**Exploring Mathematical Frontiers: Unveiling the Quotient of 1 i 3 4i**

In the realm of complex numbers, there lies a curious expression that has perplexed mathematicians for centuries: the quotient of 1+3i and 4i. This enigmatic mathematical puzzle has captivated the minds of scholars and ignited a quest for its elusive solution. Embark on an intellectual journey as we delve into the intricacies of complex numbers and unravel the secrets held within this enigmatic expression.

**Unveiling the Enigma: A Journey into Complex Numbers**

The concept of complex numbers expands the realm of real numbers by introducing imaginary numbers, denoted by the symbol ‘i’ and defined as the square root of -1. This seemingly paradoxical entity opens up a new dimension in mathematical exploration, allowing for the representation and manipulation of quantities that cannot be expressed using real numbers alone. The quotient of 1+3i and 4i becomes a gateway into this fascinating world of complex numbers.

**Unveiling the Quotient: A Formulaic Revelation**

The quest to determine the quotient of 1+3i and 4i leads us to the following formula:

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

Applying the properties of complex numbers, we can simplify this expression further:

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

Utilizing the identity i^2 = -1, we arrive at the final result:

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

```
= (-1 / 4)
```

Therefore, the quotient of 1+3i and 4i is equal to -1/4.

**Conclusion: Unveiling the Hidden Truths**

Our exploration into the quotient of 1+3i and 4i has unveiled the power and intricacies of complex numbers. We have witnessed the transformation of a看似复杂的expression into a simplified result, shedding light on the underlying mathematical principles. This journey not only provides a solution to the initial query but also opens doors to further exploration in the realm of complex numbers and beyond.

**Quotient of 1 + 3/4i:**

### Understanding Complex Numbers:

Complex numbers, denoted by the symbol ‘C’, are numbers that have both real and imaginary parts. The real part is the same as the real numbers we use in everyday life, while the imaginary part is a multiple of the imaginary unit ‘i’, which is defined as the square root of -1.

### Representation of Complex Numbers:

Complex numbers can be represented in the following form:

$$Z = a + bi$$

Where ‘a’ is the real part, ‘b’ is the imaginary part, and ‘i’ is the imaginary unit.

### Complex Numbers in Quotients:

When dividing complex numbers, we can use the concept of the complex conjugate. The complex conjugate of a complex number ‘Z = a + bi’ is given by ‘Z* = a – bi’.

### Finding the Quotient:

To find the quotient of two complex numbers, we can use the following steps:

- Multiply the numerator and denominator by the complex conjugate of the denominator.
- Simplify the resulting expression by expanding and combining like terms.
- Express the result in the standard form of a complex number (‘a + bi’).

### Example: Quotient of 1 + 3/4i:

Given the complex number ‘Z = 1 + 3/4i’, we can find its quotient as follows:

- Multiply the numerator and denominator by the complex conjugate of the denominator:

$$ frac{1 + 3/4i}{1 – 3/4i} = frac{(1 + 3/4i)(1 + 3/4i)}{(1 – 3/4i)(1 + 3/4i)} $$

- Expand and simplify the resulting expression:

$$ = frac{(1 + 3/4i)(1 + 3/4i)}{1^2 – (3/4i)^2} $$

$$ = frac{(1 + 3/4i)(1 + 3/4i)}{1 – 9/16i^2} $$

$$ = frac{1 + 3/4i + 3/4i + (9/16)i^2}{1 + 9/16} $$

$$ = frac{1 + 3/2i – 9/16}{25/16} $$

- Express the result in the standard form:

$$ = frac{16}{25} + frac{3}{2} cdot frac{16}{25}i $$

$$ = frac{16}{25} + frac{24}{25}i $$

Therefore, the quotient of 1 + 3/4i is:

$$ frac{1 + 3/4i}{1 – 3/4i} = frac{16}{25} + frac{24}{25}i $$

### Additional Points:

- The quotient of two complex numbers results in a complex number.
- The process of finding the quotient involves multiplying by the complex conjugate of the denominator.
- The simplification involves expanding, combining like terms, and expressing the result in the standard form of a complex number.

### Conclusion:

The quotient of two complex numbers can be found by multiplying the numerator and denominator by the complex conjugate of the denominator, simplifying the resulting expression, and expressing the result in the standard form of a complex number. This concept is useful in various mathematical applications, including electrical engineering, physics, and signal processing.

### FAQs:

**What is the complex conjugate of a complex number?**

- The complex conjugate of a complex number ‘Z = a + bi’ is ‘Z* = a – bi’.

**Why do we multiply by the complex conjugate in finding the quotient of complex numbers?**

- Multiplying by the complex conjugate helps simplify the expression and eliminate the imaginary part in the denominator.

**What is the quotient of 1 + 3/4i and 1 – 3/4i?**

- The quotient is (16/25) + (24/25)i.

**Can the quotient of two complex numbers be a real number?**

- Yes, the quotient of two complex numbers can be a real number if the imaginary part of the quotient is zero.

**What are some applications of complex numbers in real-world scenarios?**

- Complex numbers are used in various fields, including electrical engineering (for analyzing AC circuits), physics (for describing wave phenomena), and signal processing (for analyzing and manipulating signals).

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