Problem 36 - Algebra 2

Question

Simplify. $$ (15+3 i)-(9-3 i) $$

Step-by-Step Solution

Answer

The simplified expression is \(6 + 6i\).

Step 1: Identify Real and Imaginary Parts

Write out the expression and identify the real and imaginary parts of each complex number. We have two complex numbers here: \((15 + 3i)\) and \((9 - 3i)\). In \((15 + 3i)\), the real part is 15 and the imaginary part is \(3i\). In \((9 - 3i)\), the real part is 9 and the imaginary part is \(-3i\).

Step 2: Distribute the Subtraction Operator

Apply the subtraction operator to both the real and imaginary parts of the second complex number. This means subtracting the real parts and the imaginary parts separately: \((15 + 3i) - (9 - 3i) = (15 - 9) + (3i - (-3i))\).

Step 3: Simplify the Real Parts

Subtract the real part of the second number from the real part of the first number: \(15 - 9 = 6\).

Step 4: Simplify the Imaginary Parts

Subtract the imaginary part of the second complex number from the imaginary part of the first. Remember that subtracting a negative is the same as adding, so \(3i - (-3i) = 3i + 3i = 6i\).

Step 5: Combine the Results

Combine the simplified real and imaginary parts to form the new complex number: \(6 + 6i\).

Key Concepts

Real Part

In the realm of complex numbers, the real part holds great importance. A complex number is composed of two parts: the real and imaginary parts. To understand the real part, think of numbers on the number line you've seen since elementary school. In a complex number like \(15 + 3i\), the number 15 is what we call the "real part." It is the part of the number that doesn't involve the imaginary unit \(i\). Whenever you see a complex number, you can spot the real part because it's the regular number without the \(i\).

• For \(15 + 3i\), the real part is 15.
• In \(9 - 3i\), the real part is 9.

By identifying the real parts of different complex numbers, math becomes easier to manage, especially when performing operations like addition or subtraction.

Imaginary Part

The imaginary part of a complex number can initially seem confusing due to its unusual nature. It involves the letter \(i\), which represents the square root of -1. This is something quite different from the real number line we are familiar with.In a complex number such as \(15 + 3i\), the '\(3i\)' is the imaginary part, while in \(9 - 3i\), the imaginary part is '\(-3i\)'. Let's delve a bit deeper:

• The imaginary part \(3i\) means that we have three times the imaginary unit \(i\).
• A negative sign before the imaginary part, like in \(-3i\), simply means you're taking away 'three i's' instead of adding them.

Understanding the imaginary part often forms a new way of thinking, as you're adding a new dimension to your number line—quite literally!

Subtraction of Complex Numbers

Subtracting complex numbers might sound daunting, but it's quite similar to everyday arithmetic once you break it down into steps. To perform the subtraction \((15 + 3i) - (9 - 3i)\), follow these simple guidelines:1. **Separate the Real and Imaginary Parts**: Recognize which parts are real and which are imaginary in both numbers.2. **Distribute the Subtraction**: Apply the subtraction separately to the real parts and the imaginary parts. This means performing \((15 - 9)\) for the real parts and \(3i - (-3i)\) for the imaginary parts.3. **Calculate Real and Imaginary Differences**: For the real parts you get \(15 - 9 = 6\), and for the imaginary, \(3i - (-3i) = 3i + 3i = 6i\).4. **Combine the Results**: Bring together both simplified real and imaginary parts to form the resultant complex number, which in our case is \(6 + 6i\).Using these steps not only simplifies the process but also helps in leveraging the structure of complex numbers to your advantage.