MTH632 Final Term MCQs Data Set 2 by Abu Sultan

Module # 001

1. Complex Analysis related to complex numbers and their functions. We have different types of numbers which help us in different solutions. Natural Number, Integers, Rational Number i.e. ℚ= { 𝑎 𝑝 2. The decimal expression is non-terminating and non-periodic. 3. 4. Real numbers are not enough to solve our all problems if we consider an equation x2+2x+2=0 then we get 𝑥= −1 so we denote it 𝑖 i.e. 𝑖= −1 5. 6. “Cardano” was first to introduce complex numbers 𝑎+ −1 𝑏 into algebra but had misgivings about it.

7. “Rafael Bombelli authored I’Algebra (1572 and 1579) a set of three books “At first , the things seemed to me to be based more on sophism than on truth, but I searched until I found the proof”. 8. L.Euler (1701-1783) introduced the notation 𝑖= −1

Module # 002

9. A complex number z can be defined as an ordered pair z=(x,y) where x,y are real numbers. 10. Complex numbers can be interpreted as points in the complex plane. 11. Real numbers is a subset of complex numbers. 12. When real numbers x are displayed as points (x,0) on the real axis the set of complex numbers includes the real numbers as a subset. 13. Complex numbers of the form (0,y) corresponds to y-axis and are pure imaginary numbers when 𝑦≠0. The y-axis is then referred to the imaginary axis. 14. For complex number z=(x,y) the real numbers x,y are known as the real and imaginary parts of z respectively and we write as x=Re z and y=Im z 15. Two complex numbers z1 and z2 are equal whenever they have the same real parts and the same imaginary parts. 16. The statement z1=z2 means that z1 and z2 corresponds to same point in the complex plane or z-plane. 17. Let z1=(x1,y1), z2=(x2,y2) then z1=z2 iff x1=x2 and y1=y2.

Module # 003

18. Two ways to represent complex numbers are Addition of Complex numbers and Product of Complex Numbers. 19. Commutative Law w.r.t Addition and Associative Law w.r.t Addition holds in Complex Numbers. 20. Addition of two complex numbers: For z1=(x1,y1), z2=(x2,y2) the addition is defined as z1+z2=(x1,y1,)+(x2,y2,)=(x1+x2, y1+y2) 21. Product of two complex numbers: For z1=(x1,y1), z2=(x2,y2) the product is defined as z1.z2=(x1,y1,).(x2,y2,)=(x1x2-y1y2, x1y2+x2y1) 22. Any complex number z=(x,y) can be written as z=(x,y)=(x,0)+(0,y)

23. Any complex number z=(x,y) can be written as z=(x,y)=(x,0)+(0,y), Also we have (0,1)(y,0)=(0,y) so we can write z=(x,0)+(0,1)(y,0). The numbers of the form (x,0) are real numbers so can simply denoted by x, numbers of the form (y,0) are real numbers so can simply denoted by y and (0,1) is the number denoted by “i” then z=(x,0)+(0,1)(y,0)=x+iy 24. (0,1) is such a complex number when it is multiplied with itself then it gives -1. i.e (0,1)(0,1)=(0-1,0+0)=(-1,0)=-1or i2=-1 25. Z=x+iy is used to represent the complex numbers and this representation is due to the mathematician Gauss.

Module # 004

26. Properties of Addition of Complex Numbers: i. The addition of complex numbers follows all the properties of real numbers. ii. Additive Identity: There is a unique complex number ‘w’ such that z+w=w+z=z such number w is 0+i0 or (0,0). iii.

27. Complex numbers can be represented by two ways. i. Ordered pair representation ii. z=x+iy form. Ordered Pair Representation of complex numbers is the real foundational definition on which complex number system is built while z=x+iy is used or preferred due to simplicity.

Module # 005

28. Properties of Multiplication of Complex Numbers: i. The product of complex numbers follows all the properties of real numbers. ii. The commutative law z1z2=z2z1. The commutative implies iy=yi so z=x+iy and z=x+yi are same. iii. The associative law z1(z2z3)=(z1z2)z3. We can write z1z2z3 without parenthesis. iv. Multiplicative Identity: There is a unique complex number ‘w’ such that zw=wz=z such number w is 1+i0 or (1,0). iv. Multiplicative Inverse: For every complex number z=x+iy other than (0,0) there is a unique 𝑧−1 such that z𝑧−1 = 1, In fact 𝑧−1 = 𝑧 = 29. Division of Complex numbers can be calculated using 𝑧1 𝑧2 =

30. Distributive Law: The Operation of multiplication is distributive over addition that is z1(z2+z3)=z1z2+z1z3 31. Binomial Formula: For any two complex numbers z1, z2 we have (z1+z2)n= 𝑛 𝑘 𝑛 𝑘=0 𝑧1 𝑛 𝑘 = 𝑛!

Module # 006

32. A complex number ‘z’ can be associated with the directed line segment or vector from origin to the point of z. 33. Multiplication of two complex number is neither dot product nor vector product. 34. The vector interpretation helps us defining modulus or absolute value of complex number. 35. The modulus or absolute value of a complex number z=(x,y)=x+iy is defined and denoted as 𝑍 = 𝑥2 + 𝑦2

Module # 007-008

36. Properties of modulus of a complex number: An important consequence of triangular inequality is 𝑍1 + 𝑍2 ≥ 𝑍1 − 𝑍2 37. 38. = 𝑧1 + 𝑧2 = 𝑧1 - 𝑧2 = 𝑧1 𝑧2 and 𝑧1 𝑧2 𝑧1 𝑧2 if 𝑧2 ≠0 39. Complex conjugate of z=x+iy are Re(z)=x and Im(z)=y 40. 2𝑖 41. 42. Complex conjugate helps us in proving properties of modulus . If z=x+iy then 𝑧 = 𝑥2 + 𝑦2 43. 44.

Module # 009

45. 46. If z=0 then coordinates of 𝜃 are undefined. 47. The length of 𝑟= |𝑧| is not allowed to be negative since it is length of radius vector for z.