Math Problem Statement

Convert 1 - 3i and 5 + i into exponential form and hence calculate the modulus and argument of: (1 - 3i)/(5 + i)

Solution

Let's begin by converting the two complex numbers z1=13iz_1 = 1 - 3i and z2=5+iz_2 = 5 + i into their exponential forms. The general form for expressing a complex number z=x+iyz = x + iy in exponential form is:

z=reiθz = r e^{i\theta} where:

  • r=zr = |z| is the modulus (magnitude) of the complex number, given by r=x2+y2r = \sqrt{x^2 + y^2},
  • θ=arg(z)\theta = \arg(z) is the argument (angle) of the complex number, given by θ=tan1(yx)\theta = \tan^{-1}\left(\frac{y}{x}\right).

Step 1: Convert z1=13iz_1 = 1 - 3i into exponential form.

  • Modulus: r1=z1=12+(3)2=1+9=10r_1 = |z_1| = \sqrt{1^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10}

  • Argument: θ1=arg(z1)=tan1(31)=tan1(3)\theta_1 = \arg(z_1) = \tan^{-1}\left(\frac{-3}{1}\right) = \tan^{-1}(-3) Since the real part is positive and the imaginary part is negative, the complex number is in the fourth quadrant. The argument is: θ1=tan1(3)\theta_1 = -\tan^{-1}(3)

Thus, z1z_1 in exponential form is: z1=10eitan1(3)z_1 = \sqrt{10} e^{-i\tan^{-1}(3)}

Step 2: Convert z2=5+iz_2 = 5 + i into exponential form.

  • Modulus: r2=z2=52+12=25+1=26r_2 = |z_2| = \sqrt{5^2 + 1^2} = \sqrt{25 + 1} = \sqrt{26}

  • Argument: θ2=arg(z2)=tan1(15)\theta_2 = \arg(z_2) = \tan^{-1}\left(\frac{1}{5}\right) Since the real part is positive and the imaginary part is positive, the complex number is in the first quadrant. The argument is: θ2=tan1(15)\theta_2 = \tan^{-1}\left(\frac{1}{5}\right)

Thus, z2z_2 in exponential form is: z2=26eitan1(15)z_2 = \sqrt{26} e^{i\tan^{-1}\left(\frac{1}{5}\right)}

Step 3: Calculate z1z2\frac{z_1}{z_2}.

Now, divide z1z_1 by z2z_2: z1z2=10eitan1(3)26eitan1(15)\frac{z_1}{z_2} = \frac{\sqrt{10} e^{-i\tan^{-1}(3)}}{\sqrt{26} e^{i\tan^{-1}\left(\frac{1}{5}\right)}} This simplifies to: z1z2=1026ei(tan1(3)tan1(15))\frac{z_1}{z_2} = \frac{\sqrt{10}}{\sqrt{26}} e^{i\left(-\tan^{-1}(3) - \tan^{-1}\left(\frac{1}{5}\right)\right)}

Step 4: Modulus of z1z2\frac{z_1}{z_2}.

The modulus is simply the ratio of the moduli of z1z_1 and z2z_2: Modulus=z1z2=1026=1026=1026=55\text{Modulus} = \frac{|z_1|}{|z_2|} = \frac{\sqrt{10}}{\sqrt{26}} = \frac{\sqrt{10}}{\sqrt{26}} = \sqrt{\frac{10}{26}} = \frac{\sqrt{5}}{5}

Step 5: Argument of z1z2\frac{z_1}{z_2}.

The argument is the difference of the arguments of z1z_1 and z2z_2: Argument=arg(z1z2)=tan1(3)tan1(15)\text{Argument} = \arg\left(\frac{z_1}{z_2}\right) = -\tan^{-1}(3) - \tan^{-1}\left(\frac{1}{5}\right)

Thus, the modulus and argument of z1z2\frac{z_1}{z_2} are:

  • Modulus: 55\frac{\sqrt{5}}{5}
  • Argument: tan1(3)tan1(15)-\tan^{-1}(3) - \tan^{-1}\left(\frac{1}{5}\right)

Would you like further clarification or details on any part of this process?

Here are some follow-up questions you might find interesting:

  1. How do you calculate the real and imaginary parts from an exponential form?
  2. What is the significance of the modulus and argument in complex numbers?
  3. Can you express any complex number in polar form and vice versa?
  4. How can you geometrically interpret the division of two complex numbers?
  5. What are the applications of complex number operations in electrical engineering?

Tip: When working with arguments of complex numbers, make sure to adjust the angle depending on the quadrant where the number lies.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Complex Numbers
Exponential Form
Modulus and Argument

Formulas

Modulus: r = √(x^2 + y^2)
Argument: θ = tan⁻¹(y/x)
Exponential Form: z = r e^{iθ}

Theorems

Euler's Formula: z = r e^{iθ}
Properties of Moduli and Arguments in Complex Division

Suitable Grade Level

Grades 10-12