Everything about Magnitude Mathematics totally explained
The
magnitude of a mathematical object is its size: a property by which it can be larger or smaller than other objects of the same kind; in technical terms, an
ordering of the
class of objects to which it belongs.
The Greeks distinguished between several types of magnitude, including:
They had proven that the first two couldn't be the same, or even
isomorphic systems of magnitude. They didn't consider negative magnitudes to be meaningful, and
magnitude is still chiefly used in contexts in which zero is either the lowest size or less than all possible sizes.
Real numbers
The magnitude of a real number is usually called the
absolute value or
modulus. It is written |
x |, and is defined by:
» |
x | =
x, if
x ≥ 0
|
x | = −
x, if
x < 0
This gives the number's distance from zero on the real
number line. For example, the modulus of −5 is 5.
Complex numbers
Similarly, the magnitude of a
complex number, called the
modulus, gives the distance from zero in the
Argand diagram. The formula for the modulus is the same as that for
Pythagoras' theorem.
»
where
x = [
x1,
x2, ...,
xn]. The notation |
x| is also used for the norm. For instance, the magnitude of [4,5, 6] is √(4
2 + 5
2 + 6
2) = √77 or about 8.775.
General vector spaces
A concept of magnitude can be applied to a vector space in general. This is then called a
normed vector space. The function that maps objects to their magnitudes is called a
norm.
Practical math
A magnitude is never negative. When comparing magnitudes, it's often helpful to use a
logarithmic scale. Real-world examples include the
loudness of a
sound (
decibel), the
brightness of a
star, or the
Richter scale of earthquake intensity.
To put it another way, often it isn't meaningful to simply
add and
subtract magnitudes.
Further Information
Get more info on 'Magnitude Mathematics'.
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