Problem 25 - College Algebra with Modeling and Visualization

Question

Write the expression in standard form. $$ 2 i-(-5+23 i) $$

Step-by-Step Solution

Answer

The expression in standard form is \(5 - 21i\).

Step 1: Distribute Negative Sign

Start by distributing the negative sign through the expression \(-(-5 + 23i)\). This becomes \(2i + 5 - 23i\).

Step 2: Combine Like Terms

Combine the real parts and the imaginary parts separately. The real part is \(5\), and the imaginary parts are \(2i - 23i\), which combine to \(-21i\).

Step 3: Write in Standard Form

In standard form, a complex number is written as \(a + bi\). So, the expression becomes \(5 - 21i\).

Key Concepts

Standard Form of Complex Numbers

Complex numbers are a fascinating part of mathematics that bring together real and imaginary components into one entity. The standard form of a complex number is written as \(a + bi\), where \(a\) represents the real part and \(b\) the imaginary part. This format allows us to easily identify both components of the complex number.
The standard form is important for simplifying and accurately presenting complex numbers.
When writing a complex number in standard form, make sure that all real numbers are grouped first, followed by the imaginary numbers.
In the example given, the expression \(2i - (-5 + 23i)\) is transformed through distribution and combination of terms to result in \(5 - 21i\), showcasing the clear distinction and arrangement expected in standard form.
Using this format makes it easier to perform operations like addition, subtraction, multiplication, and division, because each part can be manipulated independently.

Imaginary Numbers

Imaginary numbers may initially seem abstract, but they are simply multiples of the imaginary unit \(i\), where \(i = \sqrt{-1}\).
This unit allows us to work with square roots of negative numbers, which are not possible with normal real numbers. Imaginary numbers expand our numerical system to include solutions to equations that don't have real solutions.
In our expression \(2i - (-5 + 23i)\), both terms \(2i\) and \(23i\) are imaginary numbers because they involve multipliers of \(i\).
When you combine these numbers, you use normal addition or subtraction, treating \(i\) like a variable or constant. Imaginary numbers allow for more complex calculations and are essential in fields like engineering and physics, where they help describe oscillations and waveforms.

Real and Imaginary Parts

Every complex number has two main components: the real part and the imaginary part. These parts can be identified directly when a complex number is in its standard form \(a + bi\). The "a" in this form represents the real part, and "b" represents the imaginary component.
In the given expression, once simplified, we obtain \(5 - 21i\). Here, \(5\) is the real part, and \(-21i\) is the imaginary part.
Understanding the role of each part is essential, as it helps in simplifying expressions and performing arithmetic operations.
This separation facilitates solving equations and provides a clear method to measure and understand complex phenomena, like electrical currents or quantum mechanics.
By separating and individually manipulating these parts, one can comprehend the overall behavior of a complex number in mathematical operations and real-world applications.