Problem 52 - A Graphical Approach to College Algebra

Question

Add or subtract as indicated. Write each sum or difference in standard form. $$(-4-i)-(2+3 i)+(-4+5 i)$$

Step-by-Step Solution

Answer

The expression simplifies to \(-10 + i\).

Step 1: Remove Parentheses

Start by writing out the expression without parentheses. When removing parentheses, remember to distribute the negative sign that is in front of any expressions. \(-4 - i - 2 - 3i - 4 + 5i\)

Step 2: Combine Like Terms (Real Parts)

Group and combine the real number terms from the expression.Real parts: \(-4 - 2 - 4 = -10\)

Step 3: Combine Like Terms (Imaginary Parts)

Group and combine the imaginary number terms from the expression.Imaginary parts:\(-i - 3i + 5i = -1i - 3i + 5i = 1i\)

Step 4: Write in Standard Form

Combine the simplified real and imaginary parts to write the final result in the standard form of a complex number, which is \(a + bi\).\(- \(-10 + 1i\)\)

Key Concepts

Standard Form

In the world of complex numbers, expressing numbers in the standard form is highly significant. The standard form is a way of writing complex numbers as a sum of their real part and imaginary part. It has the general structure - \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part.
This format helps in simplifying operations like addition, subtraction, multiplication, and division by clearly distinguishing between the two components.
For example, if you have a complex expression like - \((-4-i)-(2+3i)+(-4+5i)\), first eliminate parentheses and organize the terms.
After simplification, the expression should be written as - \(-10 + 1i\), which is now in the standard form.
This way, it’s easy to see the contribution of the real (\(-10\)) and imaginary (\(1i\)) parts separately.

Imaginary Parts

Imaginary parts of complex numbers are identified and manipulated similarly to real numbers, but they always include a factor of \(i\), the imaginary unit.
The imaginary unit \(i\) is defined by the characteristic that \(i^2 = -1\). This unique property allows us to handle expressions involving square roots of negative numbers.
When operating on complex expressions, it’s crucial to wisely group and combine the imaginary terms.In the given example, the imaginary part of each number involves terms like \(-i\), \(-3i\), and \(+5i\).
To find their total sum, you simply add these coefficients, treating \(i\) like a "variable".- So, the operation becomes: - \(-i - 3i + 5i = 1i\).
This shows that the total imaginary part left in our simplified expression is \(1i\).
Once you’ve properly combined the imaginary parts, it enhances the ease of writing the expression in standard form.

Real Parts

Real parts of complex numbers are somewhat straightforward as they involve regular numbers (without the imaginary unit \(i\)).
The real numbers are the terms within the expression that stand alone without an attached imaginary component.For instance, in the expression \((-4-i)-(2+3i)+(-4+5i)\), when the parentheses are removed, the real terms are:- \(-4, -2, -4\).
Combining these gives us: - \(-4 - 2 - 4 = -10\).
These isolated real terms are integral in forming the complete complex number in standard form.
By clearly identifying and operating on these real numbers, we achieve the real component in our standard form complex expression.
This step is essential to separating and correctly identifying the full character of the complex number we are dealing with.