Problem 7 - Intermediate Algebra

Question

Fill in the blanks. a. To add (or subtract) complex numbers, add (or subtract) their _____ parts. b. To multiply two complex numbers, such as \((2+3 i)(3+5 i),\) we can use the _____ method.

Step-by-Step Solution

Answer

a. real and imaginary; b. FOIL

Step 1: Understanding Addition and Subtraction of Complex Numbers

Complex numbers are numbers of the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. To add or subtract complex numbers, you must add or subtract their real parts and their imaginary parts separately. This means if you have two complex numbers \((a+bi)\) and \((c+di)\), their sum is \((a+c) + (b+d)i\), and their difference is \((a-c) + (b-d)i\).

Step 2: Identify the Part to Add/Subtract

For the expression 'To add (or subtract) complex numbers, add (or subtract) their _____ parts,' the blank should be filled with 'real and imaginary.' This is because both the real and imaginary parts need to be processed separately.

Step 3: Understanding Multiplication of Complex Numbers

Multiplying complex numbers \((2+3i)\) and \((3+5i)\) requires the distributive property, similar to binomial expansion. This involves multiplying each part of the first complex number by each part of the second complex number.

Step 4: Identifying the Method for Multiplication

The particular method that uses the distributive property for multiplication of binomials is the 'FOIL' method. 'FOIL' stands for First, Outer, Inner, Last, which helps remember the order in which you should multiply parts of two binomials. In the context of complex numbers, FOIL involves multiplying each component of one complex number with each component of another.

Key Concepts

Addition of Complex Numbers

Complex numbers are a fascinating part of mathematics that bring together real and imaginary worlds. When you're dealing with addition of complex numbers, you're essentially combining these two elements in a straightforward way.
In general, a complex number takes the form of \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. Now, when you encounter two complex numbers, say \((a+bi)\) and \((c+di)\), you add them by:

• Adding the real parts: \(a + c\)
• Adding the imaginary parts: \(b + d\)

Therefore, the sum will be \((a+c) + (b+d)i\). This step of separating real and imaginary components ensures clarity and correctness in addition.
Keeping the real and imaginary parts distinct is crucial because you cannot directly combine them into a simpler form. They each play their own role in the structure of complex numbers.

Subtraction of Complex Numbers

Subtraction of complex numbers follows a very similar approach to addition but with a simple reversal in operation. Understanding this concept opens up further exploration in the realm of complex arithmetic.
Like addition, each part of the complex number needs to be handled separately. Let’s say you have the complex numbers \((a+bi)\) and \((c+di)\). To subtract them:

• Subtract the real parts: \(a - c\)
• Subtract the imaginary parts: \(b - d\)

Thus, the result of the subtraction is \((a-c) + (b-d)i\).
This separation ensures that each type of number (real and imaginary) is processed correctly without any mixing, leading to accurate results. By doing so, we maintain the integrity and use of the imaginary unit \(i\), which remains distinct throughout the arithmetic operations.

Multiplication of Complex Numbers

Multiplication of complex numbers encompasses a basic yet powerful concept that involves using the distributive property, quite similar to multiplying two binomials.
Consider multiplying the complex numbers \((2+3i)\) and \((3+5i)\). Here, a particular methodological approach known as the 'FOIL' method is immensely helpful. FOIL is an acronym that stands for:

• First: Multiply the first terms of each binomial: \(2 \times 3 = 6\)
• Outer: Multiply the outer terms: \(2 \times 5i = 10i\)
• Inner: Multiply the inner terms: \(3i \times 3 = 9i\)
• Last: Multiply the last terms: \(3i \times 5i = 15i^2\)

After performing these steps, you combine all results: \(6 + 10i + 9i + 15i^2\). Remember that \(i^2 = -1\), so the expression simplifies to \(6 + 19i - 15\), which further simplifies to \(-9 + 19i\).
Using FOIL helps keep the process organized and ensures you don’t miss any component during multiplication, making it not only accurate but efficient.