Problem 46 - Algebra 2 and Trigonometry

Question

In \(46-60,\) write each quotient in \(a+b i\) form. $$ (8+4 i) \div(1+i) $$

Step-by-Step Solution

Answer

The quotient in the form \(a + bi\) is \(6 - 2i\).

Step 1: Identify the Problem

We need to express the quotient \((8 + 4i) \div (1 + i)\) in the form \(a + bi\), where \(a\) and \(b\) are real numbers.

Step 2: Multiply Numerator and Denominator by the Conjugate

To eliminate the imaginary part in the denominator, multiply both the numerator \((8 + 4i)\) and the denominator \((1 + i)\) by the conjugate of the denominator, which is \(1 - i\). This gives:\[(8 + 4i) \cdot (1 - i) \quad \text{and} \quad (1 + i) \cdot (1 - i)\]

Step 3: Simplify the Denominator

Calculate the denominator:\[(1+i)(1-i) = 1^2 - i^2 = 1 - (-1) = 2\]The denominator simplifies to 2.

Step 4: Expand the Numerator

Expand the expression \((8 + 4i)(1 - i)\):\[8 \cdot 1 + 8 \cdot (-i) + 4i \cdot 1 + 4i \cdot (-i)\]This simplifies to:\[8 - 8i + 4i - 4i^2\]Since \(i^2 = -1\), the expression simplifies to:\[8 - 4i + 4 = 12 - 4i\]

Step 5: Write the Quotient in Standard Form

Now divide each term in the numerator by the denominator:\[\frac{12 - 4i}{2} = \frac{12}{2} - \frac{4i}{2} = 6 - 2i\]So, the quotient in the form \(a + bi\) is \(6 - 2i\).

Key Concepts

Imaginary Numbers

In the world of complex numbers, imaginary numbers are the elements that help us to extend the real number system, allowing us to handle equations that don't have solutions within real numbers alone. An imaginary number is of the form \( bi \), where \( b \) is a real number, and \( i \) represents the imaginary unit. The imaginary unit \( i \) has the special property that \( i^2 = -1 \). This property is crucial because it helps us to perform algebraic operations involving square roots of negative numbers, which aren't possible using just real numbers.To have a deeper insight, consider how "imaginary" numbers allow the square root of negative numbers to exist. For instance, the square root of \(-1\) is \( i \). Consequently, imaginary numbers aren't 'imaginary' in the sense of not being real—they are vital in making a more comprehensive mathematical system, informing not just advanced mathematics, but also electrical engineering, physics, and many applied sciences.Through imaginary numbers, the complex numbers are formed, which include both real and imaginary parts. The number \( 6 - 2i \), for instance, combines a real part \( (6) \) and an imaginary part \( (-2i) \), forms the solution to our division problem from the original exercise.

Conjugate

To effectively manage divisions involving complex numbers, the concept of conjugates is essential. The conjugate of a complex number involves changing the sign of the imaginary part. For a number \( a + bi \), its conjugate is \( a - bi \). This operation is tremendously useful, particularly in the division process because it removes the imaginary part from the denominator.In our exercise, dividing the complex numbers \((8 + 4i) \div (1 + i)\) required using the conjugate of the denominator \((1 - i)\). By multiplying the numerator and the denominator by the conjugate, the imaginary components in the denominator cancel out:- The product \((1+i)(1-i)\) simplifies the denominator since \(1^2 - i^2 = 1 + 1 = 2\).By using conjugates, we manage to transform complex division problems into simpler algebraic forms. This method not only simplifies calculations but also ensures the resulting form, \( a + bi \), remains standard, a combination of real and imaginary with no imaginary denominator.

Complex Division

Complex division is straightforward when approached methodically with the right techniques, such as multiplying by the conjugate to simplify the expression. The goal is to express the division \((8 + 4i) \div (1 + i)\) in the standard form \( a + bi \), where both \( a \) and \( b \) are real numbers.First, identify the conjugate of the denominator, which is \(1 - i\), and multiply both the numerator and the denominator by this conjugate:

• The multiplication of both results in the denominator being \(2\), with the expression \((1 + i)(1 - i) = 2\).
• The numerator after multiplication becomes \(12 - 4i\).

After simplifying the expression, it becomes clear that each term in the numerator \(12 - 4i \) can be divided by \(2\), leading to the simplified result of \(6 - 2i\).By using the process of multiplying by conjugates in complex division, one effectively reduces the problem to simplified algebra while preserving the real number properties in the quotient. As such, understanding this technique is crucial for mastering complex number operations, whether in mathematics exercises or practical applications in science and engineering.