What is "magnitude"?

Question

Mathematical definiton, please.

Answer 1

Magnitude is the highest exponent in a mathematical expression.

Unfortunately I don't know how to do super script on the Q and A's or I would show you an example.

The biggest exponent is the magnitude of the expression.

Answer 2

According to Wikipedia:

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.

http://en.wikipedia.org/wiki/Magnitude_%28mathematics%29

Answer 3

Bigness!

You usually only hear that when you have a quantity that has both magnitude and direction, such as velocity

The length of the velocity vector is speed. Speed combined with direction is velocity, Same thing for forces. It makes a big difference whether you push or pull on something.

Answer 4

there are two aspects of a vector
1.the quantitative aspevt given by its MAGNITUDE and
2.the qualitative aspect given by its DIRECTION

Answer 5

(1) The expression Magnitude especially marks determinate Quantity, and is for that reason not a suitable name for Quantity in general. (2) Mathematics usually define magnitude as what can be increased or diminished. This definition has the defect of containing the thing to be defined over again: but it may serve to show that the category of magnitude is explicitly understood to be changeable and indifferent, so that, in spite of its being altered by an increased extension or intension, the thing - a house, for example - does not cease to be a house, and red to be red. (3) The Absolute is pure Quantity. This point of view is on the whole the same as when the Absolute is defined to be Matter, in which, though form undoubtedly is present, the form is a characteristic of no importance one way or another. Quantity too constitutes the main characteristic of the Absolute, when the Absolute is regarded as absolute indifference, and only admitting of quantitative distinction. Otherwise pure space, time, etc., may be taken as examples of Quantity, if we allow ourselves to regard the real as whatever fills up space and time, it matters not with what.

The mathematical definition of magnitude as what may be increased or diminished, appears at first sight to be more plausible and perspicuous than the exposition of the notion in the present section. When closely examined, however, it involves, under cover of presuppositions and images, the same elements as appear in the notion of quantity reached by the method of logical development. In other words, when we say that the notion of magnitude lies in the possibility of being increased or diminished, we state that magnitude (or more correctly, quantity), as distinguished from quality, is a characteristic of such kind that the characterised thing is not in the least affected by any change in it.

What then, it may be asked, is the fault which we have to find with this definition? It is that to increase and to diminish is the same thing as to characterise magnitude otherwise. If this aspect then were an adequate account of it, quantity would be described merely as whatever can be altered. But quality is no less than quantity open to alteration; and the distinction here given between quantity and quality is expressed by saying increase or diminution: the meaning being that, towards whatever side the determination of magnitude be altered, the thing still remains what it is.

One remark more. Throughout philosophy we do not seek merely for correct, still less for plausible definitions, whose correctness appeals directly to the popular imagination; we seek approved or verified definitions, the content of which is not assumed merely as given, but is seen and known to warrant itself, because warranted by the free self-evolution of thought. To apply this to the present case: however correct and self-evident the definition of quantity usual in Mathematics may be, it will still fail to satisfy the wish to see how far this particular thought is founded in universal thought, and in that way necessary. This difficulty, however, is not the only one.

If quantity is not reached through the action of thought, but taken uncritically from our generalised image of it, we are liable to exaggerate the range of its validity, or even to raise it to the height of an absolute category. And that such a danger is real, we see when the title of exact science is restricted to those sciences the objects of which can be submitted to mathematical calculation. Here we have another trace of the bad metaphysics (mentioned in § 98n) which replace the concrete idea by partial and inadequate categories of understanding. Our knowledge would be in a very awkward predicament if such objects as freedom, law, morality, or even God himself, because they cannot be measured and calculated, or expressed in a mathematical formula, were to be reckoned beyond the reach of exact knowledge, and we had to put up with a vague generalised image of them, leaving their details or particulars to the pleasure of each individual, to make out of them what he will. The pernicious consequences, to which such a theory gives rise in practice, are at once evident. And this mere mathematical view, which identifies with the Idea one of its special stages, viz., quantity, is no other than the principle of Materialism. Witness the history of the scientific modes of thought, especially in France since the middle of last century. Matter in the abstract is just what, though of course there is form in it, has that form only as an indifferent and external attribute.

The present explanation would be utterly misconceived if it were supposed to disparage mathematics. By calling the quantitative characteristic merely external and indifferent, we provide no excuse for indolence and superficiality, nor do we assert that quantitative characteristics may be left to mind themselves, or at least require no very careful handling. Quantity, of course, is a stage of the Idea: and as such it must have its due, first as a logical category, and then in the world of objects, natural as well as spiritual. Still, even so, there soon emerges the different importance attaching to the category of quantity according as its objects belong to the natural or to the spiritual world. For in Nature, where the form of the Idea is to be other than, and at the same time outside, itself, greater importance is for that very reason attached to quantity than in the spiritual world, the world of free inwardness. No doubt we regard even spiritual facts under a quantitative point of view; but it is at once apparent that in speaking of God as a Trinity, the number three has by no means the same prominence, as when we consider the three dimensions of space or the three sides of a triangle the fundamental feature of which last is just to be a surface bounded by three lines. Even inside the realm of Nature we find the same distinction of greater or less importance of quantitative features. In the inorganic world, Quantity plays, so to say, a more prominent part than in the organic. Even in organic nature, when we distinguish mechanical functions from what are called chemical, and in the narrower sense physical, there is the same difference. Mechanics is of all branches of science, confessedly, that in which the aid of mathematics can be least dispensed with-where indeed we cannot take one step without them. On that account mechanics is regarded, next to mathematics, as the science par excellence; which leads us to repeat the remark about the coincidence of the materialist with the exclusively mathematical point of view. After all that has been said, we cannot but hold it, in the interest of exact and thorough knowledge, one of the most hurtful prejudices, to seek all distinction and determinateness of objects merely in quantitative considerations. Mind to be sure is more than Nature and the animal is more than the plant: but we know very little of these objects and the distinction between them, if a more and less is enough for us, and if we do not proceed to comprehend them in their peculiar, that is, their qualitative character.


Although, In astronomy, absolute magnitude is the apparent magnitude, m, an object would have if it were at a standard luminosity distance away from us, in the absence of interstellar extinction. It allows the overall brightnesses of objects to be compared without regard to distance.

The absolute magnitude uses the same convention as the visual magnitude, with a ~2.512 difference in brightness between step rates (because 2.5125 ≈ 100). The Milky Way, for example, has an absolute magnitude of about -20.5. So a quasar at an absolute magnitude of -25.5 is 100 times brighter than our galaxy. If this particular quasar and our galaxy could be seen side by side at the same distance, the quasar would be 5 magnitudes (or 100 times) brighter than our galaxy. Absolute Magnitude for stars and galaxies (M)
In stellar and galactic astronomy, the standard distance is 10 parsecs (about 32.616 light years, or 3×1014 kilometres). A star at ten parsecs has a parallax of 0.1" (100 milli arc seconds).

In defining absolute magnitude it is necessary to specify the type of electromagnetic radiation being measured. When referring to total energy output, the proper term is bolometric magnitude. The dimmer an object (at a distance of 10 parsecs) would appear, the higher its absolute magnitude. The lower an object's absolute magnitude, the higher its luminosity. A mathematical equation relates apparent magnitude with absolute magnitude, via parallax.

Many stars visible to the naked eye have an absolute magnitude which is capable of casting shadows from a distance of 10 parsecs; Rigel (-7.0), Deneb (-7.2), Naos (-6.0), and Betelgeuse (-5.6).

For comparison, Sirius has an absolute magnitude of 1.4 and the Sun has an absolute visual magnitude of 4.83 (it actually serves as a reference point).

Absolute magnitudes for stars generally range from -10 to +17. The absolute magnitude for galaxies can be much lower (brighter). For example, the giant elliptical galaxy M87 has an absolute magnitude of -22.

Computation
You can compute the absolute magnitude of an object given its apparent magnitude and luminosity distance :


where is the star's luminosity distance in parsecs, which are (≈ 3.2616 light-years)

For nearby astronomical objects (such as stars in our galaxy) the luminosity distance DL is almost identical to the real distance to the object, because spacetime within our galaxy is almost Euclidean. For much more distant objects the Euclidean approximation is not valid, and General Relativity must be taken into account when calculating the luminosity distance of an object.

In the Euclidean approximation for nearby objects, the absolute magnitude of a star can be calculated from its apparent magnitude and parallax:


where π is the star's parallax in arcseconds.

You can also compute the absolute magnitude of an object given its apparent magnitude and distance modulus :

Example
Rigel has a visual magnitude of mV=0.18 and distance about 774 light-years.
MVRigel = 0.18 + 5*log10(32.616/773) = -6.7
Vega has a parallax of 0.133", and an apparent magnitude of +0.03
MVVega = 0.03 + 5*(1 + log10(0.133)) = +0.65
Alpha Centauri has a parallax of 0.750" and an apparent magnitude of -0.01
MVα Cen = -0.01 + 5*(1 + log10(0.750)) = +4.37
Black Eye Galaxy has a visual magnitude of mV=+9.36 and a distance modulus of 31.06.
MVBlack Eye Galaxy = 9.36 - 31.06 = -20.01
Apparent magnitude
Given the absolute magnitude , for objects within our galaxy you can also calculate the apparent magnitude from any distance :


For objects at very great distances (outside our galaxy) the luminosity distance DL must be used instead of d.

Absolute Magnitude for planets (H)
For planets, comets and asteroids a different definition of absolute magnitude is used which is more meaningful for nonstellar objects.

In this case, the absolute magnitude is defined as the apparent magnitude that the object would have if it were one astronomical unit (au) from both the Sun and the Earth and at a phase angle of zero degrees. This is a physical impossibility, as it requires the observing telescope to be at the centre of the Sun, but it is convenient for purposes of calculation.

To convert a stellar or galactic absolute magnitude into a planetary one, subtract 31.57. This factor also corresponds to the difference between the Sun's visual magnitude of -26.8 and its (stellar) absolute magnitude of +4.8. Thus, the Milky Way (galactic absolute magnitude -20.5) would have a planetary absolute magnitude of -52.
[edit] Apparent magnitude
The absolute magnitude can be used to help calculate the apparent magnitude of a body under different conditions.


where

is 1 au, is the phase angle, the angle between the Sun-Body and Body-Observer lines; by the law of cosines, we have:


is the phase integral (integration of reflected light; a number in the 0 to 1 range)

Example: (An ideal diffuse reflecting sphere) - A reasonable first approximation for planetary bodies


A full-phase diffuse sphere reflects ⅔ as much light as a diffuse disc of the same diameter
Distances:
is the distance between the observer and the body
is the distance between the Sun and the body
is the distance between the observer and the Sun

Example
Moon

= +0.25
= = 1 au
= 384.5 Mm = 2.57 mau
How bright is the Moon from Earth?
Full Moon: = 0, ( ≈ 2/3)

(Actual -12.7) A full Moon reflects 30% more light at full phase than a perfect diffuse reflector predicts.
Quarter Moon: = 90°, (if diffuse reflector)

(Actual approximately -11.0) The diffuse reflector formula does better for smaller phases.