Complex Numbers: Videos & Practice Problems

Complex numbers are a combination of real numbers and imaginary numbers, typically expressed in the standard form a + bi, where a represents the real part and b represents the imaginary part. The imaginary unit is denoted by i, which is defined as the square root of -1. Understanding complex numbers is crucial as they have numerous applications in various fields, including engineering, physics, and applied mathematics.

To identify the real and imaginary parts of a complex number, consider the following examples:

1. For the complex number 4 - 3i, the real part a is 4, while the imaginary part b is -3, since it is the coefficient of i.

2. In the case of 0 + 7i, the real part a is 0, and the imaginary part b is 7. Although this can be simplified to 7i, it is important to recognize that the real part is still 0.

3. For the complex number 2 + 0i, the real part a is 2, and the imaginary part b is 0. This can also be expressed simply as 2, but it is still classified as a complex number due to the presence of the imaginary part.

In summary, complex numbers are essential in mathematics, and recognizing their components—real and imaginary parts—enhances our understanding of their properties and applications.

Complex numbers can be added and subtracted using the same principles as algebraic expressions, making the process straightforward. When performing these operations, the key is to combine like terms, which include both constants and terms involving the imaginary unit i.

For example, when adding complex numbers such as \(4 + 8i + 2 + 3i\), you first remove the parentheses to simplify the expression to \(4 + 8i + 2 + 3i\). Next, combine the constants (4 and 2) to get 6, and combine the imaginary terms (8i and 3i) to yield 11i. The result is expressed in standard form, which is \(a + bi\). Thus, the final answer is \(6 + 11i\).

Subtraction of complex numbers follows a similar approach. For instance, when subtracting \(4 + 8i - (2 + 3i)\), you must distribute the negative sign across the parentheses, resulting in \(4 + 8i - 2 - 3i\). Combine the constants (4 and -2) to get 2, and the imaginary terms (8i and -3i) to get 5i. Again, ensure the answer is in standard form, leading to a final result of \(2 + 5i\).

In summary, adding and subtracting complex numbers involves combining like terms, treating the imaginary unit i as a variable, and expressing the final answer in the standard form \(a + bi\).

Complex numbers can be multiplied using methods similar to those used for algebraic expressions, specifically through distribution or the FOIL (First, Outside, Inside, Last) method. When multiplying complex numbers, it is essential to remember that the imaginary unit \( i \) satisfies the equation \( i^2 = -1 \), which will be crucial for simplifying the results.

To illustrate the multiplication of complex numbers, consider the expression \( 3i \times (7 - 2i) \). The first step involves distributing \( 3i \) across the terms in the parentheses. This results in:

\( 3i \times 7 = 21i \)

\( 3i \times (-2i) = -6i^2 \)

Next, substituting \( i^2 \) with \(-1\) gives:

\( -6i^2 = -6 \times -1 = 6 \)

Combining these results leads to:

\( 21i + 6 \)

To express the answer in standard form \( a + bi \), it is rearranged to \( 6 + 21i \).

In another example, multiplying \( (-6 + 2i) \) by \( (3 + 4i) \) using the FOIL method yields:

First: \( -6 \times 3 = -18 \)

Outside: \( -6 \times 4i = -24i \)

Inside: \( 2i \times 3 = 6i \)

Last: \( 2i \times 4i = 8i^2 \)

Substituting \( i^2 \) gives:

Thus, \( 8i^2 = 8 \times -1 = -8 \)

Combining all terms results in:

\( -18 - 8 - 24i + 6i = -26 - 18i \)

Finally, this expression is already in standard form \( a + bi \), confirming the final answer as \( -26 - 18i \).

Through these examples, it is clear that multiplying complex numbers involves careful application of distribution or FOIL, simplification using the property of \( i^2 \), and combining like terms to achieve the final result in standard form.

Understanding the concept of the complex conjugate is essential for dividing complex numbers. The complex conjugate of a complex number is formed by reversing the sign of its imaginary part. For a complex number expressed as \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part, the complex conjugate is \( a - bi \). Conversely, if you start with \( a - bi \), its conjugate will be \( a + bi \).

For example, the complex conjugate of \( 1 + 2i \) is \( 1 - 2i \), and the conjugate of \( 1 - 2i \) is \( 1 + 2i \). Similarly, for \( -1 + 2i \), the complex conjugate is \( -1 - 2i \). This process of finding the conjugate is straightforward: simply change the sign of the imaginary component while keeping the real part unchanged.

When you multiply a complex number by its conjugate, the result is always a real number. For instance, multiplying \( 2 + 3i \) by its conjugate \( 2 - 3i \) involves using the FOIL method:

1. First: \( 2 \times 2 = 4 \)

2. Outside: \( 2 \times -3i = -6i \)

3. Inside: \( 3i \times 2 = 6i \)

4. Last: \( 3i \times -3i = -9i^2 \)

Since \( i^2 = -1 \), the term \( -9i^2 \) becomes \( +9 \). Combining all these results gives:

\( 4 - 6i + 6i + 9 = 4 + 9 = 13 \)

This shows that \( (2 + 3i)(2 - 3i) = 13 \), a real number. In general, multiplying \( a + bi \) by \( a - bi \) yields \( a^2 + b^2 \). This relationship is not only useful for calculations but also serves as a helpful shortcut when working with complex numbers.

In summary, the complex conjugate is a fundamental concept that simplifies operations involving complex numbers, particularly division, which will be explored further in subsequent lessons.

Dividing complex numbers involves a systematic approach to eliminate the imaginary unit from the denominator. When faced with a fraction where the denominator contains an imaginary component, the goal is to simplify it into a form that is easier to work with. This is achieved by using the complex conjugate.

The complex conjugate of a complex number \( a + bi \) is \( a - bi \). For example, if we want to divide \( \frac{3}{1 + 2i} \), we first identify the complex conjugate of the denominator, which is \( 1 - 2i \). To eliminate the imaginary unit from the denominator, we multiply both the numerator and the denominator by this complex conjugate:

\[\frac{3}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{3(1 - 2i)}{(1 + 2i)(1 - 2i)}\]

Next, we expand both the numerator and the denominator. The numerator becomes:

\[3(1 - 2i) = 3 - 6i\]

For the denominator, we apply the FOIL method:

\[(1 + 2i)(1 - 2i) = 1 \cdot 1 + 1 \cdot (-2i) + 2i \cdot 1 + 2i \cdot (-2i) = 1 - 2i + 2i - 4i^2\]

Since \( i^2 = -1 \), we can simplify \( -4i^2 \) to \( +4 \). Thus, the denominator simplifies to:

\[1 + 4 = 5\]

Now, we can rewrite the fraction as:

\[\frac{3 - 6i}{5}\]

To separate the real and imaginary parts, we can express this as:

\[\frac{3}{5} - \frac{6}{5}i\]

This final expression, \( \frac{3}{5} - \frac{6}{5}i \), represents the quotient of the original complex division in its simplest form. Thus, the process of dividing complex numbers not only provides a solution but also reinforces the understanding of complex conjugates and their role in simplifying expressions.