Problem 28 - Algebra and Trigonometry Real Mathematics, Real People

Question

Perform the addition or subtraction and write the result in standard form. $$-(-3.7-12.8 i)-(6.1-16.3 i)$$

Step-by-Step Solution

Answer

The result is $-2.4 + 29.1i$.

Step 1: Removal of double negatives

First step is to remove the double negatives in the question. It can be turned into addition, make sure you understand that 'negative of negative' yields positive. So, the equation $-(-3.7-12.8 i)-(6.1-16.3 i)$ turns into $-(-3.7-12.8 i)$ + $(6.1-16.3 i)$ = $(3.7 + 12.8i) - (6.1 - 16.3i)$.

Step 2: Perform subtraction of complex numbers

Next, perform subtraction between these two complex numbers. Separate the real and imaginary parts: $Real = 3.7 - 6.1$ and $Imaginary = 12.8 - (-16.3)$. Proceed to evaluate each part.

Step 3: Evaluation & Final Answer

Evaluate the real and imaginary parts. So, the real part is $-2.4$ and the imaginary part is $29.1$. Now put these values into the standard form of a complex number. This gives us the final answer as $-2.4 + 29.1i$.

Key Concepts

Addition of Complex Numbers

Complex numbers are expressed as a combination of a real number and an imaginary number, typically in the form of \(a + bi\), where \(a\) stands for the real part and \(b\) is the imaginary part. When adding two complex numbers together, it's essential to combine the corresponding parts.

To add, let's consider two generic complex numbers, \((a + bi)\) and \((c + di)\). The steps are straightforward:

• Add the real parts: \(a + c\)
• Add the imaginary parts: \(b + d\)

The result will be \((a + c) + (b + d)i\).

For example, if we are given \( (3 + 2i) + (5 + 4i) \), the addition process involves combining the real parts \((3 + 5 = 8)\) and the imaginary parts \((2 + 4 = 6)\), resulting in \(8 + 6i\).

This method simplifies many calculations involving complex numbers, making it a foundational operation in complex arithmetic.

Subtraction of Complex Numbers

Subtracting complex numbers is similar in process to addition, but involves subtracting the real and imaginary components. The typical form for this operation is \((a + bi) - (c + di)\).

Here are the steps:

• Subtract the real parts: \(a - c\)
• Subtract the imaginary parts: \(b - d\)

This will produce a result of \((a - c) + (b - d)i\).

For instance, consider subtracting \((5 + 3i) - (2 + 7i)\). Subtract the real parts \((5 - 2 = 3)\) and the imaginary parts \((3 - 7 = -4)\), giving us a final result of \(3 - 4i\).

In our exercise, after handling the double negative, the problem was essentially a subtraction: \((3.7 + 12.8i) - (6.1 - 16.3i)\). By applying these steps, we separate and perform arithmetic operations properly.

Standard Form of Complex Numbers

The standard form of a complex number is \(a + bi\), where \(a\) is the real component and \(b\) is the imaginary component accompanied by \(i\), the imaginary unit, defined by the property \(i^2 = -1\). This form is crucial as it clearly distinguishes the real part from the imaginary part.

After performing operations such as addition or subtraction on complex numbers, the result should always be expressed in this standard form for clarity and consistency.

For example, if the outcome of a calculation is \(-2.4 + 29.1i\), this directly fits into the \(a + bi\) format, with \(-2.4\) being the real part and \(29.1i\) as the imaginary part. Expressing complex numbers this way ensures a standardized approach in mathematics.

This conclusion is particularly helpful in complex analysis and when performing further calculations, since it provides a clear, consistent method of representing complex values.