Here are all the Solutions and notes of all exercises of Unit 2 Real and Complex numbers;
Welcome to the Unit 2 Outlines for the course on Real Numbers and Complex Numbers. In this unit, we will delve into the fundamental concepts and properties of real numbers and explore the intriguing world of complex numbers. By the end of this unit, students will acquire a comprehensive understanding of these topics and will be able to apply their knowledge to solve various mathematical problems.
The unit is structured into several sections:
Real Numbers (2.1): We will begin by exploring the set of real numbers, understanding how it is formed as a union of rational and irrational numbers. Students will learn to depict real numbers on the number line, distinguishing between numbers with terminating and non-terminating recurring decimals.
Properties of Real Numbers (2.2): This section focuses on the essential properties of real numbers that govern their behavior in mathematical operations. By grasping these properties, students will be able to manipulate and simplify expressions involving real numbers effectively.
Radicals and Radicands (2.3): Understanding radicals and their radicands is crucial for dealing with expressions involving roots. Students will learn to differentiate between radical form and exponential form of expressions and transform between the two.
Laws of Exponents / Indices (2.4): Building upon their knowledge of exponents, students will discover the laws of exponents and learn how to apply them to simplify expressions with real exponents.
Complex Numbers (2.5): Delving into the captivating realm of complex numbers, students will learn about numbers of the form “a + ib,” where “a” and “b” are real numbers, and “i” represents the imaginary unit, √(-1). They will comprehend the structure and properties of complex numbers.
Basic Operations on Complex Numbers (2.6): This section will equip students with the skills to carry out basic operations such as addition, subtraction, multiplication, and division on complex numbers, enabling them to solve complex mathematical problems effectively.
By the end of this unit, students will be able to:
Recall and understand the set of real numbers and represent them on the number line.
Differentiate between rational and irrational numbers and provide decimal representations for both.
Grasp the properties of real numbers and apply them in mathematical operations.
Comprehend the concept of radicals and radicands and convert between radical and exponential forms.
Use the laws of exponents to simplify expressions with real exponents.
Define and recognize complex numbers, their real and imaginary parts, and their conjugates.
Carry out fundamental operations on complex numbers, including addition, subtraction, multiplication, and division.
This unit will provide a strong foundation for further exploration of advanced mathematical concepts and problem-solving skills. Let’s embark on this exciting journey of numbers and unleash the potential of real and complex mathematics.