Question

Add or subtract. Write the answer in standard form. \(-3+(8+2 i)+7 i\)

Short answer

The solution is \(5 + 9i\).

Step-by-step solution

Distribute terms

To start with, distribute the terms into real and imaginary. So, breaking down the given equation \(-3+(8+2i)+7i\), we get \(-3 + 8 + 2i + 7i\).

Summarize the terms

Next step, we summarize the real and imaginary parts separately. The real part of the equation is \(-3 + 8\), and the imaginary part of the equation is \(2i + 7i\).

Calculate the results

From the calculation of step 2, we get \(5\) as the sum of the real part and \(9i\) as the sum of the imaginary part.

Write in standard form

Lastly, we just write the complex number in standard form \(a+bi\). Therefore, the result is \(5 + 9i\).

Key concepts

Standard Form

In mathematics, particularly when dealing with complex numbers, the standard form is a neat way to express these numbers. A complex number is usually written as \(a + bi\), where \(a\) and \(b\) are real numbers. The term "\(a\)" represents the real part of the complex number, while "\(bi\)" stands for the imaginary part.

This format allows for easy visualization and understanding of the relationship between the real and imaginary components. The imaginary unit is represented by \(i\), which is defined as \(\sqrt{-1}\). When adding or subtracting complex numbers, the goal is to arrive at this standard form for clarity and consistency.

Imaginary Numbers

Imaginary numbers are formed when you take the square root of a negative number. The notation "\(i\)" is essential in this context because it allows us to handle situations where negative numbers are square rooted, something not possible with real numbers. Imaginary numbers are represented as \(bi\), where \(b\) is a real number that when multiplied by \(i\) gives you the imaginary component of the complex number.

In complex numbers, the imaginary part is vital for operations such as addition, subtraction, multiplication, and division. Applications of imaginary numbers extend into engineering, quantum physics, and signal processing, where they help to model phenomena that cannot be described using just real numbers.

Real Numbers

Real numbers are perhaps the most familiar type of numbers, encompassing all the numbers found on the number line, including positives, negatives, and zero. In the context of complex numbers, the real number part contributes to the "a" in \(a + bi\). This part of the complex number influences its position along the horizontal axis when plotted on a complex plane.

For example, in the number \(-3 + 8\) from the exercise, these are simplified as \(5\), and are entirely real. They are added together just like normal real numbers, without any imaginary units. Real numbers are critical as they form the base of complex numbers and interact with imaginary parts during arithmetic operations.

Addition and Subtraction of Complex Numbers

Adding and subtracting complex numbers involves grouping and simplifying the real and imaginary parts separately. This simplification makes it easier to arrive at the standard form. For addition, you sum the real parts and then the imaginary parts. The same goes for subtraction, where you subtract real from real and imaginary from imaginary.

In the original exercise,

• Step 1 involved distributing the parts: separating them into real and imaginary components, \(-3+8+2i+7i\).
• Step 2 combined like terms: \(-3 + 8\) for the real parts and \(2i + 7i\) for the imaginary parts.
• Step 3 summed them up to provide \(5\) and \(9i\).
• Step 4 arranged these sums into the standard form, \(5 + 9i\).

Practicing such steps with multiple examples helps solidify the understanding of complex numbers and ensures accurate computation.