Complex Numbers: Videos & Practice Problems

Complex numbers, expressed in the form a + bi, consist of a real part (a) and an imaginary part (b). Operations such as addition, subtraction, multiplication, and division can be performed similarly to algebraic expressions. The complex conjugate, a - bi, is crucial for division, allowing the elimination of the imaginary unit in the denominator. Multiplying a complex number by its conjugate results in a real number, specifically a² + b². Understanding these concepts is essential for further studies in complex number applications.

Introduction to Complex Numbers

Complex numbers are a combination of real 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 \( i \) is defined as \( i = \sqrt{-1} \). Understanding complex numbers is crucial as they have numerous applications in various fields, including engineering and physics.

In the expression \( 3 + 2i \), the real part is \( 3 \) and the imaginary part is \( 2 \). It is important to note that the imaginary part is identified as the coefficient of \( i \), which in this case is \( 2 \). When analyzing other complex numbers, such as \( 4 - 3i \), the real part \( a \) is \( 4 \) and the imaginary part \( b \) is \( -3 \), since it is the coefficient multiplying \( i \).

For the complex number \( 0 + 7i \), the real part is \( 0 \) and the imaginary part is \( 7 \). Although it can be simplified to \( 7i \), recognizing the real part as \( 0 \) is essential when considering it as a complex number. Similarly, in \( 2 + 0i \), the real part is \( 2 \) and the imaginary part is \( 0 \). While this can also be represented simply as \( 2 \), it is still classified as a complex number due to the presence of the imaginary part, albeit it being zero.

In summary, complex numbers consist of both a real and an imaginary component, and understanding how to identify these parts is fundamental for further studies in mathematics and its applications.

Identify the real and imaginary parts of each complex number. $-4-9i$

• A. $a = -9, b = -4$
• B. $a = -4, b = -9$
• C. $a = 4, b = 9$
• D. $a = -4, b = 9$

Answer: (B)

Identify the real and imaginary parts of each complex number. $3+2i\sqrt3$

• A. $a=3,b=2\sqrt3$
• B. $a = 3, b = 2$
• C. $a=2\sqrt3,b=3$
• D. $a=3,b=\sqrt3$

Answer: (A)

Write the complex number in standard form. $\frac{9+\sqrt{-16}}{3}$

• A. $3+4i$
• B. $9+16i$
• C. $3+\frac43i$
• D. $\frac{13i}{3}$

Answer: (C)

Adding and Subtracting Complex Numbers

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 \(a + bi\).

Find the difference. Express your answer in standard form. $\left(2+8i\right)-\left(4-i\right)$

• A. $-2+9i$
• B. $6+7i$
• C. $2+7i$
• D. $2-9i$

Answer: (A)

Find the sum. Express your answer in standard form. $5\left(4+7i\right)+6\left(3-2i\right)$

• A. $7+5i$
• B. $38+23i$
• C. $2+47i$
• D. $7+9i$

Answer: (B)

Multiplying Complex Numbers

Complex numbers can be multiplied using methods similar to those used for algebraic expressions, specifically through distribution or the FOIL 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:

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

Thus, combining the terms yields:

\( 21i + 6 \)

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

In another example, multiplying \( (-6 + 2i) \) by \( (3 + 4i) \) requires the FOIL method since both complex numbers have two terms. The calculations proceed as follows:

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

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

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

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

Now, substituting \( i^2 \) gives:

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

Combining all terms results in:

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

Finally, the answer is confirmed to be in standard form as \( -26 - 18i \).

Through these examples, it is clear that multiplying complex numbers involves careful distribution or FOIL application, followed by simplification using the property of \( i^2 \). This process ensures that the final result is expressed in the standard form of complex numbers.

Perform the indicated operation. Express your answer in standard form. $\left(3+8i\right)^2$

• A. $-55+48i$
• B. $9+64i$
• C. $24i$
• D. $9+24i$

Answer: (A)

Find the product. Express your answer in standard form. $2i\left(9-4i\right)\left(6+5i\right)$

• A. $8+18i$
• B. $54-20i$
• C. $54-40i$
• D. $-42+148i$

Answer: (D)

Complex Conjugates

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 the complex number is \( a - bi \), its conjugate is \( a + bi \).

For example, the complex conjugate of \( 1 + 2i \) is \( 1 - 2i \), and the conjugate of \( 1 - 2i \) is \( 1 + 2i \). Similarly, for the complex number \( -1 + 2i \), the conjugate is \( -1 - 2i \). This process of finding the conjugate is straightforward and involves only the imaginary part of the number.

When multiplying a complex number by its conjugate, the result is always a real number. For instance, if we take \( 2 + 3i \) and multiply it by its conjugate \( 2 - 3i \), we can apply the FOIL method:

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

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

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

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

Combining these results gives:

\( 4 - 6i + 6i - 9(-1) = 4 + 9 = 13 \)

Notice that the imaginary parts cancel each other out, leaving a real number. This pattern holds true for any complex number multiplied by its conjugate, resulting in \( a^2 + b^2 \), where \( a \) and \( b \) are the real and imaginary parts, respectively. This relationship can serve as a useful shortcut when working with complex numbers.

In summary, the complex conjugate is a fundamental concept that not only aids in division but also simplifies multiplication, ensuring that the product of a complex number and its conjugate yields a real number.

Find the product of the given complex number and its conjugate. $4-5i$

• A. $16$
• B. $41$
• C. $25$
• D. $20$

Answer: (B)

Find the product of the given complex number and its conjugate. $-7-i$

• A. $50$
• B. $14$
• C. $49$
• D. $1$

Answer: (A)

Dividing Complex Numbers

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 interpret. 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:

\[(1 + 2i)(1 - 2i) = 1^2 - (2i)^2 = 1 - 4(-1) = 1 + 4 = 5\]

Now, substituting back into our fraction gives us:

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

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

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

This expression is now in its simplest form, with the real part \( \frac{3}{5} \) and the imaginary part \( -\frac{6}{5}i \). Thus, the final result of dividing the complex numbers is:

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

By following these steps—multiplying by the complex conjugate, simplifying, and separating the real and imaginary parts—you can effectively divide complex numbers and express the result in a clear and concise manner.

Find the quotient. Express your answer in standard form.  
$\frac{6+i}{4-2i}$

• A. $\frac{11}{10}+\frac45i$
• B. $\frac65+\frac45i$
• C. $\frac{11}{10}-\frac45i$
• D. $22+16i$

Answer: (A)

Find the quotient. Express your answer in standard form. 
$\frac{-5+3i}{-7-4i}$

• A. $\frac35+\frac45i$
• B. $18i$
• C. $23-41i$
• D. $\frac{23}{65}-\frac{41}{65}i$

Answer: (D)

Here’s what students ask on this topic:

What is a complex number and how is it represented?

A complex number is a number that combines a real part and an imaginary part. It is represented in the form $a+bi$, where $a$ is the real part and $b$ is the imaginary part. The imaginary unit $i$ is defined as $i^2=-1$. For example, in the complex number $3+2i$, 3 is the real part and 2 is the imaginary part.

How do you add and subtract complex numbers?

To add or subtract complex numbers, you combine their real parts and their imaginary parts separately. For addition, if you have $a+bi$ and $c+di$, the sum is $\left(a+c\right)+\left(b+d\right)i$. For subtraction, the difference is $\left(a-c\right)+\left(b-d\right)i$. For example, $4+3i+2+5i=6+8i$ and $4+3i-2+5i=2-2i$.

How do you multiply complex numbers?

To multiply complex numbers, you use the distributive property or the FOIL method. For example, to multiply $\left(2+3i\right)\left(4+5i\right)$, you calculate: $2\times 4+2\times 5i+3i\times 4+3i\times 5i$. This simplifies to $8+10i+12i+15i^2$. Since $i^2=-1$, the expression becomes $8+22i-15$, which simplifies to $-7+22i$.

What is the complex conjugate and how is it used in division?

The complex conjugate of a complex number $a+bi$ is $a-bi$. It is used in division to eliminate the imaginary unit from the denominator. For example, to divide $3$ by $1+2i$, multiply both the numerator and the denominator by the conjugate of the denominator: $3\times 1-2i$ and $\left(1+2i\right)\left(1-2i\right)$. This results in $3-6i$ over $1+4$, which simplifies to $3-6i$ over $5$, or $3/5-6/5i$.

How do you divide complex numbers?

To divide complex numbers, multiply the numerator and the denominator by the conjugate of the denominator. For example, to divide $3$ by $1+2i$, multiply both by the conjugate $1-2i$. This gives $3\times 1-2i$ over $\left(1+2i\right)\left(1-2i\right)$. Simplify to get $3-6i$ over $1+4$, which is $3-6i$ over $5$. Finally, split into real and imaginary parts: $3/5-6/5i$.