Question

what's the mod value of (3+i)×(1+2i)/(2-i)×(2+3i)?
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Answer 1

Let z = x + iy where x and y are real and i = √-1. Then the non negative square root of (x22+ y 22) is called the modulus or absolute value of z (or x + iy).

Modulus of a complex number z = x + iy, denoted by mod(z) or |z| or |x + iy|, is defined as |z|[or mod z or |x + iy|] = + x2+y2−−−−−−√x2+y2 ,where a = Re(z), b = Im(z)

i.e., + Re(z)2+Im(z)2−−−−−−−−−−−−−√Re(z)2+Im(z)2

Sometimes, |z| is called absolute value of z. Clearly, |z| ≥ 0 for all zϵ C.

For example:

(i) If z = 6 + 8i then |z| = 62+82−−−−−−√62+82 = √100 = 10.

(ii) If z = -6 + 8i then |z| = (−6)2+82−−−−−−−−−√(−6)2+82 = √100 = 10.

(iii) If z = 6 - 8i then |z| = 62+(−8)2−−−−−−−−−√62+(−8)2 = √100 = 10.

(iv) If z = √2 - 3i then |z| = (√2)2+(−3)2−−−−−−−−−−−−√(√2)2+(−3)2 = √11.

(v) If z = -√2 - 3i then |z| = (−√2)2+(−3)2