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### Complex Numbers

Complex numbers are the numbers of the form a+ib where i=√(-1) and a and b are real numbers.

Definition:- Complex numbers are defined as an ordered pair of real numbers like (x,y) where

z=(x,y)=x+iy

and both x and y are real numbers and x is known as real part of complex number and y is known as imaginary part of the complex number.

Let z1=x1+iy1 and z2=x2+iy2 then

z1+z2=(x1+x2)+i(y1+y2)

Subtraction

z1-z2=(x1-x2)+i(y1-y2)

Multiplication

(z1.z2)=(x1+iy1).(x2+iy2)

Division

To divide complex number by another , first write quotient as a fraction. Then reduce the fraction to rectangular form by multiplying the numerator and denominator by the complex conjugate of the denominatormaking the denominator real.

Points to remember

1. Complex conjugate of z=x+iy is $\overline{z}=x-\mathrm{iy}$

2. Modulus of the absolute value of z is denoted by |z| and is defined by √(x2+y2)

3. Let r be any non negative number and θ any real number. If we take x=rcosθ and y=rsinθ then,

$r=\sqrt{{x}^{2}+{y}^{2}}$ which is the modulus of z and $\theta ={\mathrm{tan}}^{-1}\frac{y}{x}$ which is the argument or amplitude of z and is denoted by arg.z

we also have x+iy=r(cosθ+isinθ)=r[cos(2nπ+θ)+isin(2nπ+θ)] , where n=0, ±1, ±2, ....

4. Argument of a complex number is not unique since if θ is the value of argument then 2nπ+θ (n=0, ±1, ±2, ....) are also values of the argument. Thus, argument of complex number can have infinite number of values which differ from each other by any multiple of 2π.

5. Arg(0) is not defined.

6. argument of positive real number is zero.

7. argument of negative real number is ±π

8. Properties of moduli:-

• |z1+z2|≤|z1|+|z2|
• |z1-z2|≥|z1|-|z2|
• |z1•z2|=|z1|•|z2|
• |z1/z2|=|z1|/|z2|

### EDUCATIONAL OPPORTUNITIES for physics students

Good career opportunities for qualified physicists are available both as teachers and researchers. For teaching jobs in universities and colleges, one has to qualify in the UGC – CSIR NET. Every year a large number of candidates are recruited by several research institutions.

(1) Laboratory Assistant: This job revolves around taking care of the laboratory and its equipment. The lab-assistant arranges instruments and apparatus foe various experiments to be conducted in the lab. A PG Diploma in Medical Lab Technology would be better option though.
Scientific Assistant: A scientific assistant works under a scientist and may have responsibilities like recording routine readings of instruments , scanning books, internet and journals for reference material , compiling working notes , etc.
(2) Forestry: Science graduates are eligible for the post of Forester and Forest Ranger. Candidates qualifying in the written test and fulfilling prescribed physical requirements are sent to the Indian Forest Research Institute and College at Dehradun or Coimbatore for a 2-year training course in Forestry.
(3) Defense Services: Science graduates and postgraduates who meet the prescribed physical and medical requirements and clear the entrance tests are eligible for appointment in the Indian Army, Indian Navy and Air Force, and in the Defence Science Service.
(4) Other Entry Occupations: Physics graduates can always try for other options open to graduates for all streams. These include various competitive exams, government jobs, subordinate executive and clerical posts, etc.
(5) Sales: Science graduates are specially suited for sales f commercial products like pharmaceuticals, scientific instruments, biotechnology products, etc. Medical representations are hired by all pharmaceutical and medical equipment manufacturers for promoting and marketing their products to doctors and hospitals.
(6) Apprenticeship: Many industrial undertakings recruit science graduates for paid apprenticeship in the chemical, mechanical, or other relevant engineering department. They are paid a stipend during training and may afterwards be absorbed in the factories and laboratories.
(7) Management Trainees: On the basis of a selection test and interview, many business houses employ promising science graduates as management trainees. On successful completion of training, they are employed as executives.
A degree in Physics can start you on a career in research and the building of knowledge in any particular area of subject. Most students of physics work in research and development, engineering, and information technology fields. Some physicists work on problems at the frontier of knowledge, in fields such as nuclear physics, astrophysics and so on. The main areas of research in physics include astronomy and astrophysics, condensed matter physics and material science.

### Question on motion in one dimension

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