Modulus

Modulus

The term modulus has several meanings in different contexts, primarily in mathematics and physics. It is a fundamental concept used in various branches of science and engineering.

Mathematics

In mathematics, the modulus is a function that measures the size or magnitude of a number. The most common types of modulus functions are:

• Absolute value: The absolute value of a real number is its distance from zero on the number line. For a real number \( x \), the absolute value is denoted as \( |x| \).
• Modular arithmetic: In modular arithmetic, the modulus is the number at which we start back at zero. For example, in arithmetic modulo \( n \), numbers wrap around after reaching \( n \). The notation \( a \mod n \) represents the remainder when \( a \) is divided by \( n \).

Physics

In physics, the term modulus is often used to describe material properties. The most common types of moduli in this context are:

• Young's modulus: A measure of the stiffness of a solid material. It is defined as the ratio of stress (force per unit area) to strain (proportional deformation) in a material in the linear elasticity regime of a uniaxial deformation.
• Shear modulus: Also known as the modulus of rigidity, it describes the material's response to shear stress. It is defined as the ratio of shear stress to the shear strain.
• Bulk modulus: A measure of a substance's resistance to uniform compression. It is defined as the ratio of the infinitesimal pressure increase to the resulting relative decrease of the volume.

Complex Numbers

In the context of complex numbers, the modulus of a complex number \( z = a + bi \) (where \( a \) and \( b \) are real numbers and \( i \) is the imaginary unit) is the distance from the origin to the point \( (a, b) \) in the complex plane. It is given by the formula:

\[ |z| = \sqrt{a^2 + b^2} \]

Related Pages

• Absolute value
• Modular arithmetic
• Young's modulus
• Shear modulus
• Bulk modulus
• Complex number