Problem 88 - Intermediate Algebra

Question

Divide. Write all answers in the form \(a+b i.\) $$ -\frac{8}{5 i} $$

Step-by-Step Solution

Answer

The solution is \(0 - \frac{8}{5}i\).

Step 1: Identify the Division

The problem involves dividing \(-\frac{8}{5i}\). We'll need to simplify this expression to the form \(a + bi\), where \(a\) and \(b\) are real numbers.

Step 2: Multiply by the Conjugate

To eliminate the imaginary denominator, we'll multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \(5i\) is \(-5i\). Now multiply: \[-\frac{8}{5i} \times \frac{-5i}{-5i} = \frac{8 \times (-5i)}{5i \times (-5i)}\].

Step 3: Simplify the Numerator

Multiply the numerators: \[8 \times (-5i) = -40i\]. So, the numerator of the new fraction is \(-40i\).

Step 4: Simplify the Denominator

Multiply the denominators: \[5i \times (-5i) = -25i^2\]. Since \(i^2 = -1\), substitute \(-1\) for \(i^2\), yielding: \(-25(-1) = 25\). So, the denominator is \(25\).

Step 5: Write the Fraction in Standard Form

Now, we have: \[\frac{-40i}{25}\]. Divide the numerator and the denominator by 5: \[= \frac{-8i}{5}\]. This can be written in the form \(0 - \frac{8}{5}i\).

Key Concepts

Imaginary Numbers

Imaginary numbers are numbers that give a real number when squared results in a negative number. The unit of an imaginary number is represented by the symbol \(i\), and it is defined as the square root of -1. Mathematically, this is written as \(i = \sqrt{-1}\).
So, when squared, \(i^2 = -1\), which is a crucial identity used in manipulating complex numbers.

Imaginary numbers are used in various applications, including engineering and physics, where they solve problems involving waveforms, electrical currents, and control systems.

• They extend the concept of number beyond the real numbers.
• In combination with real numbers, they form complex numbers.
• Remember that \(i^2 = -1\), which is essential in calculations.

understanding imaginary numbers simplify the manipulation of complex divisions, especially when eliminating imaginary components from denominators.

Division of Complex Numbers

Dividing complex numbers might seem challenging, but it becomes straightforward once you understand the process. When you encounter a division involving a complex number as the denominator, like dividing by \(5i\), it helps to multiply both numerator and denominator by the conjugate of the denominator.
This process is key because it removes the imaginary part from the denominator, allowing for a standard division.

The reason why using the conjugate works is because multiplying a number by its conjugate forms a real number. Check out this quick list:

• Always identify the conjugate of the denominator first.
• Multiply both the numerator and denominator by this conjugate.
• This results in a real number as the denominator.

This multiplication helps in expressing the result of a division in the standard form \(a+bi\). The steps follow a logical sequence, helping solve complex expressions easily.

Conjugate of a Complex Number

A complex number like \(z = a + bi\) has its conjugate defined as \(z^* = a - bi\). The concept of conjugates is vital, particularly when it comes to division in complex numbers. It helps eliminate imaginary parts from the denominator.
Consider you have a complex number \(5i\) as in our example, its conjugate would be \(-5i\). By multiplying by the conjugate:

• The product of the imaginary component and its conjugate results in a real number because of the identity \(i^2 = -1\).
• This eliminates the imaginary part present in the denominator completely, allowing for algebraic simplification.
• Use conjugates in the division process to transform complex fractions into more manageable forms.

It simplifies the calculation significantly, turning complex expressions into a recognizable form \(a+bi\). By understanding conjugates, you unlock a critical tool for managing complex calculations.