Mathematics Lab Activity-10 Class XI

Mathematics Laboratory Activities on Complex Numbers for class XI students Non-Medical. These activities are strictly according to the CBSE syllabus

Chapter - 4 | Complex Numbers

Activity - 10

Objective

To interpret geometrically the meaning of $i=\sqrt{-1}$ and its integral powers

Material Required

Card Board, White Chart paper, Pieces of thread, Nail, Pencil etc

Theory

1Complex Number: A number of the form a +ib, where a and b are real numbers, is defined to be a complex number.

Here a is called the real part.

b is called the imaginary part

i is called iota.

$i=\sqrt{-1}$ ⇒ i² = -1

Other powers of i are as follows

$i^{3}=i^{2}\times i=-1\times i=-i$

$i^{4}=\left(i^{2}\right)^{2}=\left(-1\right)^{2}=1$

$i^{5}=i^{4}\times i=1\times i=i$

$i^{10}=i^{8}\times i^{2}=(i^{4})^{2}\times-1=1^{2}\times-1=-1$

……………………………………………….

………………………………………….

Procedure

1. Take a card board of appropriate size and paste a white chart paper on it .

2. Draw two mutually perpendicular lines XOX’ and YOY’ which cut each other at point O. as shown in figure 11.1

3. Take a thread of unit length representing the number 1 along OX i.e. OA = 1

4. Fix one end of the thread to the nail at O and the other end at A as shown in figure .

Fig 10.1

5. Set free, the other end of the thread at A and rotate it through angles of 90^o, 180^o, 270^o, and 360^o and mark these points as A₁, A₂, A₃, A₄ as shown in figure

Observations

1. In the Argand plane , OA represents = 1

OA₁ represents = i = 1 x i

OA₂ represents = -1 = 1 x i x i = i²

OA₃ represents = – i = 1 x i x i x i = i³

OA₄ represents = 1 = 1 x i x i x i x i = i⁴ = OA

2. From the above explanation we observe that : Each time , rotation of OA by 90^o is equivalent to multiplication by .

3. Therefore, i is referred to as the multiplication factor for a rotation of 90^o.

4. On rotation OA through n- right angles

OA_n = 1 x i x i x i x i ……………n times =1 x iⁿ

Result

i (iota) is called the multiplication factor for the rotation of 90^o.

Applications

This activity can be used to evaluate any integral power of I.

VIVA – VOICE

Q. 1 What is the value of i ?

Ans. $i=\sqrt{-1}$

Q. 2. If z = a +i b, then what is the value of |z| ?

Ans. $|z|=\sqrt{a^{2}+b^{2}}$

Q. 3. What is the value of | i |

Ans. 1

Q. 4. What is the Argand plane?

Ans. The plane having a complex number assigned to each of its point is called the complex plane or Argand plane.

Q. 5. What is Argand diagram?

Ans. An Argand plane is a plot of complex numbers as points in the complex plane using the x-axis as the real axis and y-axis as the imaginary axis.

Search This Blog

Featured Posts on Lesson Plan

Mathematics Lab Activity-20 Class X | Probability