Problem 88 - Calculus for Biology and Medicine

Question

Simplify each expression and write it in the standard form \(a+b i\). \((6-4 i)+(2+5 i)\)

Step-by-Step Solution

Answer

The simplified form is \(8 + i\).

Step 1: Identify the Real Parts

In the expression \((6-4i)+(2+5i)\), first identify the real parts which are \(6\) and \(2\).

Step 2: Identify the Imaginary Parts

Next, identify the imaginary parts from the expression. These are \(-4i\) and \(5i\).

Step 3: Add the Real Parts

Add the real parts together: \(6 + 2 = 8\).

Step 4: Add the Imaginary Parts

Add the imaginary parts together: \(-4i + 5i = 1i\).

Step 5: Combine the Results

Combine the results from Steps 3 and 4 to form the standard form of the complex number, \(8 + 1i\).

Key Concepts

Addition of Complex Numbers

Adding complex numbers might seem challenging at first, but it's quite similar to adding simple algebraic expressions. Complex numbers have two parts: a real part and an imaginary part. When adding them together, the primary goal is to add these parts separately, as they each follow distinct rules.

Let's look at an example

• (6 - 4i) + (2 + 5i) where we clearly have two complex numbers.
• First, we take the real parts of each number, which are 6 and 2.
• Then, we add the imaginary parts: -4i and 5i.

By adding the real parts, we obtain 6 + 2 = 8 and by adding the imaginary parts, we get -4i + 5i = i. Combine these results to achieve the final sum: 8 + i.

Real and Imaginary Parts

As we delve into complex numbers, it's essential to differentiate between their two components, the real and imaginary parts. A complex number is usually expressed as a + bi, where a represents the real part and bi represents the imaginary part. Both parts play crucial roles in arithmetic operations involving complex numbers.

• The real part, a, is simply the number that is not multiplied by the imaginary unit, i.
• The imaginary part, on the other hand, includes the component multiplied by i.

In the expression 6 - 4i, the real part is 6, and the imaginary part is -4i. Recognizing these components allows us to perform operations, such as addition, more efficiently by combining like terms.

Standard Form of Complex Numbers

Expressing a complex number in its standard form is essential for clarity, simplicity, and consistent mathematical communication. The standard form of a complex number is a + bi, where a and b are real numbers. Here, a is the real part, and b is the coefficient of the imaginary part, i.

For example, if we add the complex numbers (6 - 4i) and (2 + 5i), the result should always be written as 8 + i.

• This format ensures that both components are clearly visible and distinct.
• It's important to write the imaginary part last, even if it is negative, to maintain consistency.

The standard form offers a straightforward way to represent complex numbers, making further mathematical operations more seamless and intuitive.