Introduction to Complex Numbers: Videos & Practice Problems

Complex numbers combine real and imaginary parts in the form $a+bi$, where $a$ is the real part and $b$ the coefficient of the imaginary unit $i$, defined as $\sqrt{-1}=i$. Simplifying square roots of negative numbers involves factoring out $\sqrt{-1}$ as $i$. Addition and subtraction of complex numbers follow combining like terms, treating $i$ like a variable, and results are expressed in standard form $a+bi$. This foundational understanding supports operations on polynomials and exponential expressions involving imaginary components.

Square Roots of Negative Numbers

In mathematics, the concept of square roots extends beyond positive numbers to include negative numbers through the use of the imaginary unit, denoted as i. The imaginary unit is defined as the square root of negative one, or i = √(-1). This allows us to evaluate the square roots of negative numbers, which are not real numbers.

To simplify the square root of a negative number, we can factor it into the product of the square root of negative one and the square root of a positive number. For example, to find the square root of -4, we can express it as:

√(-4) = √(-1) × √(4) = i × 2 = 2i.

It is customary to write the result as 2i rather than i × 2 for clarity. This method can be applied to any negative number. For a general negative number -b, where b is positive, the square root can be expressed as:

√(-b) = √(-1) × √(b) = i × √(b).

When dealing with square roots of negative numbers, it is important to maintain the order of terms. For instance, if we evaluate the square root of -32, we can simplify it as follows:

√(-32) = √(-1) × √(32) = i × √(16 × 2) = i × 4√(2) = 4i√(2).

In this case, the final answer is presented as 4i√(2), with the whole number first, followed by the imaginary unit, and then the radical. All results that include the imaginary unit are classified as imaginary numbers.

Understanding how to work with imaginary numbers is crucial, as they play a significant role in various mathematical applications, including complex numbers, which combine real and imaginary parts.

Simplify the given square root.
$\sqrt{-75}$

• A. $25i\sqrt3$
• B. $5i\sqrt3$
• C. $3i\sqrt5$
• D. $75i$

Answer: (B)

Introduction to Complex Numbers

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.

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.
$3+2i\sqrt{3}$

• 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.
$(2+8i)-(4-i)$

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

Answer: (A)

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

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

Answer: (B)

Here’s what students ask on this topic:

What is the imaginary unit and how is it used to simplify square roots of negative numbers?

The imaginary unit, denoted as $i$, is defined as the square root of negative one, $\sqrt{-1}=i$. It was created to help evaluate square roots of negative numbers, which are not real. To simplify the square root of a negative number, you factor out $\sqrt{-1}$ as $i$ and then take the square root of the positive part separately. For example, $\sqrt{-4}=\sqrt{-1}\sqrt{4}=i2=2i$. This method works for any negative number, allowing us to express the result as an imaginary number involving $i$.

What is the standard form of a complex number and how do you identify its real and imaginary parts?

A complex number is expressed in standard form as $a+bi$, where $a$ is the real part and $b$ is the coefficient of the imaginary part. The real part $a$ is the number not multiplied by $i$, while the imaginary part $b$ is the number multiplying $i$. For example, in $4-3i$, the real part is $4$ and the imaginary part is $-3$. Even if one part is zero, such as $0\; +\; 7i$ or $2\; +\; 0i$, it is still considered a complex number with the zero part explicitly noted.

How do you add and subtract complex numbers?

Adding and subtracting complex numbers is similar to combining like terms in algebra. Treat the imaginary unit $i$ like a variable. For addition, combine the real parts and the imaginary parts separately. For example, $(4\; +\; 8i)\; +\; (2\; +\; 3i)\; =\; (4\; +\; 2)\; +\; (8i+\; 3i)\; =\; 6\; +\; 11i$. For subtraction, distribute the negative sign and then combine like terms: $(4\; +\; 8i)\; -\; (2\; +\; 3i)\; =\; (4\; -\; 2)\; +\; (8i-\; 3i)\; =\; 2\; +\; 5i$. Always express the final answer in standard form $a+bi$.

Why can't the square root of a negative number be a real number?

The square root of a negative number cannot be a real number because no real number multiplied by itself results in a negative number. For example, $(-2)\; imes\; (-2)\; =\; 4$, which is positive, and $2\; imes\; 2\; =\; 4$ as well. Since squaring any real number always gives a non-negative result, the square root of a negative number is not defined within the real numbers. This limitation led to the introduction of the imaginary unit $i$, which allows us to work with square roots of negative numbers by defining $\sqrt{-1}=i$.

How do you simplify the square root of a negative number like $\sqrt{-32}$?

To simplify $\sqrt{-32}$, first factor it as $\sqrt{-1}\sqrt{32}=i\sqrt{32}$. Next, simplify the positive square root: $\sqrt{32}=\sqrt{16imes2}=\sqrt{16}\sqrt{2}=\; 4\sqrt{2}$. Combine this with $i$ to get $4i\sqrt{2}$. The standard way to write this is with the whole number first, then $i$, and then the radical: $4i\sqrt{2}$.