Modulus

Actually, the modulus of a number x, witten as |x| can be loosely defined as it's length. So for a real number, it is defined as:

|x| = x if x > 0
|x| = -x if x < 0

so |5| = 5 and |-3| = 3

For complex numbers, the situation is more, erm, complex. The modulus is the distance between the origin and the point in the complex plane that represents that number. This can be written as:

|z| = sqrt(x² + y²)

where z = x+iy (think pythagoras and that'll make sense). So |1+i| = 1.141.... and |-3-4i| = 5

The modulus, and the fact it is always positive is very important in many proofs in analysis. In fact, the whole of complex analysis, and in turn much of the study of calculus, is based in one way or another around this concept.

Note that the modulus, or absolute value, of x ∈ R is equivalent to √(x²), where √ denotes the non-negative square root, as

√(x2) = x if x ≥ 0,
       -x if x < 0.

This equivalency leads to the following results for all a,b ∈ R:

1. |-a| =|a|
2. |ab| =|a| * |b|
3. |a/b| = |a|/|b| provided b ≠ 0.
4. -|a| ≤ a ≤ |a|
5. |a| < b ⇔ -b < a < b
6. |a| ≤ b ⇔ -b ≤ a ≤ b

1,2 and 3 follow from |a| = √(a²), as for each the left and right hand side expressions are greater than or equal to 0 (as with all moduli) and their squares are equivalent.

To prove 4, consider positive and negative values of a:
When a ≥ 0, |a| =a ≥ 0 and -|a| = -a ≤ 0, hence -|a| = -a ≤ 0 ≤ a = |a| ⇒ - |a| ≤ a ≤ |a| as desired.
When a < 0, |a| = -a > 0 and -|a| = a < 0, hence -|a| = a < 0 < -a = |a| ⇒ -|a| = a < |a| ⇒ -|a| ≤ a ≤ |a| more generally as desired.

5 (and by similar argument, 6) can be shown as follows:

First, we'll show |a| < b ⇒ -b < a < b.
We have |a| < b ⇒ -b < -|a| and by 4, ⇒ -b < -|a| ≤ a ≤ |a| < b which simplifies to -b < a < b as required.

Having proved this from left to right, it is necessary to prove in the opposite direction for an iff statement to be valid.
So to show -b < a < b ⇒ |a| < b, consider that multiplication by -1 (which switches the inequality signs) yields -b < -a < b. So -b < max{a,-a} < b. But max{a,-a} is yet another way of writing |a|, so |a| < b.

Mod"u*lus (?), n.; pl. Moduli (#). [L., a small measure. See Module, n.] Math., Mech., & Physics

A quantity or coefficient, or constant, which expresses the measure of some specified force, property, or quality, as of elasticity, strength, efficiency, etc.; a parameter.

Modulus of a machine, a formula expressing the work which a given machine can perform under the conditions involved in its construction; the relation between the work done upon a machine by the moving power, and that yielded at the working points, either constantly, if its motion be uniform, or in the interval of time which it occupies in passing from any given velocity to the same velocity again, if its motion be variable; -- called also the efficiency of the machine. Mosley. Rankine.
-- Modulus of a system of logarithms Math., a number by which all the Napierian logarithms must be multiplied to obtain the logarithms in another system.
-- Modulus of elasticity. (a) The measure of the elastic force of any substance, expressed by the ratio of a stress on a given unit of the substance to the accompanying distortion, or strain. (b) An expression of the force (usually in terms of the height in feet or weight in pounds of a column of the same body) which would be necessary to elongate a prismatic body of a transverse section equal to a given unit, as a square inch or foot, to double, or to compress it to half, its original length, were that degree of elongation or compression possible, or within the limits of elasticity; -- called also Young's modulus.
-- Modulus of rupture, the measure of the force necessary to break a given substance across, as a beam, expressed by eighteen times the load which is required to break a bar of one inch square, supported flatwise at two points one foot apart, and loaded in the middle between the points of support. Rankine.

© Webster 1913.