Problem 69 - Algebra 2

Question

Simplify. \((4-3 i)-(5-6 i)\)

Step-by-Step Solution

Answer

The simplified expression is \(-1 + 3i\).

Step 1: Identify the Real and Imaginary Parts

For the expression \((4 - 3i) - (5 - 6i)\), identify the real and imaginary components for both complex numbers. The first number \(4 - 3i\) has a real part of 4 and an imaginary part of -3, while the second number \(5 - 6i\) has a real part of 5 and an imaginary part of -6.

Step 2: Subtract the Real Parts

Subtract the real parts of the two complex numbers: \(4 - 5\). This results in \(-1\).

Step 3: Subtract the Imaginary Parts

Perform the subtraction on the imaginary components: \((-3) - (-6)\) or \(-3 + 6\). This simplifies to \(3\).

Step 4: Combine the Results

Combine the simplified real part and the imaginary part to form the resultant simplified complex number. Thus, the answer is \(-1 + 3i\).

Key Concepts

Real Part

Every complex number is composed of two parts: a real part and an imaginary part. The real part is the component of the complex number that is not attached to the imaginary unit, \(i\). Simply put, it is the regular number that you see in the complex format.
For example, in the complex number \(4 - 3i\), the real part is 4. It is the part you would get if you ignored the \(-3i\) entirely.
Identifying the real parts of complex numbers is crucial when performing arithmetic operations, such as addition and subtraction, because these parts are operated separately from the imaginary parts.

Imaginary Part

The imaginary part of a complex number involves the imaginary unit \(i\), which is defined as the square root of \(-1\). It essentially adds an extra dimension to our regular, real numbers.
When you see a complex number like \(4 - 3i\), the \(-3i\) represents the imaginary part. The coefficient of \(i\) (in this case, \(-3\)) is often referred to informally as "the imaginary part," though technically the full imaginary term is \(-3i\).
Understanding both the real and imaginary parts allows us to handle operations on complex numbers effectively, as each component must be treated separately.

Simplifying Expressions

Simplifying expressions involving complex numbers takes several steps. Each part of the complex number (real and imaginary) should be treated separately.

• First, identify both real and imaginary components in all complex numbers.
• Next, you perform the necessary arithmetic operation separately on these parts, such as addition or subtraction.
• Finally, you'll combine the simplified real and imaginary results into a single complex number again.

For example, to simplify \((4 - 3i) - (5 - 6i)\), you subtract the real parts \(4 - 5\), and then the imaginary parts \(-3i - (-6i)\), resulting in the new simplified complex number \(-1 + 3i\).
Each step ensures that the numbers are properly managed, making sure we don't confuse real numbers with imaginary components.

Subtracting Complex Numbers

Subtracting complex numbers is akin to subtracting regular numbers, with the key difference being the separate treatment of real and imaginary components.

• The process begins by identifying the parts of each complex number.
• Next, subtract the real parts to get the result for the real term.
• Independently, subtract the imaginary parts to handle the \(i\) term.

An example can be seen with the subtraction of \((4 - 3i)\) and \((5 - 6i)\). Perform the real subtraction: \(4 - 5\), leading to \(-1\). Then subtract the imaginaries: \(-3 - (-6)\), simplifying to \(3\). Hence, the result of this operation is the complex number \(-1 + 3i\).
By breaking down the process, students can easily grasp how complex numbers can effectively be subtracted.