Modulus

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Modulus[edit | edit source]

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[edit | edit source]

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[edit | edit source]

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[edit | edit source]

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:

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

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