What is the quotient of the complex number 4-3i divided by its conjugate

Quotient of $4 - 3i$ by its Conjugate

To find the quotient of the complex number $4 - 3i$ divided by its conjugate, follow these steps:

1. Identify the conjugate of the complex number $4 - 3i$.

   The conjugate of $4 - 3i$ is $4 + 3i$.

2. Express the division as a fraction:

   $$\frac{4 - 3i}{4 + 3i}$$

3. Multiply numerator and denominator by the conjugate of the denominator:

   $$\frac{(4 - 3i)(4 - 3i)}{(4 + 3i)(4 - 3i)}$$

4. Calculate the numerator:

Expanding the terms:

Since $i^2 = -1$:

5. Calculate the denominator:

Expanding the terms:

Again, since $i^2 = -1$:

6. Express the final result:

   $$\frac{7 - 24i}{25} = \frac{7}{25} - \frac{24i}{25}$$

Hence, the quotient of the complex number $\mathbf{4 - 3i}$ divided by its conjugate is:

In the context provided, this corresponds to the third option: $\mathbf{\frac{7}{25} - \frac{24}{25}i}$.

Explanation

The conjugate of a complex number is an important concept in complex number theory. Given a complex number, its conjugate can be easily determined and used in various mathematical operations such as simplifying division or finding the magnitude.

Definition

If $z$ is a complex number represented as:

where $a$ and $b$ are real numbers, and $i$ is the imaginary unit such that $i^2 = -1$, then the conjugate of $z$ is denoted by $\overline{z}$ and is defined as:

Key Properties

1. Real Part Unchanged: The real part $a$ of the complex number remains unchanged.
2. Imaginary Part Sign Change: The imaginary part $b$ changes its sign.

Example

For a complex number $z = 3 + 4i$:

Usage in Operations

1. Magnitude Calculation: The magnitude (or modulus) of a complex number $z$ is given by:

   $$|z| = \sqrt{z \cdot \overline{z}}$$

   For $z = a + bi$:

   $$|z| = \sqrt{(a + bi)(a - bi)} = \sqrt{a^2 + b^2}$$

2. Division of Complex Numbers: When dividing by a complex number, multiplying the numerator and the denominator by the conjugate of the denominator simplifies the expression:

   $$\frac{z_1}{z_2} \cdot \frac{\overline{z_2}}{\overline{z_2}} = \frac{z_1 \cdot \overline{z_2}}{|z_2|^2}$$

Concluding Remarks

Understanding the conjugate of a complex number aids significantly in complex number arithmetic and has applications in solving equations, electrical engineering, and quantum physics where complex numbers are frequently used. The conjugate simplifies many mathematical processes involving complex numbers by neutralizing the imaginary part when necessary.

Overview

The multiplication of complex numbers follows the distributive property. Complex numbers are expressed in the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit with the property that $i^2 = -1$.

Steps for Multiplying Complex Numbers

1. Distribute each term: Multiply each part of the first complex number by each part of the second complex number.
2. Combine like terms: Group and combine the real parts and the imaginary parts.
3. Simplify: Use the property $i^2 = -1$ to simplify the expression.

Illustration with an Example

Suppose we want to multiply the complex numbers $(3 + 4i)$ and $(2 + 5i)$.

1. Distribute using the FOIL method:

2. Multiply each pair:

   $$= 6 + 15i + 8i + 20i^2$$

3. Combine like terms and use $i^2 = -1$ to simplify:

   $$6 + 23i + 20(-1)$$ $$= 6 + 23i - 20$$

4. Final result:

   $$= -14 + 23i$$

Summary

• Distribute terms and apply the property $i^2 = -1$.
• Combine like terms for your final result.
• The output will be in the form $a + bi$, representing a complex number.

The multiplication of complex numbers is fundamentally an application of the distributive property combined with the unique properties of the imaginary unit $i$.