Question

Explain to a friend how you would add two complex numbers and how you would multiply two complex numbers. Explain any differences between the two explanations.

Short answer

Add the real parts and imaginary parts separately for addition. Use the distributive property and \(i^2 = -1\) when multiplying.

Step-by-step solution

Understanding Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part. They are usually written in the form: \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.

Adding Two Complex Numbers

To add two complex numbers, simply add their corresponding real and imaginary parts separately. For example, if you have two complex numbers, \((a + bi)\) and \((c + di)\): \[(a + bi) + (c + di) = (a + c) + (b + d)i\]

Example of Adding

Let's add \(3 + 4i\) and \(1 + 2i\): \[(3 + 4i) + (1 + 2i) = (3 + 1) + (4 + 2)i = 4 + 6i\]

Multiplying Two Complex Numbers

To multiply two complex numbers, use the distributive property (also known as the FOIL method). For two complex numbers \((a + bi)\) and \((c + di)\): \[(a + bi) \times (c + di) = ac + adi + bic + bidi = (ac - bd) + (ad + bc)i\] Remember that \(i^2 = -1\).

Example of Multiplying

Let's multiply \(3 + 4i\) and \(1 + 2i\): \[(3 + 4i) \times (1 + 2i) = 3 \times 1 + 3 \times 2i + 4i \times 1 + 4i \times 2i = 3 + 6i + 4i + 8i^2\]Since \(i^2 = -1\): \[3 + 6i + 4i + 8(-1) = 3 + 10i - 8 = -5 + 10i\]

Differences Between Addition and Multiplication

The main difference is that addition involves adding the real parts and the imaginary parts separately, while multiplication involves using the distributive property and combining terms, making sure to utilize the property \(i^2 = -1\).

Key concepts

Complex Number Addition

Adding complex numbers might sound tricky at first, but it's quite straightforward once you get the hang of it. A complex number is written in the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.
To add two complex numbers, you simply add their corresponding parts together:

• Add the real parts (the terms without \(i\)) together.
• Add the imaginary parts (the terms with \(i\)) together.

Let's say you have two complex numbers \(a + bi\) and \(c + di\). When you add these together, the result is \( (a + c) + (b + d)i \).
For example, if you add \(3 + 4i\) and \(1 + 2i\), you would calculate as follows:
\[ (3 + 4i) + (1 + 2i) = (3 + 1) + (4 + 2)i = 4 + 6i \]
This is the new complex number.

Complex Number Multiplication

Multiplying complex numbers is a tad more complex (pun intended). It involves using the distributive property, often called the FOIL (First, Outer, Inner, Last) method.
Let’s break it down step by step. Given two complex numbers \(a + bi\) and \(c + di\), you need to multiply them as follows:

• First: Multiply the first terms: \(ac\)
• Outer: Multiply the outer terms: \(adi\)
• Inner: Multiply the inner terms: \(bic\)
• Last: Multiply the last terms: \(bidi\)
• Combine all these products and use the fact that \(i^2 = -1\)

So, \( (a + bi)(c + di) = ac + adi + bic + bidi = ac + adi + bic + bdi^2 \)
Given that \ i^2 = -1 \:
\[ ac + adi + bic + bdi^2 \] simplifies to \( ac + adi + bic - bd \)
Combine like terms to get \( (ac - bd) + (ad + bc)i \).
For instance, let's multiply \( 3 + 4i \) and \( 1 + 2i \):
\[ (3 + 4i) \times (1 + 2i) = 3 \times 1 + 3 \times 2i + 4i \times 1 + 4i \times 2i = 3 + 6i + 4i + 8i^2 \].
Since \(i^2 = -1\), the expression becomes:
\[ 3 + 6i + 4i + 8(-1) = 3 + 10i - 8 = -5 + 10i \].

Imaginary Unit

The imaginary unit, represented by \(i\), is a mathematical concept used to extend the real numbers.
It is defined by \(i^2 = -1\). This property is crucial in operations involving complex numbers.
For instance, when multiplying complex numbers, you will often encounter terms like \(i^2\). Remembering that \(i^2 = -1\) will help you simplify these terms correctly.
The imaginary unit allows us to express numbers that would otherwise be impossible to represent within the real number system, particularly those equations where we include the square root of a negative number.
This expands our ability to solve problems and understand phenomena in both mathematics and engineering.
In summary, the imaginary unit \(i\) plays a pivotal role in both the addition and multiplication of complex numbers. It helps us manage the imaginary parts and correctly simplify expressions when necessary.