key term - Modulus

Definition

The modulus of a complex number is the absolute value or magnitude of that number, representing its distance from the origin on the complex plane. It is a fundamental concept in the study of complex numbers and their properties.

5 Must Know Facts For Your Next Test

1. The modulus of a complex number $z = a + bi$ is calculated as $|z| = \sqrt{a^2 + b^2}$, which represents the distance of the point $(a, b)$ from the origin on the complex plane.
2. The modulus of a complex number is always a non-negative real number, and it is often denoted as $|z|$ or $\mathrm{mod}(z)$.
3. The modulus of a complex number is an important property in the polar form of complex numbers, where it represents the magnitude or length of the vector representing the complex number.
4. The modulus of a complex number is a key component in the multiplication and division of complex numbers, as it affects the overall magnitude of the result.
5. The modulus of a complex number is also used in the calculation of the distance between two complex numbers on the complex plane.

Review Questions

• Explain the relationship between the modulus of a complex number and its representation on the complex plane.
  • The modulus of a complex number $z = a + bi$ represents the distance of the point $(a, b)$ from the origin on the complex plane. This distance is calculated as $\sqrt{a^2 + b^2}$, which is the length of the vector representing the complex number. The modulus is a non-negative real number and is a fundamental property of complex numbers, as it describes the magnitude or absolute value of the number.
• Describe how the modulus of a complex number is used in the polar form of complex numbers.
  • In the polar form of a complex number $z = r(\cos\theta + i\sin\theta)$, the modulus $r$ represents the magnitude or length of the vector representing the complex number on the complex plane. The modulus is the absolute value or magnitude of the complex number, and it is a key component in the polar form, which also includes the argument (angle) $\theta$. The polar form allows for easier manipulation of complex numbers, particularly in multiplication and division, where the moduli and arguments can be combined in a straightforward manner.
• Analyze the importance of the modulus of a complex number in the context of complex number operations.
  • The modulus of a complex number is crucial in the various operations performed on complex numbers. In multiplication, the modulus of the product is the product of the moduli of the individual complex numbers. In division, the modulus of the quotient is the ratio of the moduli of the dividend and divisor. The modulus also plays a role in the addition and subtraction of complex numbers, as it affects the overall magnitude of the result. Additionally, the modulus is used to calculate the distance between two complex numbers on the complex plane, which is an important concept in the study of complex numbers and their properties.

Related terms

Complex Number: A complex number is a number that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, satisfying the equation $i^2 = -1$.

Argument:

The argument of a complex number is the angle (in radians) between the positive real axis and the line segment connecting the origin to the point representing the complex number on the complex plane.

Polar Form:

The polar form of a complex number is a representation that uses the modulus and argument of the number, expressed as $r(\cos\theta + i\sin\theta)$, where $r$ is the modulus and $\theta$ is the argument.