However, it is only so in appearance, not in complex algebraic operation. It is actually the following complex conjugate expression in disguise $$(z_1-z_2)(\bar z_1-\bar z_2 )= (z_2-z_3)(\bar z_2-\bar z_3 ) =( z_3-z_1)(\bar z_3-\bar z_1 )\tag3$$ which is more involved than (1). Besides, the expression (2) is in fact a system of two equations But first equality of complex numbers must be defined. If two complex numbers, say a +bi, c +di are equal, then both their real and imaginary parts are equal; a +bi =c +di ⇒ a =c and b =d Addition and subtraction Addition of complex numbers is defined by separately adding real and imaginary parts; so if z =a +bi, w =c +di then z +w =(a +c)+(b complex-numbers; Share. Cite. Follow edited Nov 24, 2017 at 13:15. vidyarthi. 6,926 2 2 gold badges 19 19 silver badges 55 55 bronze badges. asked Sep 2, 2015 at 2:36. Therefore not linear (consider the bar on the right of w and z as it is on the upper). Share. Cite. Follow Also, z and \[\bar{z}\] are called the complex conjugate pair. For example, z = x + iy is a complex number that is inclined on the real axis making an angle of α, and \[\bar{z}\] = x - iy which is inclined to the real axis making an angle -α. (Image will be uploaded soon) Practice set 1: Finding absolute value. To find the absolute value of a complex number, we take the square root of the sum of the squares of the parts (this is a direct result of the Pythagorean theorem): | a + b i | = a 2 + b 2. For example, the absolute value of 3 + 4 i is 3 2 + 4 2 = 25 = 5 . Problem 1.1. A complex number is of the form a + ib and is usually represented by z. Here both a and b are real numbers. The value 'a' is called the real part which is denoted by Re (z), and 'b' is called the imaginary part Im (z). Also, ib is called an imaginary number. Definition of Modulus of a Complex Number: Let z = x + iy where x and y are real and i = √-1. Then the non negative square root of (x^2 + y^2) is called the modulus or absolute value of z (or x + iy). $\begingroup$ I guess I'm still confused because these examples are given, without proof (and usually Rudin sketches a proof, at least in the earlier chapters) before the middle of Chapter 2. The definition of continuity and proofs using it don't occur until Chapter 4, but should I be able, in the middle of Chapter 2, to work out a proof that apparently uses material from much further on in The complex numbers $c+di$ and $c-di$ are called complex conjugates. If $z=c+di$, we use $\overline{z}$ to denote $c-di$. If $z=c+di$, we use $\overline{z}$ to denote $c-di$. Viewing $z=a+bi$ as a vector in the complex plane, it has magnitude $$ |z| = \sqrt{a^2+b^2}, $$ which we call the modulus or absolute value of $z$. The absolute value measures the distance between two complex numbers. Thus, z 1 and z 2 are close when jz 1 z 2jis small. We can then de ne the limit of a complex function f(z) as follows: we write lim z!c f(z) = L; where cand Lare understood to be complex numbers, if the distance from f(z) to L, jf(z) Lj, is small whenever jz cjis small. More kKxAO.