Real Part

Definition

The real part of a complex number is the component of the number that represents the real number portion, as opposed to the imaginary number portion. It is a crucial concept in the study of complex numbers.

5 Must Know Facts For Your Next Test

1. The real part of a complex number is always a real number, while the imaginary part is always an imaginary number.
2. The real part of a complex number is represented by the coefficient of the $1$ term, while the imaginary part is represented by the coefficient of the $i$ term.
3. The real part of a complex number can be positive, negative, or zero, depending on the value of the real number component.
4. The real part of a complex number is essential in performing operations such as addition, subtraction, multiplication, and division of complex numbers.
5. The real part of a complex number is often denoted using the symbol $\text{Re}(z)$, where $z$ is the complex number.

Review Questions

• Explain the relationship between the real part and the imaginary part of a complex number.
  • The real part and the imaginary part of a complex number are the two components that make up the number. The real part represents the real number portion, while the imaginary part represents the imaginary number portion. These two parts work together to define the complete complex number, with the real part and imaginary part being equally important in the representation and manipulation of complex numbers.
• Describe how the real part of a complex number is used in performing operations with complex numbers.
  • The real part of a complex number is essential in performing various operations, such as addition, subtraction, multiplication, and division. When adding or subtracting complex numbers, the real parts are added or subtracted, respectively. When multiplying complex numbers, the real part is involved in the calculation of the new real part. Similarly, when dividing complex numbers, the real part is used in the division process. The real part of a complex number is a crucial component in these operations, as it allows for the proper manipulation and representation of complex numbers.
• Analyze the significance of the real part of a complex number in the context of the complex plane.
  • On the complex plane, the real part of a complex number $a + bi$ corresponds to the horizontal axis, representing the real number line. The real part determines the position of the complex number along the x-axis, while the imaginary part determines the position along the y-axis. The real part is essential in visualizing and understanding the location of a complex number on the complex plane, as it provides information about the real number component of the complex number. This understanding of the real part's role in the complex plane is crucial for analyzing and working with complex numbers in geometric and graphical representations.

Related terms

Complex Number: A complex number is a number that can be expressed in the form $a + bi$, where $a$ is the real part and $b$ is the imaginary part.

Imaginary Part:

The imaginary part of a complex number is the component of the number that represents the imaginary number portion, as opposed to the real number portion.

Modulus:

The modulus of a complex number is the distance of the complex number from the origin on the complex plane, and is calculated as $\sqrt{a^2 + b^2}$, where $a$ is the real part and $b$ is the imaginary part.