What Is The Quotient Of The Complex Number 4-3i

Diving into the world of complex numbers often feels like stepping into a realm of mathematical artistry. The complex number 4-3i isn't just a random expression; it's a fundamental building block in various fields, from electrical engineering to quantum physics. Understanding its quotient is crucial for manipulating and solving problems involving complex numbers.

Unveiling Complex Numbers

Before we dive into finding the quotient, let's briefly recap what complex numbers are. A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as the square root of -1. The a part is called the real part, and the b part is called the imaginary part.

So, in our case, 4-3i has a real part of 4 and an imaginary part of -3. They're not just abstract concepts; complex numbers provide solutions to equations that real numbers can't handle, expanding the scope of mathematical possibilities.

What Exactly is a Quotient in This Context?

Here's where things get interesting. When we talk about the quotient of a single complex number like 4-3i, we're not referring to division in the traditional sense. Instead, the term "quotient" is often used loosely, and most likely the question is related to one of these concepts:

• The Complex Conjugate: This is the most common interpretation. The complex conjugate of 4-3i is 4+3i.
• Division Involving the Complex Number: This means dividing 1 by the complex number (finding its reciprocal), or dividing another complex number by it. This requires a bit more work.
• Polar Form Representation: While not directly a quotient, converting to polar form involves finding the magnitude and angle, which can be seen as components that define its position in the complex plane.

Let's explore each of these in detail.

1. The Complex Conjugate: A Mirror Image

The complex conjugate is formed by simply changing the sign of the imaginary part. For a complex number a + bi, its conjugate is a - bi. Geometrically, the conjugate is a reflection of the original complex number across the real axis in the complex plane.

Finding the Complex Conjugate of 4-3i

This is straightforward:

The complex conjugate of 4 - 3i is 4 + 3*i.

Why is the Conjugate Important?

• Rationalizing Denominators: Multiplying a complex number by its conjugate results in a real number. This is crucial for dividing complex numbers, as it eliminates the imaginary part from the denominator.
• Finding Modulus (Magnitude): The modulus of a complex number z is denoted as |z| and is calculated as √(z * conjugate of z).
• Solving Equations: Complex conjugates appear in solutions to polynomial equations with real coefficients. If a + bi is a root, then a - bi is also a root.
• Electrical Engineering: In circuit analysis, complex conjugates are used to calculate impedance and power.

2. Division Involving the Complex Number: Finding the Reciprocal

Finding the reciprocal of a complex number, or dividing 1 by it, involves a clever trick using the complex conjugate.

Finding the Reciprocal of 4-3i

To find the reciprocal of 4 - 3i, which is 1 / (4 - 3i), we need to get rid of the imaginary part in the denominator. We do this by multiplying both the numerator and the denominator by the complex conjugate of the denominator:

1 / (4 - 3i) = (1 / (4 - 3i)) * ((4 + 3i) / (4 + 3i))

Let's break down the multiplication:

• Numerator: 1 * (4 + 3i) = 4 + 3i
• Denominator: (4 - 3i) * (4 + 3i) = 44 + 4(3i) - (3i)4 - (3i*)(3i*) = 16 + 12i - 12i - 9i² Since i² = -1, this becomes: = 16 - 9*(-1) = 16 + 9 = 25

Therefore, the reciprocal of 4 - 3i is (4 + 3i) / 25, which can be written as:

Reciprocal = 4/25 + (3/25)i

Dividing a Complex Number by 4-3i

Let's say we want to divide the complex number a + bi by 4 - 3i. We use the same principle as finding the reciprocal:

(a + bi) / (4 - 3i) = ((a + bi) / (4 - 3i)) * ((4 + 3i) / (4 + 3i))

Multiplying this out:

• Numerator: (a + bi) * (4 + 3i) = 4a + 3ai + 4bi* + 3bi*² = 4a + 3ai + 4bi* - 3b = (4a - 3b) + (3a + 4*b)i
• Denominator: As we calculated before, (4 - 3i) * (4 + 3i) = 25

Therefore:

(a + bi) / (4 - 3i) = ((4a - 3b) / 25) + ((3a + 4b) / 25)i

Example:

Let's divide 2 + i by 4 - 3i:

Using the formula above:

• a = 2, b = 1
• Real part: (42 - 31) / 25 = (8 - 3) / 25 = 5/25 = 1/5
• Imaginary part: (32 + 41) / 25 = (6 + 4) / 25 = 10/25 = 2/5

So, (2 + i) / (4 - 3i) = 1/5 + (2/5)i

3. Polar Form Representation: A Different Perspective

While not a "quotient" in the strict division sense, representing a complex number in polar form provides a different way to understand its components and can be useful in certain calculations. The polar form expresses the complex number in terms of its magnitude (distance from the origin in the complex plane) and its angle (the angle it makes with the positive real axis).

A complex number z = a + bi can be represented in polar form as:

z = r (cos θ + i sin θ)

Where:

• r is the magnitude or modulus: r = √(a² + b²)
• θ is the argument or angle: θ = arctan(b / a) (paying attention to the quadrant of the complex number).

Converting 4-3i to Polar Form

1. Find the Magnitude (r):

   r = √(4² + (-3)²) = √(16 + 9) = √25 = 5

2. Find the Angle (θ):

   θ = arctan(-3 / 4)

   Using a calculator, arctan(-3/4) ≈ -0.6435 radians or -36.87 degrees. Since 4-3i lies in the fourth quadrant, this angle is correct.

Therefore, the polar form of 4 - 3i is:

4 - 3i = 5 (cos(-0.6435) + i sin(-0.6435)) or 5 (cos(-36.87°) + i sin(-36.87°))

Why Polar Form Matters:

• Multiplication and Division Made Easy: Multiplying complex numbers in polar form involves multiplying their magnitudes and adding their angles. Division involves dividing magnitudes and subtracting angles.
• Finding Powers and Roots: De Moivre's Theorem provides a straightforward way to calculate powers and roots of complex numbers in polar form.
• Visual Representation: Polar form provides a clear geometric interpretation of the complex number's position in the complex plane.

Further Considerations and Advanced Applications

Understanding these different interpretations of "quotient" in the context of complex numbers unlocks a wide range of possibilities.

• Complex Functions: In complex analysis, functions can take complex numbers as inputs and produce complex numbers as outputs. Understanding quotients and reciprocals is fundamental to analyzing the behavior of these functions.
• Impedance in AC Circuits: Electrical engineers use complex numbers to represent impedance in AC circuits. Calculations involving series and parallel impedances often require finding reciprocals and performing complex division.
• Quantum Mechanics: Complex numbers are essential in quantum mechanics, where wave functions are complex-valued. Calculations involving probabilities and expected values rely on understanding complex conjugates and moduli.
• Signal Processing: Complex numbers are used to represent signals in signal processing. Techniques like Fourier analysis rely heavily on manipulating complex exponentials, which are closely related to the polar form of complex numbers.

Common Mistakes to Avoid

Working with complex numbers can be tricky, so here are some common pitfalls to watch out for:

• Forgetting to Multiply by the Conjugate: When dividing complex numbers, always remember to multiply both the numerator and denominator by the complex conjugate of the denominator. Multiplying only the denominator will not eliminate the imaginary part.
• Incorrectly Calculating the Conjugate: Double-check that you've only changed the sign of the imaginary part when finding the conjugate.
• Ignoring the Quadrant When Finding the Angle: When converting to polar form, use the arctangent function carefully and consider the quadrant of the complex number to determine the correct angle. The arctangent function has a limited range, so you might need to add or subtract 180 degrees (π radians) to get the correct angle.
• Confusing Modulus and Conjugate: The modulus is a real number representing the magnitude, while the conjugate is another complex number.
• Incorrectly Applying De Moivre's Theorem: When using De Moivre's Theorem, make sure you're working with the polar form of the complex number and that you correctly apply the power to both the magnitude and the angle.

Examples and Practice Problems

Let's reinforce our understanding with some examples and practice problems.

Example 1: Find the reciprocal of 1 + i

1. Multiply by the conjugate: (1 / (1 + i)) * ((1 - i) / (1 - i))
2. Simplify: (1 - i) / (1 - i²) = (1 - i) / (1 - (-1)) = (1 - i) / 2
3. Result: 1/2 - (1/2)i

Example 2: Divide 5 - i by 2 + 3i

1. Multiply by the conjugate: ((5 - i) / (2 + 3i)) * ((2 - 3i) / (2 - 3i))
2. Simplify numerator: (5 - i) * (2 - 3i) = 10 - 15i - 2i + 3i² = 10 - 17i - 3 = 7 - 17i
3. Simplify denominator: (2 + 3i) * (2 - 3i) = 4 - 9i² = 4 + 9 = 13
4. Result: (7/13) - (17/13)i

Practice Problems:

1. Find the complex conjugate of -2 + 5i.
2. Find the reciprocal of 3 - 4i.
3. Divide 1 + i by 1 - i.
4. Convert -1 - i to polar form.
5. What is the modulus of 6 + 8i?

Conclusion: Mastering Complex Number Quotients

Understanding the "quotient" of a complex number, whether it's interpreted as the complex conjugate, the reciprocal, or a component of its polar form, is essential for working with these fascinating mathematical entities. By mastering these concepts, you unlock the ability to manipulate complex numbers, solve complex equations, and apply them to various fields of science and engineering. The journey into complex numbers might seem daunting at first, but with practice and a solid understanding of the fundamentals, you can confidently navigate this rich and rewarding area of mathematics. Remember to always double-check your work, pay attention to the details, and don't be afraid to explore the many applications of complex numbers in the real world.