Problem 11 - College Algebra and Calculus: An Applied Approach

Question

Write the complex number in standard form and find its complex conjugate. $$-21$$

Step-by-Step Solution

Answer

The complex number in standard form is \(-21 + 0i\) and its complex conjugate is \(-21 - 0i\).

Step 1: Write the complex number in standard form

Write the given number as a complex number by adding an imaginary part. Since the number \(-21\) is a real number, its imaginary part is \(0i\). So the number in standard form is \(-21 + 0i\).

Step 2: Compute the complex conjugate

The complex conjugate is found by changing the sign of the imaginary part. Since the imaginary part in this case is zero, changing the sign wouldn't make any difference. Therefore, the complex conjugate of \(-21 + 0i\) is \(-21 - 0i\).

Key Concepts

Standard Form of Complex Numbers

Understanding the standard form of a complex number is foundational in working with complex numbers. A complex number is composed of two parts: a real part and an imaginary part. The standard form is typically written as \( a + bi \) where \( a \) is the real component, and \( b \) is the imaginary component multiplied by \( i \), which represents the square root of \( -1 \).

For instance, if we have a real number like \( -21 \), to express it in standard form, we simply add \( 0i \) to show that it has an imaginary part with a value of zero. So, \( -21 \) in standard form is \( -21 + 0i \). This notation highlights that even real numbers can be considered as complex numbers with an imaginary part of zero. When working with complex numbers, it’s important to ensure they are expressed in standard form, as it allows for the consistent application of complex numbers algebra.

Imaginary Numbers

Imaginary numbers are an extension of the real numbers we are accustomed to. They were invented to provide solutions to equations that do not have solutions within the set of real numbers, such as the square root of a negative number. The imaginary unit is denoted as \( i \), and by definition, it satisfies the equation \( i^2 = -1 \).

Any number that can be written in the form of \( bi \) where \( b \) is a real number, is referred to as an imaginary number. For example, \( 5i \), \( -3i \), and \( 0i \) are all imaginary numbers. It is important to note that \( 0i \) is considered imaginary and real at the same time because it does not introduce any imaginary component when added to a real number since its value is zero.

Complex Numbers Algebra

The algebra of complex numbers allows us to perform arithmetic operations such as addition, subtraction, multiplication, and division, just as we do with real numbers. The key to complex numbers algebra is the manipulation of both the real and imaginary components according to algebraic rules. One crucial rule to remember is that \( i^2 \) should always be replaced with \( -1 \) when it appears during multiplication or raising to a power.

Additionally, the concept of a complex conjugate is significant in complex numbers algebra. The complex conjugate of a complex number is obtained by changing the sign of the imaginary part only. This is particularly useful in division, as multiplying a complex number by its conjugate provides a real number. This simplifies many calculations, especially when it comes to expressions that require the denominator to be rationalized.