Problem 41 - Intermediate Algebra

Question

Perform the operations. Write all answers in the form \(a+b i.\) $$ (7-3 i)-(4+2 i) $$

Step-by-Step Solution

Answer

The solution is \(3 - 5i\).

Step 1: Identify the Expression

The given expression is \((7-3i) - (4+2i)\). We need to simplify this expression to find the result in the form \(a + bi\).

Step 2: Distribute the Negative Sign

We start by distributing the negative sign across the second complex number: \( (4 + 2i) \). This gives us \( -4 - 2i \).

Step 3: Combine Like Terms

Now combine the real parts and the imaginary parts separately. For real parts: \(7 - 4 = 3\). For imaginary parts: \(-3i - 2i = -5i\).

Step 4: Construct the Final Complex Number

Combine the results from the previous step to write the final complex number in the form \(a + bi\): \(3 - 5i\).

Key Concepts

Addition and Subtraction of Complex Numbers

Understanding how to add and subtract complex numbers is a basic yet essential concept in complex number arithmetic. Complex numbers have two parts: a real part and an imaginary part. They are usually expressed in the form \(a + bi\), where \(a\) is the real component and \(b\) is the coefficient of the imaginary component \(i\).

When adding or subtracting complex numbers, the process involves handling the real and imaginary parts separately:

• **Adding Complex Numbers:** For addition, add the real parts and then add the imaginary parts separately. For instance, for two complex numbers \((x + yi)\) and \((u + vi)\), the sum is \((x + u) + (y + v)i\).
• **Subtracting Complex Numbers:** Similarly, for subtraction, subtract the real parts and then subtract the imaginary parts. In our exercise, subtract \(4 + 2i\) from \(7 - 3i\), leading to the new expression \((7 - 3i) - (4 + 2i)\).
  
  

This approach simplifies calculations and allows complex numbers to be manipulated just like other numerical systems, keeping computations straightforward.

Distributive Property

The distributive property is one of the foundational properties of arithmetic operations and applies to complex numbers as well. In the context of complex numbers, particularly when subtracting, it becomes crucial to distribute the negative sign across the components of the number being subtracted.

For the expression \((7 - 3i) - (4 + 2i)\), distributing the negative sign means:

• Changing the sign of the real component: \(-4\)
• Changing the sign of the imaginary component: \(-2i\)

This transforms the entire expression into adding \(-4 - 2i\) to \(7 - 3i\). Thanks to the distributive property, we can clearly see how each part of the complex numbers changes, simplifying computation. Always remember the importance of properly applying the distributive property before proceeding with addition or subtraction of complex numbers.

Imaginary Numbers

Imaginary numbers are a fascinating concept in mathematics, characterized by the presence of the imaginary unit \(i\), where \(i\) is defined as \(\sqrt{-1}\). This concept is essential for understanding complex numbers, especially when dealing with operations involving them.

The distinguishing feature of imaginary numbers is that they allow for the consideration of roots of negative numbers, which are not possible within the realm of real numbers. The imaginary part of a complex number is denoted as \(bi\), where \(b\) is a real number multiplier of \(i\).

When performing arithmetic with complex numbers, it's crucial to handle the imaginary parts as carefully as the real parts. This ensures the integrity of the mathematical operations and leads to an accurate result. In simplified form, after performing arithmetic operations, the complex number maintains its form \(a + bi\), illustrating the coexistence of real and imaginary components in one entity.