Problem 16 - Algebra and Trigonometry

Question

Evaluate the expression and write the result in the form \(a+b i .\) $$ (-4+i)-(2-5 i) $$

Step-by-Step Solution

Answer

The expression evaluates to \(-6 + 6i\).

Step 1: Distribute Negative Sign to the Second Term

We begin by distributing the negative sign to each term within the second expression, transforming the subtraction into addition. This changes the expression from \[(-4+i)-(2-5i)\]to\[(-4+i)+(-2+5i).\]

Step 2: Separate the Real and Imaginary Parts

Next, we group the real parts and imaginary parts separately for addition. The real parts are \[-4\text{ and } -2,\]and the imaginary parts are \[i \text{ and } 5i.\]

Step 3: Add the Real Parts

Find the sum of the real parts: \[-4 + (-2) = -6.\]

Step 4: Add the Imaginary Parts

Add the imaginary parts together: \[i + 5i = 6i.\]

Step 5: Combine the Results

Finally, combine the results of the real parts and imaginary parts to express the answer in the form \(a+bi\):\[-6 + 6i.\]

Key Concepts

Adding Complex Numbers

Complex numbers are numbers that consist of two components: a real part and an imaginary part. When adding complex numbers, the key is to add the real parts together, and separately add the imaginary parts together. Let's illustrate this using our example expression

• First complex number:
  • Real part: -4
  • Imaginary part: +1
• Second complex number:
  • Real part: 2
  • Imaginary part: -5

To add these, first ensure any negative signs are properly distributed, which we'll explore later. By aligning each part in columns, you can see that adding them is straightforward. Just follow these steps:

• Add the real parts: (-4) + (-2) = -6
• Add the imaginary parts: (i) + (5i) = 6i

The final sum is (-6 + 6i), which is in standard complex form, denoted as (a + bi) where a = real part and  = imaginary part.

Real Parts

The real parts of complex numbers are similar to ordinary numbers. They don’t involve any imaginary unit, such as 'i'. When dealing with complex numbers, you treat real parts much like regular algebraic terms.

• In the expression (-4+i), the real part is -4
• In the expression (2-5i), the real part is 2

When adding these, you simply perform a normal addition or subtraction as you would with any integer or real number. In our example, (-4) + (-2) = -6.
By separating the real parts from any imaginary components, calculations become straightforward and manageable.

Imaginary Parts

Imaginary parts require a bit more understanding because they involve the imaginary unit 'i', which is defined as the square root of (-1). In a complex number, the imaginary part is the coefficient of 'i'. When adding, you apply the same principles as you do with the real parts:

• In (-4+i), the imaginary part is the coefficient 1 (the number 1 in front of 'i')
• In (2-5i), the imaginary part is -5

To add them, (1i) + (5i) = 6i. These coefficients are treated just like any numbers you'd add or subtract, but the final result is always multiplied by 'i' to remain as part of the imaginary number.

Distributing Negative Signs

One crucial element in manipulating complex number expressions involves distributing negative signs. This step often helps eliminate confusion and prevent mistakes in calculations. Behind every subtraction is the addition of a negative, which can be applied in complex arithmetic as well:

• With (-4+i)-(2-5i), the negative sign affects everything inside the parentheses (2-5i)
• Distributing it yields: (-2+5i) which transforms our expression into addition: (-4+i)+(-2+5i)

This conversion allows all terms to be clearly and easily grouped, removing potential errors. Once this distribution is complete, add the corresponding parts without worrying about subtracting anything directly.