Problem 3 - Precalculus

Question

For Exercises specify the real and imaginary parts of each complex number. (a) \(4+5 i\) (b) \(4-5 i\) (c) \(\frac{1}{2}-i\) (d) \(16 i\)

Step-by-Step Solution

Answer

(a) Real: 4, Imaginary: 5; (b) Real: 4, Imaginary: -5; (c) Real: \(\frac{1}{2}\), Imaginary: -1; (d) Real: 0, Imaginary: 16.

Step 1: Identify Real and Imaginary Parts of (a)

The given complex number is \(4 + 5i\). In this expression, the real part is 4, and the imaginary part is represented by the coefficient of \(i\), which is 5.

Step 2: Identify Real and Imaginary Parts of (b)

The complex number given is \(4 - 5i\). Here, the real part is 4, and the imaginary part, represented by the coefficient of \(i\), is -5.

Step 3: Identify Real and Imaginary Parts of (c)

For the complex number \(\frac{1}{2} - i\), the real part is \(\frac{1}{2}\), and the imaginary part, represented by the coefficient of \(i\), is -1.

Step 4: Identify Real and Imaginary Parts of (d)

The given number is \(16i\). This can also be expressed as \(0 + 16i\). Therefore, the real part is 0, and the imaginary part is 16.

Key Concepts

Real Part of Complex Numbers

Complex numbers are a fascinating area of mathematics where numbers take on a more flexible form. The real part of a complex number is the component that does not involve the imaginary unit, typically denoted as 'i'.

• In a complex number like 4 + 5i, the real part is 4.
• For the number 4 - 5i, the real part remains 4.
• Even if the complex number includes fractions, like \( \frac{1}{2} - i \), the real part is still straightforward: it's \( \frac{1}{2} \).
• Sometimes, a complex number might have a real part of zero, seen in 16i where it can be rewritten as 0 + 16i. Here, the real part is 0.

Understanding the real part helps in performing operations such as addition and subtraction of complex numbers and is fundamental for manipulating these numbers in equations.

Imaginary Part of Complex Numbers

The imaginary part of a complex number might sound mysterious, but it's simply the part of the number that includes the imaginary unit 'i'. This part is crucial for understanding the nature of complex numbers.

• In 4 + 5i, the imaginary part is designated by the coefficient of 'i', which is 5.
• For 4 - 5i, the imaginary part changes to -5, reflecting how subtraction impacts it.
• In the fraction example \( \frac{1}{2} - i \), the imaginary part is noted as -1, as 'i' stands alone without another number preceding it.
• When only the imaginary part is present, like in 16i, the imaginary component is 16.

The imaginary part often helps represent phenomena in physics and engineering that cannot be explained with real numbers alone. It's used to describe electrical currents, signal processing, and more, making it invaluable across scientific disciplines.

Complex Number Notation

Complex numbers are typically written in the form \( a + bi \), where 'a' is the real part and 'b' is the imaginary part, with 'i' representing the square root of -1. This notation helps keep calculations organized and aligns with algebraic operations.When writing complex numbers:

• Ensure the real part is clear, even if it's zero, as in "0 + 16i".
• Use brackets if involving expressions like fractions, \( \left( \frac{1}{2} \right) - i \), to avoid confusion.
• Always follow the a + bi format to maintain consistency across mathematical problems.
• Remember that 'i' squared is -1, which links the real and imaginary components in complex arithmetic.

Learning and using complex number notation is vital for solving equations integrating both real and imaginary parts. This form assures clarity and precision in mathematical computations and applications.