Square Root of 1-i: A Comprehensive Guide

The square root of a complex number can be found using various methods. Here, we will find the square root of the complex number \(1 - i\).

Method 1: Algebraic Approach

To find the square root of \(1 - i\), we assume it can be expressed in the form \(a + bi\). Thus, we need to solve:

\[(a + bi)^2 = 1 - i\]

Expanding the left-hand side, we get:

\[a^2 + 2abi + (bi)^2 = 1 - i\]

Simplifying, we have:

\[a^2 - b^2 + 2abi = 1 - i\]

This results in the system of equations:

• \(a^2 - b^2 = 1\)
• \(2ab = -1\)

From the second equation, we can express \(b\) in terms of \(a\):

\[b = -\frac{1}{2a}\]

Substituting this into the first equation, we get:

\[a^2 - \left(-\frac{1}{2a}\right)^2 = 1\]

Which simplifies to:

\[a^4 - a^2 - \frac{1}{4} = 0\]

Solving this quadratic equation in \(a^2\), we find:

\[a^2 = \frac{1}{2} \left(1 \pm \sqrt{5}\right)\]

Since \(a\) must be real, we select the positive root. Thus:

\[a = \pm \sqrt{\frac{1 + \sqrt{5}}{2}}\]

And correspondingly:

\[b = \mp \sqrt{\frac{\sqrt{5} - 1}{2}}\]

Hence, the two square roots of \(1 - i\) are:

• \(\sqrt{\frac{1 + \sqrt{5}}{2}} - i\sqrt{\frac{\sqrt{5} - 1}{2}}\)
• \(-\sqrt{\frac{1 + \sqrt{5}}{2}} + i\sqrt{\frac{\sqrt{5} - 1}{2}}\)

Method 2: Polar Form

Alternatively, we can find the square root using the polar form of complex numbers. First, convert \(1 - i\) to its polar form:

\[1 - i = \sqrt{2} e^{-i\pi/4}\]

The square roots are then given by:

\[\sqrt{\sqrt{2}} e^{-i\pi/8} \text{ and } -\sqrt{\sqrt{2}} e^{-i\pi/8}\]

Simplifying, we get the same roots as before:

• \(\sqrt[4]{2} \left(\cos\left(-\frac{\pi}{8}\right) + i \sin\left(-\frac{\pi}{8}\right)\right)\)
• \(-\sqrt[4]{2} \left(\cos\left(-\frac{\pi}{8}\right) + i \sin\left(-\frac{\pi}{8}\right)\right)\)

Verification

To verify, we can square these results and confirm they equal \(1 - i\).

This comprehensive approach covers both algebraic and polar methods for finding the square root of \(1 - i\).

The square root of a complex number, such as \(1 - i\), is an intriguing concept that combines elements of algebra, geometry, and complex analysis. Understanding how to calculate the square root of complex numbers can be vital in various fields, including engineering, physics, and applied mathematics. This section will introduce the basic concepts and significance of finding the square root of \(1 - i\), setting the stage for more detailed explorations in subsequent sections.

Complex numbers are numbers that have both a real and an imaginary part, typically expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with the property that \(i^2 = -1\). The number \(1 - i\) is a specific example of a complex number with a real part of 1 and an imaginary part of -1.

Calculating the square root of a complex number involves finding another complex number which, when multiplied by itself, results in the original number. For \(1 - i\), this means finding a complex number \(z = x + yi\) such that:

\[
(x + yi)^2 = 1 - i
\]

This process can be approached using different methods, including the rectangular coordinates method and the polar coordinates method, each providing unique insights and applications. Through these methods, one can not only find the square roots of complex numbers but also gain a deeper understanding of the behavior and properties of these fascinating mathematical entities.

In this guide, we will explore the necessary definitions, properties, and formulas, followed by detailed, step-by-step calculation examples. Whether you're a student, educator, or professional, this comprehensive overview will enhance your understanding and ability to work with complex numbers and their square roots.

Complex numbers are an extension of the real numbers and are fundamental in various branches of mathematics and engineering. A complex number is typically written in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with the property that \(i^2 = -1\).

The number \(a\) is known as the real part, and \(b\) is the imaginary part. For example, in the complex number \(1 - i\), the real part is 1, and the imaginary part is -1.

To understand complex numbers better, it is useful to visualize them on the complex plane. The complex plane is similar to the Cartesian coordinate system but is specifically designed for complex numbers. The horizontal axis (x-axis) represents the real part, while the vertical axis (y-axis) represents the imaginary part.

Here is a summary of key concepts related to complex numbers:

• Real Part: The real component of a complex number, denoted as \(a\) in \(a + bi\).
• Imaginary Part: The coefficient of the imaginary unit \(i\), denoted as \(b\) in \(a + bi\).
• Imaginary Unit: Denoted by \(i\), it satisfies \(i^2 = -1\).
• Complex Conjugate: For a complex number \(z = a + bi\), the complex conjugate is \( \overline{z} = a - bi\).
• Magnitude (or Modulus): The distance of the complex number from the origin in the complex plane, calculated as \(\sqrt{a^2 + b^2}\).
• Argument (or Phase): The angle formed with the positive real axis, usually denoted as \(\theta\) and calculated using \(\tan^{-1}(b/a)\).

Complex numbers can be added, subtracted, multiplied, and divided using specific rules:

• Addition: \((a + bi) + (c + di) = (a + c) + (b + d)i\)
• Subtraction: \((a + bi) - (c + di) = (a - c) + (b - d)i\)
• Multiplication: \((a + bi) \cdot (c + di) = (ac - bd) + (ad + bc)i\)
• Division: \(\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{c^2 + d^2} = \frac{ac + bd}{c^2 + d^2} + \frac{bc - ad}{c^2 + d^2}i\)

These operations allow for a wide range of applications, from solving polynomial equations to analyzing electrical circuits. By mastering the basics of complex numbers, one can unlock a deeper understanding of many mathematical and physical phenomena.

The square root of a complex number involves finding another complex number that, when multiplied by itself, yields the original complex number. For a complex number \(z = a + bi\), its square root can be represented as \(w = x + yi\) such that:

\[
(x + yi)^2 = a + bi
\]

Expanding the left-hand side, we get:

\[
x^2 + 2xyi + (yi)^2 = a + bi
\]

Since \(i^2 = -1\), this can be rewritten as:

\[
x^2 - y^2 + 2xyi = a + bi
\]

For the equality to hold, the real and imaginary parts must be equal on both sides, giving us two equations:

• Real part: \(x^2 - y^2 = a\)
• Imaginary part: \(2xy = b\)

Solving these equations simultaneously will yield the values of \(x\) and \(y\) which form the square roots of the complex number \(a + bi\).

Key properties of the square root of a complex number include:

• Multiple Solutions: Every non-zero complex number has two square roots. If \(w\) is a square root of \(z\), then \(-w\) is also a square root of \(z\).
• Principal Square Root: By convention, the principal square root of a complex number is the one with the non-negative real part. If both square roots have the same real part, the one with the non-negative imaginary part is chosen.
• Conjugate Property: The square roots of a complex number and its conjugate are conjugates of each other. If \(w\) is a square root of \(z\), then \(\overline{w}\) is a square root of \(\overline{z}\).
• Magnitude: The magnitudes of the square roots of \(z = a + bi\) are given by \(\sqrt{r}\), where \(r = |z|\). This is because \(|w|^2 = |z|\).
• Argument: The arguments of the square roots of \(z\) are \(\frac{\theta}{2}\) and \(\frac{\theta}{2} + \pi\), where \(\theta\) is the argument of \(z\).

To find the square roots of a complex number, two primary methods are often used: the rectangular coordinates method and the polar coordinates method. Both approaches yield the same results but offer different perspectives and computational advantages.

Understanding these properties and methods allows for effective calculation and application of complex number square roots in various mathematical and engineering contexts.

Calculating the square root of a complex number can be approached using either the rectangular coordinates method or the polar coordinates method. Each method has its own set of formulas and steps, providing different insights and computational techniques.

Rectangular Coordinates Method

Given a complex number \(z = a + bi\), the goal is to find \(w = x + yi\) such that:

\[
(x + yi)^2 = a + bi
\]

This equation leads to two simultaneous equations:

• Real part: \(x^2 - y^2 = a\)
• Imaginary part: \(2xy = b\)

To solve these, follow these steps:

1. Compute the magnitude \(r\) of the complex number \(z\):

\[
r = \sqrt{a^2 + b^2}
\]

2. Find \(x\) and \(y\) using the following formulas:
• \[ x = \pm \sqrt{\frac{r + a}{2}} \]
• \[ y = \pm \sqrt{\frac{r - a}{2}} \]
3. Select the correct signs for \(x\) and \(y\) to satisfy \(2xy = b\).

Polar Coordinates Method

Using the polar form of a complex number \(z\), which is written as \(z = re^{i\theta}\), where \(r\) is the magnitude and \(\theta\) is the argument, the square root can be calculated as follows:

Given \(z = re^{i\theta}\), the square roots are:

• \[ w_1 = \sqrt{r}e^{i\theta/2} \]
• \[ w_2 = \sqrt{r}e^{i(\theta/2 + \pi)} \]

These formulas yield the two square roots of the complex number.

Example: Square Root of \(1 - i\)

Using the rectangular coordinates method for \(z = 1 - i\):

• Compute the magnitude:

\[
r = \sqrt{1^2 + (-1)^2} = \sqrt{2}
\]

• Find \(x\) and \(y\):
• \[ x = \pm \sqrt{\frac{\sqrt{2} + 1}{2}} \]
• \[ y = \pm \sqrt{\frac{\sqrt{2} - 1}{2}} \]
• Select the correct signs for \(x\) and \(y\) to satisfy \(2xy = -1\).

Using the polar coordinates method for \(z = 1 - i\):

• Convert \(1 - i\) to polar form:

\[
r = \sqrt{2}, \quad \theta = -\frac{\pi}{4}
\]

• Compute the square roots:
• \[ w_1 = \sqrt{\sqrt{2}} e^{-i\pi/8} \]
• \[ w_2 = \sqrt{\sqrt{2}} e^{i(3\pi/8)} \]

Both methods yield the same roots, providing a comprehensive understanding of how to calculate the square root of a complex number.

The rectangular coordinates method involves working directly with the real and imaginary parts of the complex number to find its square roots. This method is particularly useful for understanding the algebraic relationships between these parts.

Given a complex number \( z = a + bi \), we want to find \( w = x + yi \) such that:

\[
(x + yi)^2 = a + bi
\]

Expanding and equating the real and imaginary parts gives us two equations:

• Real part: \( x^2 - y^2 = a \)
• Imaginary part: \( 2xy = b \)

Follow these steps to solve for \(x\) and \(y\):

1. Calculate the magnitude \( r \) of \( z \):

\[
r = \sqrt{a^2 + b^2}
\]

2. Find \( x \) and \( y \) using the formulas:
• \[ x = \pm \sqrt{\frac{r + a}{2}} \]
• \[ y = \pm \sqrt{\frac{r - a}{2}} \]
3. Select the correct signs for \( x \) and \( y \) to satisfy \( 2xy = b \).

Let's apply this method to find the square roots of \( 1 - i \).

Example: Square Root of \( 1 - i \)

Given \( z = 1 - i \):

1. Calculate the magnitude \( r \):

\[
r = \sqrt{1^2 + (-1)^2} = \sqrt{2}
\]

2. Find \( x \) and \( y \) using the formulas:
• \[ x = \pm \sqrt{\frac{\sqrt{2} + 1}{2}} \]
• \[ y = \pm \sqrt{\frac{\sqrt{2} - 1}{2}} \]
3. Determine the correct signs for \( x \) and \( y \):

To satisfy \( 2xy = -1 \), we find:

• For \( x = \sqrt{\frac{\sqrt{2} + 1}{2}} \) and \( y = -\sqrt{\frac{\sqrt{2} - 1}{2}} \):

\[
2 \left(\sqrt{\frac{\sqrt{2} + 1}{2}}\right) \left(-\sqrt{\frac{\sqrt{2} - 1}{2}}\right) = -1
\]

Thus, the square roots of \( 1 - i \) are:

• \[ w_1 = \sqrt{\frac{\sqrt{2} + 1}{2}} - \sqrt{\frac{\sqrt{2} - 1}{2}}i \]
• \[ w_2 = -\sqrt{\frac{\sqrt{2} + 1}{2}} + \sqrt{\frac{\sqrt{2} - 1}{2}}i \]

The rectangular coordinates method provides a clear algebraic approach to finding the square roots of complex numbers, ensuring a comprehensive understanding of the relationships between their real and imaginary parts.

The polar coordinates method involves expressing a complex number in its polar form and then finding its square roots. This method provides a geometric perspective and can simplify calculations, especially when dealing with multiplication and division of complex numbers.

Given a complex number \( z = a + bi \), it can be represented in polar form as \( z = re^{i\theta} \), where \( r \) is the magnitude (or modulus) and \( \theta \) is the argument (or angle) of the complex number.

The steps to find the square roots using the polar coordinates method are as follows:

1. Calculate the magnitude \( r \) of \( z \):

\[
r = \sqrt{a^2 + b^2}
\]

2. Determine the argument \( \theta \) of \( z \):

\[
\theta = \tan^{-1}\left(\frac{b}{a}\right)
\]

3. Compute the magnitudes of the square roots:

\[
\sqrt{r}
\]

4. Find the arguments of the square roots. There are two square roots, corresponding to the arguments:
• \[ \theta_1 = \frac{\theta}{2} \]
• \[ \theta_2 = \frac{\theta}{2} + \pi \]
5. Express the square roots in polar form:
• \[ w_1 = \sqrt{r}e^{i\theta_1} \]
• \[ w_2 = \sqrt{r}e^{i\theta_2} \]

Example: Square Root of \( 1 - i \)

Given \( z = 1 - i \), follow these steps to find its square roots:

1. Calculate the magnitude \( r \):

\[
r = \sqrt{1^2 + (-1)^2} = \sqrt{2}
\]

2. Determine the argument \( \theta \):

\[
\theta = \tan^{-1}\left(\frac{-1}{1}\right) = -\frac{\pi}{4}
\]

3. Compute the magnitudes of the square roots:

\[
\sqrt{\sqrt{2}}
\]

4. Find the arguments of the square roots:
• \[ \theta_1 = -\frac{\pi}{8} \]
• \[ \theta_2 = -\frac{\pi}{8} + \pi = \frac{7\pi}{8} \]
5. Express the square roots in polar form:
• \[ w_1 = \sqrt{\sqrt{2}}e^{-i\pi/8} \]
• \[ w_2 = \sqrt{\sqrt{2}}e^{i7\pi/8} \]

Convert these results back to rectangular form for verification:

• For \( w_1 \):

\[
w_1 = \sqrt{\sqrt{2}} \left( \cos\left(-\frac{\pi}{8}\right) + i\sin\left(-\frac{\pi}{8}\right) \right)
\]

• For \( w_2 \):

\[
w_2 = \sqrt{\sqrt{2}} \left( \cos\left(\frac{7\pi}{8}\right) + i\sin\left(\frac{7\pi}{8}\right) \right)
\]

Using the polar coordinates method, we gain a geometric understanding of the square roots of complex numbers and can easily perform the calculations, especially for numbers with significant imaginary components.

Let's explore step-by-step calculations for finding the square root of \(1-i\).

1. Express the complex number in polar form:

   \(1 - i\)	= \(r (\cos \theta + i \sin \theta)\)
   	= \(r \operatorname{cis} \theta\)

2. Calculate the modulus \(r\):
   • Modulus (\(r\)) = \(\sqrt{a^2 + b^2}\) where \(a = 1\) and \(b = -1\)
   • Modulus (\(r\)) = \(\sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2}\)
3. Calculate the argument (\(\theta\)):
   • Argument (\(\theta\)) = \(\tan^{-1}\left(\frac{b}{a}\right)\) where \(a = 1\) and \(b = -1\)
   • Argument (\(\theta\)) = \(\tan^{-1}\left(\frac{-1}{1}\right) = \tan^{-1}(-1)\)
   • Argument (\(\theta\)) = \(-\frac{\pi}{4}\)
4. Write the complex number in polar form:
   • \(1 - i = \sqrt{2} \operatorname{cis} \left(-\frac{\pi}{4}\right)\)
5. Find the principal square root:
   • The principal square root is given by \(\sqrt{r} \operatorname{cis} \left(\frac{\theta}{2}\right)\)
   • Substitute \(r = \sqrt{2}\) and \(\theta = -\frac{\pi}{4}\) into the formula
   • The principal square root of \(1 - i\) is \(\sqrt{\sqrt{2}} \operatorname{cis} \left(\frac{-\pi}{8}\right)\)
   • For numerical approximation, \(\sqrt{\sqrt{2}} \approx 1.189\)

Let's demonstrate how to find the square root of \(1-i\) using rectangular coordinates:

1. Express the complex number \(1 - i\) in rectangular form:

1. Apply the square root formula for rectangular coordinates:

1. Calculate \(x\) and \(y\) using the formulas:

1. Perform the calculations to find \(x\) and \(y\):

1. Write the square root in rectangular form:

Let's illustrate how to find the square root of \(1-i\) using polar coordinates:

1. Express the complex number \(1 - i\) in polar form:

1. Calculate the modulus \(r\):

1. Calculate the argument \(\theta\):

1. Write the complex number in polar form:

1. Find the principal square root:

Let's explore some applications of complex number square roots:

1. Engineering and Physics: Complex numbers are extensively used in engineering and physics, especially in fields like electrical engineering and quantum mechanics. Square roots of complex numbers are crucial in solving various equations and analyzing circuit behaviors.
2. Signal Processing: Complex numbers play a fundamental role in signal processing applications such as Fourier analysis and digital signal processing. Square roots of complex numbers are utilized in understanding and manipulating signals.
3. Control Systems: Complex numbers are indispensable in control systems engineering. Square roots of complex numbers help in analyzing stability and performance of control systems, designing controllers, and predicting system responses.
4. Quantum Computing: Complex numbers are fundamental in quantum computing, where quantum bits (qubits) can be represented by complex vectors. Square roots of complex numbers are essential in quantum algorithms and quantum information processing.
5. Mathematical Modeling: Complex numbers are used in mathematical modeling to describe physical phenomena with real and imaginary components. Square roots of complex numbers aid in solving differential equations, modeling wave phenomena, and analyzing dynamical systems.

Here are some commonly asked questions about the square root of \(1-i\):

1. What is the square root of \(1-i\)?

The square root of \(1-i\) is a complex number that, when squared, equals \(1-i\). It can be expressed in both rectangular and polar forms.

2. How do you find the square root of \(1-i\)?

To find the square root of \(1-i\), you can use either rectangular or polar coordinates. In rectangular coordinates, you express \(1-i\) as \(a + bi\) and apply the square root formula. In polar coordinates, you express \(1-i\) as \(r \operatorname{cis} \theta\) and calculate the modulus \(r\) and argument \(\theta\) to find the principal square root.

3. What are the applications of the square root of \(1-i\)?

The square root of \(1-i\) has various applications across different fields such as engineering, physics, signal processing, control systems, quantum computing, and mathematical modeling. It is used in solving equations, analyzing circuit behaviors, processing signals, designing controllers, quantum algorithms, and modeling physical phenomena.

4. Why is the square root of \(1-i\) important?

The square root of \(1-i\) is important because it represents a fundamental concept in complex numbers and has wide-ranging applications in science, engineering, and mathematics. Understanding how to calculate and manipulate complex number square roots is essential for solving problems in various domains.

In conclusion, understanding the square root of \(1-i\) and its calculation methods is essential for various mathematical and practical applications. Whether using rectangular or polar coordinates, the process involves expressing the complex number and determining its modulus and argument to find the principal square root.

The applications of complex number square roots span across disciplines such as engineering, physics, signal processing, control systems, quantum computing, and mathematical modeling. Mastery of complex number operations, including square roots, empowers individuals to solve intricate problems and advance scientific knowledge.

By exploring the step-by-step calculation examples, applications, and frequently asked questions, it's evident that the square root of \(1-i\) is a foundational concept with profound implications. Embracing its complexity leads to a deeper understanding of mathematical principles and fosters innovation in various fields.