Problem 173 - A Complete Resource Book in Mathematics for JEE Main

Question

The roots of the equation \(z^{4}+1=0\) are (A) \((\pm 1 \pm i)\) (B) \((\pm 2 \pm 2 i)\) (C) \(\frac{1}{\sqrt{2}}(\pm 1 \pm i)\) (D) None of these

Step-by-Step Solution

Answer

(C) \(\frac{1}{\sqrt{2}}(\pm 1 \pm i)\)

Step 1: Set the Equation to Standard Form

The given equation is \(z^4 + 1 = 0\). This can be rearranged to \(z^4 = -1\).

Step 2: Express \(-1\) in Polar Form

In polar form, \(-1\) can be expressed as \(e^{i\pi}\). Thus, we can write \(-1 = e^{i(\pi + 2k\pi)}\), where \(k\) is any integer.

Step 3: Find the Fourth Roots of \(e^{i\pi}\)

To find the roots of \(z^4 = -1\), we need to take the fourth root of \(e^{i\pi}\). The fourth roots can be given by \(z = e^{i((\pi + 2k\pi)/4)}\) for \(k = 0, 1, 2, 3\).

Step 4: Calculate the Angles for Each Root

Calculate the angles for the four roots:- For \(k=0\): \(z = e^{i\pi/4}\)- For \(k=1\): \(z = e^{i3\pi/4}\)- For \(k=2\): \(z = e^{i5\pi/4}\)- For \(k=3\): \(z = e^{i7\pi/4}\)

Step 5: Convert the Complex Numbers to Rectangular Form

Converting from polar to rectangular form using \(z = \text{Re} + i\text{Im}\), where \(\text{Re} = \cos\theta\) and \(\text{Im} = \sin\theta\):- \(e^{i\pi/4} = \frac{1}{\sqrt{2}} + \frac{i}{\sqrt{2}}\)- \(e^{i3\pi/4} = -\frac{1}{\sqrt{2}} + \frac{i}{\sqrt{2}}\)- \(e^{i5\pi/4} = -\frac{1}{\sqrt{2}} - \frac{i}{\sqrt{2}}\)- \(e^{i7\pi/4} = \frac{1}{\sqrt{2}} - \frac{i}{\sqrt{2}}\)

Step 6: Compare with the Given Options

The roots of the equation are \(\frac{1}{\sqrt{2}}(1+i),\frac{1}{\sqrt{2}}(-1+i),\frac{1}{\sqrt{2}}(-1-i),\and \frac{1}{\sqrt{2}}(1-i)\). Compare these to the given options to identify the correct answer, which corresponds to option (C).

Key Concepts

Polar Form

When dealing with complex numbers, an alternative to the traditional rectangular form is the polar form. This approach is particularly useful for simplifying products and powers of complex numbers.

To represent a complex number in polar form, we use a magnitude, denoted by \(r\), and an angle, \(\theta\), which is the argument of the complex number. The polar form is given by the expression \(z = re^{i\theta}\), where \(r = |z|\) is the modulus of the complex number and \(\theta\) is the argument.

• The modulus \(r\) is the distance from the origin to the complex number in the complex plane. It can be calculated using \(r = \sqrt{x^2 + y^2}\), where \(x\) and \(y\) are the real and imaginary parts of the number, respectively.
• The argument \(\theta\) is the angle formed with the positive direction of the real axis. It can be found using \(\theta = \tan^{-1}(y/x)\).

The polar form becomes especially handy when you're calculating powers or roots of complex numbers by utilizing properties of exponents and dealing with angles instead of components.

Roots of Unity

The 'Roots of Unity' are the solutions to the equation \(z^n = 1\), representing points evenly distributed around a circle in the complex plane at intervals of \(\frac{2\pi}{n}\). This idea extends naturally when solving equations like \(z^4 = -1\), where roots are distributed around a shifted unit circle by an angle such as \(\pi\) or \(\pi/4\).

For instance, for the equation \(z^4 = -1\), we express \(-1\) as \(e^{i\pi}\), transforming the equation into a problem of locating fourth roots:

• Calculate the principal angle: \(\pi\)
• Spread angles equally: \(\pi + 2k\pi\)
• Divide by the exponent dimension: \((\pi + 2k\pi)/4\)
• Compute for \(k = 0, 1, 2, 3\).

These concepts illustrate the power of polar form to visually and algebraically simplify complexities inherent in calculating roots.

Rectangular Form

The rectangular form of complex numbers, denoted as \(z = x + yi\), is the most straightforward way of expressing complex numbers. It shows the number plainly as a sum of real and imaginary components. This form is very useful in basic arithmetic operations such as addition and subtraction.

To switch a complex number from polar to rectangular form, you perform a straightforward transformation using trigonometric functions:

• The real part \(x = r \cos\theta\)
• The imaginary part \(y = r \sin\theta\)

Consider an example where you have the complex number in polar form \(z = e^{i\pi/4}\). The equivalent rectangular form is \(z = \frac{1}{\sqrt{2}} + \frac{i}{\sqrt{2}}\). By using this form, we simplify comparing solutions with more common answer formats.

This familiarity makes rectangular forms a convenient choice for identifying roots on the coordinate axis, while still benefiting from the computational efficiency of polar form for multiplication and division.