# Modulus

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## modulus

**1.**

*Physics*a coefficient expressing a specified property of a specified substance

**3.**

*Maths*the number by which a logarithm to one base is multiplied to give the corresponding logarithm to another base

**4.**

*Maths*an integer that can be divided exactly into the difference between two other integers

## Modulus

a geometric parameter of gears. For cylindrical spur gears the modulus *m* is equal to the ratio of the diameter *d _{p}* of the pitch circle to the number of teeth

*z*or the ratio of the pitch

*t*along the pitch circle to the number

*π: m = d*. For cylindrical helical gears a distinction is made among the transverse modulus

_{p}/z = t/π*m*=

_{s}= d_{p}/z*t*the normal modulus

_{s}/∊*m*and the axial modulus

_{s}= tn/∊*m*, where

_{a}= /t_{a}/∊*t*, and

_{s}, t_{n}*t*are, respectively, the transverse, normal, and axial pitches along the pitch cylinder. The values of gear moduli are standardized, which is the basis for standardization of the other gear parameters (the geometric dimensions of gears are made proportional to the modulus) and gear-cutting tools.

_{a}## Modulus

(mathematics). (1) The modulus (or absolute value) of a complex number *z = x + iy* is the number *r* = (the root is taken with the plus sign). If a complex number *z* is represented in trigonometric form, *z = r(cos ϕ + i* sin ϕ), then the modulus is the real number *r*. The modulus may be geometrically interpreted in the following way. A complex number *z = x + iy* can be represented by a vector issuing from the origin of a rectangular coordinate system and ending at the point with coordinates (*x, y*). The length of this vector is the modulus of the complex number *z*.

(2) The modulus for converting from a logarithm system with base *a* to a logarithm system with base *b* is the number *M* = 1/log_{a}b. To obtain the logarithms of the numbers *x* to the base *b*, if the logarithms of these numbers are known to the base *a*, it is necessary to multiply the latter by the modulus: log_{b} = *M log _{a}x*.