Question

find the modulus & argument of the complex number 1+i​

Answer 1

Answer:

Step-by-step explanation:

ANSWER

Let z=

1−i

1+i

​

Rationalizing the same,

z=

1−i

1+i

​

∗

1+i

1+i

​

z=

(1−i)(1+i)

(1+i)(1+i)

​

Using(a−b)(a+b)=a

2

−b

2

                         =

1

2

−i

2

(1+i)

2

​

Using(a+b)

2=

a

2

+b

2

+2ab

                             =

1

2

−i

2

1

2

+i

2

+2i

​

Puttingi

2

=−1

                                    =

1−(−1)

1

2

+(−1)+2i

​

                                    =

2

2i

​

                                    =i

                                    =0+i

                     Hencez=(0+i)

To calculate modulus of z,

z=(0+i)

Complexnumberzisoftheformx+iy

Hencex=0andy=1

Modulus of z=

x

2

+y

2

​

z=

0

2

+1

2

​

z=

0+1

​

z=

1

​

To find the argument,

0+i=rcosθ+irsinθ

Comparing real part,

0=rcosθ

Put r=1

0=1∗cosθ

0=cosθ

cosθ=0

Comparing imaginary part,

1=rsinθ

Put r=1

1=1∗sinθ

1=sinθ

sinθ=1

Hencecosθ=0andsinθ=1

Answer 2

$\rm \large let \: { z = 1 + i} \\ \\ \\ \rightarrow\rm |z| = | 1 + i|\\ \\ \\ \rightarrow\rm \large |z| = \sqrt{ {1}^{2} + {1}^{2} } \\ \\ \\ \large \rightarrow\rm |z| = \sqrt{ {1} + {1} } \\ \\ \\ \large \rightarrow\rm |z| = \sqrt{ 2}$

Now , we have to find out argument of the complex number (1+i).

$\large \rm \theta = arg(z) \\ \\ \\\large \rm \longrightarrow tan \theta = \bigg | \dfrac{img(z)}{real(z)} \bigg| \\ \\ \\\large \rm img(z) = 1 \: \: and \: \: real(z) = 1\\ \\ \large \rm \: clearly \: the \: point \: (1,1) \: represents \: that \: z \: lies \: in \: the \: first \: quadrant.\\ \\ \\ \large \rm \longrightarrow tan \theta = \bigg | \dfrac{1}{1} \bigg| \\ \\ \\ \large \rm \longrightarrow tan \theta = 1 \\ \\ \\\large \rm \theta = \dfrac{\pi}{4} \\ \\ \\ \large\rm \theta = arg(z) = \dfrac{\pi}{4}$

Answer :

Modulus of 1+i = √2

argument of 1+i = π/4

Additional Information :

Properties of Modules :-

If X = a+bi then ,

1) | X | = |- X |

2) | X₁ X₂ | = | X₁ | | X₂ |

3) | X₁ / X₂ | = | X₁ | / | X₂ |

4) | X₁ + X₂ | ≠ | X₁ | + | X₂ |

Properties of argument :-

1) Arg(0) = not defined

2) If X is purely imaginary number then , arg(X) = ± (π/2)

3) Arg( X₁ X₂) = Arg( X₁ ) + Arg( X₂) + 2mπ

4) Arg( X₁ - X₂) = Arg( X₁ ) - Arg( X₂) + 2mπ

5) Arg( Xⁿ) = n Arg( Xⁿ) + 2mπ