were already familiar.




Подайте заявку сейчас на любой интересующий Вас курс переподготовки, чтобы получить диплом со скидкой 50% уже осенью 2017 года.


Выберите специальность, которую Вы хотите получить:

Обучение проходит дистанционно на сайте проекта "Инфоурок".
По итогам обучения слушателям выдаются печатные дипломы установленного образца.

ПЕРЕЙТИ В КАТАЛОГ КУРСОВ

Автор
Дата добавления 10.09.2016
Раздел Математика
Подраздел Конспекты
Просмотров53
Номер материала ДБ-183415
Получить свидетельство о публикации
Похожие материалы

Включите уведомления прямо сейчас и мы сразу сообщим Вам о важных новостях. Не волнуйтесь, мы будем отправлять только самое главное.
Специальное предложение
Вверх