Question

Find the modulus of (1+i)(2+i)/(3+i)

Please Explain Step By Step ​

Answer 1

Step-by-step explanation:

Factorial of 100 has 158 digits. It is not possible to store these many digits even if we use long long int. Following is a simple solution where we use an array to store individual digits of the result. The idea is to use basic mathematics for multiplication

Answer 2

Solution:

$\longmapsto\tt{z =\dfrac{(1 + 2i)}{(1 - 3i)}}$

$\longmapsto\tt{\dfrac{(1 + 2i)(1 +3i)}{(1 - 3i)(1 + 3i)}}$

$\longmapsto\tt{\dfrac{{1(1 + 3i)+2i(1 +3i)}}{ {1²-(3i)²} \: 5}}$

$\longmapsto\tt{\dfrac{ ( 1 +3i +2i + 6i²)}{(1 + 9)}}$

$\longmapsto\tt{\dfrac{(-5 + 5i)}{10}}$

$\longmapsto\tt{(\dfrac{-1}{2}) + (\dfrac{1}{2})i}$

$\sf\red{now, \: modulus \: of \: the \: complex \: number \: is}$

$\longmapsto\tt{l(\dfrac{-1}{2}) +( \dfrac{1}{2})il}$

$\longmapsto\tt{√{(\dfrac{-1}{2})^2 + (\dfrac{1}{2})^2}}$

$\longmapsto\tt{√{\dfrac{1}{4} + \dfrac{1}{4}}}$

$\longmapsto\tt{\dfrac{1}{√2}}$

$\sf\red{now, tan∅ = |Im(z)/Re(z)|}$

$\longmapsto\tt{|(\dfrac{1}{2})(\dfrac{-1}{2})| = 1}$

$\longmapsto\tt{tan∅ =\dfrac{tanπ}{4}}$

$\longmapsto\tt{∅ = \dfrac{π}{4}}$

$\sf\red{∅ \: lies \: on \: 2nd \: quadrant \: so,}$

$\longmapsto\tt{arg(z) = π - ∅}$

$\longmapsto\tt{π -\dfrac{π}{4}}$

$\longmapsto\tt{\dfrac{3π}{4}}$

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