Problem 18 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the complex number in standard form. $$-i-(\sqrt{-23})^{2}$$

Step-by-Step Solution

Answer

The complex number in standard form is \(-23 - i\).

Step 1: Simplify Squared Expression

Firstly, simplify \((\sqrt{-23})^{2}\). As this involves the square root of a negative number, it is an imaginary number. Because the square of the square root of any number equals the original number, this simplifies to \(-23\).

Step 2: Combine with Imaginary Part

Next, combine this with \(-i\). In complex number arithmetic, real and imaginary parts cannot be combined, so the expression remains as \(-23 - i\).

Step 3: Write in Standard Form

The complex number is now simplified and in standard form. The real part 'a' is \(-23\) and the imaginary part 'b' is \(-1\). The standard form is typically expressed as 'a + bi', so the final form is \(-23 - i\).

Key Concepts

Imaginary Numbers

Imaginary numbers are numbers that when squared, yield a negative result. They involve the imaginary unit, represented by the symbol "i", which is defined as the square root of -1. Imaginary numbers are crucial to the formation of complex numbers.

• Imaginary Unit (i): The base of an imaginary number, where \(i^2 = -1\).
• Imaginary Numbers Addition: The sum of two imaginary numbers only involves adding their coefficients.
• Example: If you have \(5i + 3i\), the sum is \((5+3)i = 8i\).

These numbers appear frequently in scenarios involving square roots of negative numbers, as seen when simplifying expressions such as \(\sqrt{-23}\). Remember, when dealing with the square root of a negative number, always consider the imaginary unit, which turns the expression into an imaginary form.

Standard Form of Complex Numbers

The standard form of a complex number is \(a + bi\), where \(a\) and \(b\) are real numbers. Here, \(a\) is known as the real part, and \(b\) is the imaginary part. This form clearly separates the two components and allows for easy computation.

• Real Part: The real number component of the complex number.
• Imaginary Part: The coefficient of the imaginary unit \(i\).
• Example: For the complex number \(3 + 4i\), \(3\) is the real part and \(4\) is the imaginary part.

In the provided exercise, the number is given as \(-23 - i\). Here, \(a\) is \(-23\), and \(b\) is \(-1\), positioning it perfectly in the standard form. Understanding this form is crucial for performing operations like addition or multiplication with complex numbers.

Complex Number Arithmetic

Complex number arithmetic involves the operations of addition, subtraction, multiplication, and division with complex numbers. Each operation adheres to specific rules to maintain the integrity of the real and imaginary parts separately.

• Addition/Subtraction: Combine like parts; real components with real and imaginary with imaginary. For instance, \((3 + 2i) + (1 + 4i) = 4 + 6i\).
• Multiplication: Use distributive property and apply \(i^2 = -1\) when needed. Example: \((1 + i)(1 - i) = 1^2 - i^2 = 1 + 1 = 2\). This simplifies the expression effectively.
• Division: Involves multiplying the numerator and the denominator by the conjugate of the denominator to rid the expression of 'i' in the denominator.

In the given problem, combining the real part \(-23\) with the imaginary product resulted in the complex number \(-23 - i\), illustrating how distinct these components remain during arithmetic operations. Grasping these rules allows for the seamless execution of operations with complex numbers.