Problem 30 - College Algebra

Question

For the following exercises, perform the indicated operation and express the result as a simplified complex number. $$ \frac{-5+3 i}{2 i} $$

Step-by-Step Solution

Answer

The simplified complex number is \(\frac{3}{2} + \frac{5i}{2}\).

Step 1: Express the Denominator as a Real Number

We first multiply both the numerator and the denominator by the complex conjugate of the denominator. Here, the denominator is \(2i\), and its complex conjugate is \(-2i\). Multiply both the numerator and the denominator by \(-2i\):\[\frac{-5 + 3i}{2i} \times \frac{-2i}{-2i} = \frac{(-5 + 3i)(-2i)}{(2i)(-2i)}\]

Step 2: Multiply the Numerator

Next, we need to multiply the numerator:\[(-5 + 3i)(-2i) = -5(-2i) + 3i(-2i) = 10i - 6i^2\]Since \(i^2 = -1\), the expression simplifies to:\[10i - 6(-1) = 10i + 6\]

Step 3: Multiply and Simplify the Denominator

Now, multiply the denominator:\[(2i)(-2i) = -4i^2\]Again, replacing \(i^2 = -1\), we have:\[-4(-1) = 4\]

Step 4: Simplify the Fraction

Put the simplified results from the numerator and denominator together:\[\frac{10i + 6}{4}\]Split the fraction to simplify:\[\frac{6}{4} + \frac{10i}{4} = \frac{3}{2} + \frac{5i}{2}\]

Step 5: Final Expression as a Complex Number

The expression \(\frac{3}{2} + \frac{5i}{2}\) is now written in the standard form of a complex number, \(a + bi\), where \(a = \frac{3}{2}\) and \(b = \frac{5}{2}\).

Key Concepts

Complex Conjugate

The complex conjugate is a fundamental concept when dealing with complex numbers. It is used to simplify expressions, particularly those involving division of complex numbers. A complex conjugate is formed by changing the sign of the imaginary part of a complex number. For a complex number of the form \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit, the complex conjugate would be \(a - bi\).

In the exercise given, the complex number is just \(2i\) which is purely imaginary, meaning its real part is zero. Its complex conjugate is \(-2i\). Multiplying a complex number by its conjugate transforms it into a real number. This is particularly useful in simplifying complex fractions, as it allows us to eliminate the imaginary component from the denominator.

Simplifying Fractions

Simplifying complex fractions involves reducing them to a simpler form. When a fraction has a complex number in the denominator, the goal is to make the denominator real. This is achieved by multiplying the numerator and the denominator by the complex conjugate of the denominator.

Consider the fraction \(\frac{-5 + 3i}{2i}\). Here, multiplying both the numerator and the denominator by \(-2i\), the complex conjugate of \(2i\), allows us to simplify the denominator to a real number \(4\). After performing the multiplication and combining like terms, the numerator becomes \(10i + 6\). The result is finally split into real and imaginary components, giving us \(\frac{3}{2} + \frac{5i}{2}\). This process simplifies the calculation and expresses the fraction in a cleaner form.

Imaginary Unit

The imaginary unit \(i\) is a crucial element in complex numbers, defined as \(i = \sqrt{-1}\). This definition leads to the identity \(i^2 = -1\), which is frequently used in calculations involving complex numbers.

In our solution, multiplying \(3i\) by \(-2i\) yields \(-6i^2\). Given that \(i^2 = -1\), this simplifies to \(6\). The property \(i^2 = -1\) is pivotal in simplifying expressions that involve powers of \(i\). Understanding \(i\) and its properties ensures that complex calculations are accurate and manageable.

Standard Form of Complex Numbers

The standard form of a complex number is expressed as \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part. This form provides a clear, structured presentation of complex numbers, facilitating arithmetic operations and comparisons.

In our exercise, we simplified \(\frac{-5 + 3i}{2i}\) to the standard form \(\frac{3}{2} + \frac{5i}{2}\). The real part \(\frac{3}{2}\) and the imaginary part \(\frac{5i}{2}\) illustrate the result in a recognizable and functional format. Presenting complex numbers in standard form is crucial for clarity and understanding, making further analysis or computation much more straightforward.