Absolute magnitude (M) is a measure of the luminosity of a celestial object, on an inverse logarithmicastronomical magnitude scale. An object's absolute magnitude is defined to be equal to the apparent magnitude that the object would have if it were viewed from a distance of exactly 10.0 parsecs (32.6 light-years), without extinction (or dimming) of its light due to absorption by interstellar matter and cosmic dust. By hypothetically placing all objects at a standard reference distance from the observer, their luminosities can be directly compared on a magnitude scale.
As with all astronomical magnitudes, the absolute magnitude can be specified for different wavelength ranges corresponding to specified filter bands or passbands; for stars a commonly quoted absolute magnitude is the absolute visual magnitude, which uses the visual (V) band of the spectrum (in the UBV photometric system). Absolute magnitudes are denoted by a capital M, with a subscript representing the filter band used for measurement, such as MV for absolute magnitude in the V band.
The more luminous an object, the smaller the numerical value of its absolute magnitude. A difference of 5 magnitudes between the absolute magnitudes of two objects corresponds to a ratio of 100 in their luminosities, and a difference of n magnitudes in absolute magnitude corresponds to a luminosity ratio of 100(n/5). For example, a star of absolute magnitude MV=3.0 would be 100 times more luminous than a star of absolute magnitude MV=8.0 as measured in the V filter band. The Sun has absolute magnitude MV=+4.83.[1] Highly luminous objects can have negative absolute magnitudes: for example, the Milky Way galaxy has an absolute B magnitude of about â20.8.[2]
An object's absolute bolometric magnitude (Mbol) represents its total luminosity over all wavelengths, rather than in a single filter band, as expressed on a logarithmic magnitude scale. To convert from an absolute magnitude in a specific filter band to absolute bolometric magnitude, a bolometric correction (BC) is applied.[3]
For Solar System bodies that shine in reflected light, a different definition of absolute magnitude (H) is used, based on a standard reference distance of one astronomical unit.
Stars and galaxies[edit]
In stellar and galactic astronomy, the standard distance is 10 parsecs (about 32.616 light-years, 308.57 petameters or 308.57 trillion kilometres). A star at 10 parsecs has a parallax of 0.1â³ (100 milliarcseconds). Galaxies (and other extended objects) are much larger than 10 parsecs, their light is radiated over an extended patch of sky, and their overall brightness cannot be directly observed from relatively short distances, but the same convention is used. A galaxy's magnitude is defined by measuring all the light radiated over the entire object, treating that integrated brightness as the brightness of a single point-like or star-like source, and computing the magnitude of that point-like source as it would appear if observed at the standard 10 parsecs distance. Consequently, the absolute magnitude of any object equals the apparent magnitude it would have if it were 10 parsecs away.
The measurement of absolute magnitude is made with an instrument called a bolometer. When using an absolute magnitude, one must specify the type of electromagnetic radiation being measured. When referring to total energy output, the proper term is bolometric magnitude. The bolometric magnitude usually is computed from the visual magnitude plus a bolometric correction, Mbol = MV + BC. This correction is needed because very hot stars radiate mostly ultraviolet radiation, whereas very cool stars radiate mostly infrared radiation (see Planck's law).
Some stars visible to the naked eye have such a low absolute magnitude that they would appear bright enough to outshine the planets and cast shadows if they were at 10 parsecs from the Earth. Examples include Rigel (â7.0), Deneb (â7.2), Naos (â6.0), and Betelgeuse (â5.6). For comparison, Sirius has an absolute magnitude of 1.4, which is brighter than the Sun, whose absolute visual magnitude is 4.83 (it actually serves as a reference point). The Sun's absolute bolometric magnitude is set arbitrarily, usually at 4.75.[4][5]Absolute magnitudes of stars generally range from â10 to +17. The absolute magnitudes of galaxies can be much lower (brighter). For example, the giant elliptical galaxy M87 has an absolute magnitude of â22 (i.e. as bright as about 60,000 stars of magnitude â10).
Apparent magnitude[edit]
The Greek astronomer Hipparchus established a numerical scale to describe the brightness of each star appearing in the sky. The brightest stars in the sky were assigned an apparent magnitude m = 1, and the dimmest stars visible to the naked eye are assigned m = 6.[6] The difference between them corresponds to a factor of 100 in brightness. For objects within the immediate neighborhood of the Sun, the absolute magnitude M and apparent magnitude m from any distance d (in parsecs) is related by:
where F is the radiant flux measured at distance d (in parsecs), F10 the radiant flux measured at distance 10 pc. The relation can be written in terms of logarithm:
where the insignificance of extinction by gas and dust is assumed. Typical extinction rates within the galaxy are 1 to 2 magnitudes per kiloparsec, when dark clouds are taken into account.[7]
For objects at very large distances (outside the Milky Way) the luminosity distance dL (distance defined using luminosity measurements) must be used instead of d (in parsecs), because the Euclidean approximation is invalid for distant objects and general relativity must be taken into account. Moreover, the cosmological redshift complicates the relation between absolute and apparent magnitude, because the radiation observed was shifted into the red range of the spectrum. To compare the magnitudes of very distant objects with those of local objects, a K correction might have to be applied to the magnitudes of the distant objects.
The absolute magnitude M can also be approximated using apparent magnitude m and stellar parallaxp:
or using apparent magnitude m and distance modulusμ:
Examples[edit]
Rigel has a visual magnitude mV of 0.12 and distance about 860 light-years
Vega has a parallax p of 0.129â³, and an apparent magnitude mV of 0.03
The Black Eye Galaxy has a visual magnitude mV of 9.36 and a distance modulus μ of 31.06
Bolometric magnitude[edit]
The bolometric magnitude Mbol, takes into account electromagnetic radiation at all wavelengths. It includes those unobserved due to instrumental passband, the Earth's atmospheric absorption, and extinction by interstellar dust. It is defined based on the luminosity of the stars. In the case of stars with few observations, it must be computed assuming an effective temperature.
Classically, the difference in bolometric magnitude is related to the luminosity ratio according to:[6]
which makes by inversion:
where
In August 2015, the International Astronomical Union passed Resolution B2[8] defining the zero points of the absolute and apparent bolometric magnitude scales in SI units for power (watts) and irradiance (W/m2), respectively. Although bolometric magnitudes had been used by astronomers for many decades, there had been systematic differences in the absolute magnitude-luminosity scales presented in various astronomical references, and no international standardization. This led to systematic differences in bolometric corrections scales.[9] Combined with incorrect assumed absolute bolometric magnitudes for the Sun could lead to systematic errors in estimated stellar luminosities (and stellar properties calculated which rely on stellar luminosity, such as radii, ages, and so on).
Resolution B2 defines an absolute bolometric magnitude scale where Mbol = 0 corresponds to luminosity L0 = 3.0128Ã1028 W, with the zero point luminosityL0 set such that the Sun (with nominal luminosity 3.828Ã1026 W) corresponds to absolute bolometric magnitudeMbol,â = 4.74. Placing a radiation source (e.g. star) at the standard distance of 10 parsecs, it follows that the zero point of the apparent bolometric magnitude scale mbol = 0 corresponds to irradiancef0 = 2.518021002Ã10â8 W/m2. Using the IAU 2015 scale, the nominal total solar irradiance ('solar constant') measured at 1 astronomical unit (1361 W/m2) corresponds to an apparent bolometric magnitude of the Sun of mbol,â = â26.832.[9]
Following Resolution B2, the relation between a star's absolute bolometric magnitude and its luminosity is no longer directly tied to the Sun's (variable) luminosity:
where
The new IAU absolute magnitude scale permanently disconnects the scale from the variable Sun. However, on this SI power scale, the nominal solar luminosity corresponds closely to Mbol = 4.74, a value that was commonly adopted by astronomers before the 2015 IAU resolution.[9]
The luminosity of the star in watts can be calculated as a function of its absolute bolometric magnitude Mbol as:
using the variables as defined previously.
Solar System bodies (H)[edit]
For planets and asteroids a definition of absolute magnitude that is more meaningful for non-stellar objects is used. The absolute magnitude, commonly called H{displaystyle H}, is defined as the apparent magnitude that the object would have if it were one Astronomical Unit (AU) from both the Sun and the observer, and in conditions of ideal solar opposition (an arrangement that is impossible in practice).[10] Solar System bodies are illuminated by the Sun, therefore the magnitude varies as a function of illumination conditions, described by the phase angle. This relationship is referred to as the phase curve. The absolute magnitude is the brightness at phase angle zero, an arrangement known as opposition.
Apparent magnitude[edit]
The phase angle α{displaystyle alpha } can be calculated from the distances body-sun, observer-sun and observer-body, using the law of cosines.
The absolute magnitude H{displaystyle H} can be used to calculate the apparent magnitude m{displaystyle m} of a body. For an object reflecting sunlight, H{displaystyle H} and m{displaystyle m} are connected by the relation
where α{displaystyle alpha } is the phase angle, the angle between the body-Sun and bodyâobserver lines. q(α){displaystyle q(alpha )} is the phase integral (the integration of reflected light; a number in the 0 to 1 range).[11]
By the law of cosines, we have:
Distances:
Approximations for phase integral q(α){displaystyle q(alpha )}[edit]
The value of q(α){displaystyle q(alpha )} depends on the properties of the reflecting surface, in particular on its roughness. In practice, different approximations are used based on the known or assumed properties of the surface.[11]
Planets[edit]
Diffuse reflection on sphere and flat disk
Brightness with phase for diffuse reflection models. The sphere is 2/3 as bright at zero phase, while the disk can't be seen beyond 90 degrees.
Planetary bodies can be approximated reasonably well as ideal diffuse reflectingspheres. Let α{displaystyle alpha } be the phase angle in degrees, then[12]
A full-phase diffuse sphere reflects two-thirds as much light as a diffuse flat disk of the same diameter. A quarter phase (α=90â{displaystyle alpha =90^{circ }}) has 1Ï{displaystyle {frac {1}{pi }}} as much light as full phase (α=0â{displaystyle alpha =0^{circ }}).
For contrast, a diffuse disk reflector model is simply q(α)=cosâ¡Î±{displaystyle q(alpha )=cos {alpha }}, which isn't realistic, but it does represent the opposition surge for rough surfaces that reflect more uniform light back at low phase angles.
The definition of the geometric albedop{displaystyle p}, a measure for the reflectivity of planetary surfaces, is based on the diffuse disk reflector model. The absolute magnitude H{displaystyle H}, diameter D{displaystyle D} (in kilometers) and geometric albedo p{displaystyle p} of a body are related by[13][14]
Example: The Moon's absolute magnitude H{displaystyle H} can be calculated from its diameter D=3474 km{displaystyle D=3474{text{ km}}} and geometric albedop=0.113{displaystyle p=0.113}:[15]
We have dBS=1 AU{displaystyle d_{BS}=1{text{ AU}}}, dBO=384400 km=0.00257 AU.{displaystyle d_{BO}=384400{text{ km}}=0.00257{text{ AU}}.}At quarter phase, q(α)â23Ï{displaystyle q(alpha )approx {frac {2}{3pi }}} (according to the diffuse reflector model), this yields an apparent magnitude of m=+0.28+5log10â¡(1â
0.00257)â2.5log10â¡(23Ï)=â10.99.{displaystyle m=+0.28+5log _{10}{left(1cdot 0.00257right)}-2.5log _{10}{left({frac {2}{3pi }}right)}=-10.99.} The actual value is somewhat lower than that, m=â10.0.{displaystyle m=-10.0.} The phase curve of the Moon is too complicated for the diffuse reflector model.[16]
More advanced models[edit]
Because Solar System bodies are never perfect diffuse reflectors, astronomers use different models to predict apparent magnitudes based on known or assumed properties of the body.[11] For planets, approximations for the correction term â2.5log10â¡q(α){displaystyle -2.5log _{10}{q(alpha )}} in the formula for m have been derived empirically, to match observations at different phase angles. The approximations recommended by the Astronomical Almanac[17] are (with α{displaystyle alpha } in degrees):
Here β{displaystyle beta } is the effective inclination of Saturn's rings (their tilt relative to the observer), which as seen from Earth varies between 0° and 27° over the course of one Saturn orbit, and Ïâ²{displaystyle phi '} is a small correction term depending on Uranus' sub-Earth and sub-solar latitudes. t{displaystyle t} is the Common Era year. Neptune's absolute magnitude is changing slowly due to seasonal effects as the planet moves along its 165-year orbit around the Sun, and the approximation above is only valid after the year 2000. For some circumstances, like αâ¥179â{displaystyle alpha geq 179^{circ }} for Venus, no observations are available, and the phase curve is unknown in those cases.
Example: On 1 January 2019, Venus was dBS=0.719 AU{displaystyle d_{BS}=0.719{text{ AU}}} from the Sun, and dBO=0.645 AU{displaystyle d_{BO}=0.645{text{ AU}}} from Earth, at a phase angle of α=93.0â{displaystyle alpha =93.0^{circ }} (near quarter phase). Under full-phase conditions, Venus would have been visible at m=â4.384+5log10â¡(0.719â
0.645)=â6.09.{displaystyle m=-4.384+5log _{10}{left(0.719cdot 0.645right)}=-6.09.} Accounting for the high phase angle, the correction term above yields an actual apparent magnitude of m=â6.09+(â1.044Ã10â3â
93.0+3.687Ã10â4â
93.02â2.814Ã10â6â
93.03+8.938Ã10â9â
93.04)=â4.59.{displaystyle m=-6.09+left(-1.044times 10^{-3}cdot 93.0+3.687times 10^{-4}cdot 93.0^{2}-2.814times 10^{-6}cdot 93.0^{3}+8.938times 10^{-9}cdot 93.0^{4}right)=-4.59.} This is close to the value of m=â4.62{displaystyle m=-4.62} predicted by the Jet Propulsion Laboratory.[18]
Earth's albedo varies by a factor of 6, from 0.12 in the cloud-free case to 0.76 in the case of altostratus cloud. The absolute magnitude here corresponds to an albedo of 0.434. Earth's apparent magnitude cannot be predicted as accurately as that of most other planets.[17]
Asteroids[edit]
Asteroid 1 Ceres, imaged by the Dawn spacecraft at phase angles of 0°, 7° and 33°. The left image at 0° phase angle shows the brightness surge due to the opposition effect.
Phase integrals for various values of G
Relation between the slope parameter G{displaystyle G} and the opposition surge. Larger values of G{displaystyle G} correspond to a less pronounced opposition effect. For most asteroids, a value of G=0.15{displaystyle G=0.15} is assumed, corresponding to an opposition surge of 0.3 mag{displaystyle 0.3{text{ mag}}}.
If an object has an atmosphere, it reflects light more or less isotropically in all directions, and its brightness can be modelled as a diffuse reflector. Atmosphereless bodies, like asteroids or moons, tend to reflect light more strongly to the direction of the incident light, and their brightness increases rapidly as the phase angle approaches 0â{displaystyle 0^{circ }}. This rapid brightening near opposition is called the opposition effect. Its strength depends on the physical properties of the body's surface, and hence it differs from asteroid to asteroid.[11]
In 1985, the IAU adopted the semi-empiricalHG{displaystyle HG}-system, based on two parameters H{displaystyle H} and G{displaystyle G} called absolute magnitude and slope, to model the opposition effect for the ephemerides published by the Minor Planet Center.[19]
where
and
This relation is valid for phase angles α<120â{displaystyle alpha <120^{circ }}, and works best when α<20â{displaystyle alpha <20^{circ }}.[21]
The slope parameter G{displaystyle G} relates to the surge in brightness, typically 0.3 mag, when the object is near opposition. It is known accurately only for a small number of asteroids, hence for most asteroids a value of G=0.15{displaystyle G=0.15} is assumed.[21] In rare cases, G{displaystyle G} can be negative.[20][22] An example is 101955 Bennu, with G=â0.08{displaystyle G=-0.08}.[23]
In 2012, the HG{displaystyle HG}-system was officially replaced by an improved system with three parameters H{displaystyle H}, G1{displaystyle G_{1}} and G2{displaystyle G_{2}}, which produces more satisfactory results if the opposition effect is very small or restricted to very small phase angles. However, as of 2019, this HG1G2{displaystyle HG_{1}G_{2}}-system has not been adopted by either the Minor Planet Center nor Jet Propulsion Laboratory.[11][24]
The apparent magnitude of asteroids varies as they rotate, on time scales of seconds to weeks depending on their rotation period, by up to 2 mag{displaystyle 2{text{ mag}}} or more.[25] In addition, their absolute magnitude can vary with the viewing direction, depending on their axial tilt. In many cases, neither the rotation period nor the axial tilt are known, limiting the predictability. The models presented here do not capture those effects.[21][11]
Cometary magnitudes[edit]
The brightness of comets is given separately as total magnitude (m1{displaystyle m_{1}}, the brightness integrated over the entire visible extend of the coma) and nuclear magnitude (m2{displaystyle m_{2}}, the brightness of the core region alone).[26] Both are different scales than the magnitude scale used for planets and asteroids, and can not be used for a size comparison with an asteroid's absolute magnitude H.
The activity of comets varies with their distance from the Sun. Their brightness can be approximated as
where m1,2{displaystyle m_{1,2}} are the total and nuclear apparent magnitudes of the comet, respectively, M1,2{displaystyle M_{1,2}} are its 'absolute' total and nuclear magnitudes, dBS{displaystyle d_{BS}} and dBO{displaystyle d_{BO}} are the body-sun and body-observer distances, d0{displaystyle d_{0}} is the Astronomical Unit, and K1,2{displaystyle K_{1,2}} are the slope parameters characterising the comet's activity. For K=2{displaystyle K=2}, this reduces to the formula for a purely reflecting body.[27]
For example, the lightcurve of comet C/2011 L4 (PANSTARRS) can be approximated by M1=5.41, K1=3.69.{displaystyle M_{1}=5.41{text{, }}K_{1}=3.69.}[28] On the day of its perihelion passage, 10 March 2013, comet PANSTARRS was 0.302 AU{displaystyle 0.302{text{ AU}}} from the Sun and 1.109 AU{displaystyle 1.109{text{ AU}}} from Earth. The total apparent magnitude m1{displaystyle m_{1}} is predicted to have been m1=5.41+2.5â
3.69â
log10â¡(0.302)+5log10â¡(1.109)=+0.8{displaystyle m_{1}=5.41+2.5cdot 3.69cdot log _{10}{left(0.302right)}+5log _{10}{left(1.109right)}=+0.8} at that time. The Minor Planet Center gives a value close to that, m1=+0.5{displaystyle m_{1}=+0.5}.[29]
At the same distance, Comet Hale-Bopp is about 130 times brighter than Comet Halley.[30]
The absolute magnitude of any given comet can vary dramatically. It can change as the comet becomes more or less active over time, or if it undergoes an outburst. This makes it difficult to use the absolute magnitude for a size estimate. When comet 289P/Blanpain was discovered in 1819, its absolute magnitude was estimated as M1=8.5{displaystyle M_{1}=8.5}.[32] It was subsequently lost, and was only rediscovered in 2003. At that time, its absolute magnitude had decreased to M1=22.9{displaystyle M_{1}=22.9},[34] and it was realised that the 1819 apparition coincided with an outburst. 289P/Blanpain reached naked eye brightness (5-8 mag) in 1819, even though it is the comet with the smallest nucleus that has ever been physically characterised, and usually doesn't become brighter than 18 mag.[32][33]
For some comets that have been observed at heliocentric distances large enough to distinguish between light reflected from the coma, and light from the nucleus itself, an absolute magnitude analogous to that used for asteroids has been calculated, allowing to estimate the sizes of their nuclei.[35]
Meteors[edit]
For a meteor, the standard distance for measurement of magnitudes is at an altitude of 100 km (62 mi) at the observer's zenith.[36][37]
See also[edit]
References[edit]
External links[edit]
Retrieved from 'https://en.wikipedia.org/w/index.php?title=Absolute_magnitude&oldid=899765021'
The graph of the absolute value function for real numbers
The absolute value of a number may be thought of as its distance from zero.
![]()
In mathematics, the absolute value or modulus|x| of a real numberx is the non-negative value of x without regard to its sign. Namely, |x| = x for a positivex, |x| = âx for a negativex (in which case âx is positive), and |0| = 0. For example, the absolute value of 3 is 3, and the absolute value of â3 is also 3. The absolute value of a number may be thought of as its distance from zero.
Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces. The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical and physical contexts.
Terminology and notation[edit]
In 1806, Jean-Robert Argand introduced the term module, meaning unit of measure in French, specifically for the complex absolute value,[1][2] and it was borrowed into English in 1866 as the Latin equivalent modulus.[1] The term absolute value has been used in this sense from at least 1806 in French[3] and 1857 in English.[4] The notation |x|, with a vertical bar on each side, was introduced by Karl Weierstrass in 1841.[5] Other names for absolute value include numerical value[1] and magnitude.[1] In programming languages and computational software packages, the absolute value of x is generally represented by abs(x), or a similar expression.
The vertical bar notation also appears in a number of other mathematical contexts: for example, when applied to a set, it denotes its cardinality; when applied to a matrix, it denotes its determinant. Vertical bars denote the absolute value only for algebraic objects for which the notion of an absolute value is defined, notably an element of a normed division algebra, for example a real number, a complex number, or a quaternion. A closely related but distinct notation is the use of vertical bars for either the euclidean norm[6] or sup norm[7] of a vector in Rn{displaystyle mathbb {R} ^{n}}, although double vertical bars with subscripts (||â
||2{displaystyle ||cdot ||_{2}} and ||â
||â{displaystyle ||cdot ||_{infty }}, respectively) are a more common and less ambiguous notation.
Definition and properties[edit]Real numbers[edit]
For any real numberx, the absolute value or modulus of x is denoted by |x| (a vertical bar on each side of the quantity) and is defined as[8]
The absolute value of x is thus always either positive or zero, but never negative: when x itself is negative (x < 0), then its absolute value is necessarily positive (|x| = âx > 0).
From an analytic geometry point of view, the absolute value of a real number is that number's distance from zero along the real number line, and more generally the absolute value of the difference of two real numbers is the distance between them. Indeed, the notion of an abstract distance function in mathematics can be seen to be a generalisation of the absolute value of the difference (see 'Distance' below).
Since the square root symbol represents the unique positive square root (when applied to a positive number), it follows that
is equivalent to the definition above, and may be used as an alternative definition of the absolute value of real numbers.[9]
The absolute value has the following four fundamental properties (a, b are real numbers), that are used for generalization of this notion to other domains:
Non-negativity, positive definiteness, and multiplicativity are readily apparent from the definition. To see that subadditivity holds, first note that one of the two alternatives of taking s as either â1 or +1 guarantees that sâ
(a+b)=|a+b|â¥0.{displaystyle scdot (a+b)=|a+b|geq 0.} Now, since â1â
xâ¤|x|{displaystyle -1cdot xleq |x|} and +1â
xâ¤|x|{displaystyle +1cdot xleq |x|}, it follows that, whichever is the value of s, one has sâ
xâ¤|x|{displaystyle scdot xleq |x|} for all real x{displaystyle x}. Consequently, |a+b|=sâ
(a+b)=sâ
a+sâ
bâ¤|a|+|b|{displaystyle |a+b|=scdot (a+b)=scdot a+scdot bleq |a|+|b|}, as desired. (For a generalization of this argument to complex numbers, see 'Proof of the triangle inequality for complex numbers' below.)
Some additional useful properties are given below. These are either immediate consequences of the definition or implied by the four fundamental properties above.
Two other useful properties concerning inequalities are:
These relations may be used to solve inequalities involving absolute values. For example:
The absolute value, as 'distance from zero', is used to define the absolute difference between arbitrary real numbers, the standard metric on the real numbers.
Complex numbers[edit]
The absolute value of a complex number z{displaystyle z} is the distance r{displaystyle r} of z{displaystyle z} from the origin. It is also seen in the picture that z{displaystyle z} and its complex conjugatez¯{displaystyle {bar {z}}} have the same absolute value.
Since the complex numbers are not ordered, the definition given at the top for the real absolute value cannot be directly applied to complex numbers. However, the geometric interpretation of the absolute value of a real number as its distance from 0 can be generalised. The absolute value of a complex number is defined by the Euclidean distance of its corresponding point in the complex plane from the origin. This can be computed using the Pythagorean theorem: for any complex number
where x and y are real numbers, the absolute value or modulus of z is denoted |z| and is defined by[10]
where Re(z) = x and Im(z) = y denote the real and imaginary parts of z, respectively. When the imaginary part y is zero, this coincides with the definition of the absolute value of the real number x.
When a complex number z is expressed in its polar form as
with r=[Re(z)]2+[Im(z)]2â¥0{displaystyle r={sqrt {[mathrm {Re} (z)]^{2}+[mathrm {Im} (z)]^{2}}}geq 0} (and θ â arg(z) is the argument (or phase) of z), its absolute value is
Since the product of any complex number z and its complex conjugatez¯=xâiy,{displaystyle {bar {z}}=x-iy,} with the same absolute value, is always the non-negative real number (x2+y2){displaystyle (x^{2}+y^{2})}, the absolute value of a complex number can be conveniently expressed as
resembling the alternative definition for reals: |x|=xâ
x.{displaystyle |x|={sqrt {xcdot x}}.}
The complex absolute value shares the four fundamental properties given above for the real absolute value.
In the language of group theory, the multiplicative property may be rephrased as follows: the absolute value is a group homomorphism from the multiplicative group of the complex numbers onto the group under multiplication of positive real numbers.[11]
Importantly, the property of subadditivity ('triangle inequality') extends to any finite collection of n complex numbers (zk)k=1n{textstyle (z_{k})_{k=1}^{n}} as
This inequality also applies to infinite families, provided that the infinite seriesâk=1âzk{textstyle sum _{k=1}^{infty }z_{k}} is absolutely convergent. If Lebesgue integration is viewed as the continuous analog of summation, then this inequality is analogously obeyed by complex-valued, measurable functionsf:RâC{displaystyle f:mathbb {R} to mathbb {C} } when integrated over a measurable subsetE{displaystyle E}:
(This includes Riemann-integrable functions over a bounded interval [a,b]{displaystyle [a,b]} as a special case.)
Proof of the complex triangle inequality[edit]
The triangle inequality, as given by (â){displaystyle (*)}, can be demonstrated by applying three easily verified properties of the complex numbers: Namely, for every complex number zâC{displaystyle zin mathbb {C} },
Also, for a family of complex numbers (wk)k=1n{displaystyle (w_{k})_{k=1}^{n}}, âkwk=âkRe(wk)+iâkIm(wk){textstyle sum _{k}w_{k}=sum _{k}mathrm {Re} (w_{k})+isum _{k}mathrm {Im} (w_{k})}. In particular,
Proof of(â){displaystyle (*)}: Choose câC{displaystyle cin mathbb {C} } such that |c|=1{displaystyle |c|=1} and |âkzk|=c(âkzk){textstyle {big |}sum _{k}z_{k}{big |}=c{big (}sum _{k}z_{k}{big )}} (summed over k=1,â¦,n{displaystyle k=1,ldots ,n}). The following computation then affords the desired inequality:
It is clear from this proof that equality holds in (â){displaystyle (*)} exactly if all the czk{displaystyle cz_{k}} are non-negative real numbers, which in turn occurs exactly if all nonzero zk{displaystyle z_{k}} have the same argument, i.e., zk=akζ{displaystyle z_{k}=a_{k}zeta } for a complex constant ζ{displaystyle zeta } and real constants akâ¥0{displaystyle a_{k}geq 0} for 1â¤kâ¤n{displaystyle 1leq kleq n}.
Since f{displaystyle f} measurable implies that |f|{displaystyle |f|} is also measurable, the proof of the inequality (ââ){displaystyle (**)} proceeds via the same technique, by replacing âk(â
){textstyle sum _{k}(cdot )} with â«E(â
)dx{textstyle int _{E}(cdot ),dx} and zk{displaystyle z_{k}} with f(x){displaystyle f(x)}.[12]
Absolute value function[edit]
The graph of the absolute value function for real numbers
Composition of absolute value with a cubic function in different orders
The real absolute value function is continuous everywhere. It is differentiable everywhere except for x = 0. It is monotonically decreasing on the interval (ââ,0] and monotonically increasing on the interval [0,+â). Since a real number and its opposite have the same absolute value, it is an even function, and is hence not invertible. The real absolute value function is a piecewise linear, convex function.
Both the real and complex functions are idempotent.
Relationship to the sign function[edit]
The absolute value function of a real number returns its value irrespective of its sign, whereas the sign (or signum) function returns a number's sign irrespective of its value. The following equations show the relationship between these two functions:
or
and for x â 0,
Derivative[edit]
The real absolute value function has a derivative for every x â 0, but is not differentiable at x = 0. Its derivative for x â 0 is given by the step function:[13][14]
The subdifferential of |x| at x = 0 is the interval[â1,1].[15]
The complex absolute value function is continuous everywhere but complex differentiablenowhere because it violates the CauchyâRiemann equations.[13]
The second derivative of |x| with respect to x is zero everywhere except zero, where it does not exist. As a generalised function, the second derivative may be taken as two times the Dirac delta function.
Antiderivative[edit]
The antiderivative (indefinite integral) of the real absolute value function is
where C is an arbitrary constant of integration. This is not a complex antiderivative because complex antiderivatives can only exist for complex-differentiable (holomorphic) functions, which the complex absolute value function is not.
Distance[edit]
The absolute value is closely related to the idea of distance. As noted above, the absolute value of a real or complex number is the distance from that number to the origin, along the real number line, for real numbers, or in the complex plane, for complex numbers, and more generally, the absolute value of the difference of two real or complex numbers is the distance between them.
The standard Euclidean distance between two points
and
in Euclidean n-space is defined as:
This can be seen as a generalisation, since for a1{displaystyle a_{1}} and b1{displaystyle b_{1}} real, i.e. in a 1-space, according to the alternative definition of the absolute value,
and for a=a1+ia2{displaystyle a=a_{1}+ia_{2}} and b=b1+ib2{displaystyle b=b_{1}+ib_{2}} complex numbers, i.e. in a 2-space,
The above shows that the 'absolute value'-distance, for real and complex numbers, agrees with the standard Euclidean distance, which they inherit as a result of considering them as one and two-dimensional Euclidean spaces, respectively.
The properties of the absolute value of the difference of two real or complex numbers: non-negativity, identity of indiscernibles, symmetry and the triangle inequality given above, can be seen to motivate the more general notion of a distance function as follows:
A real valued function d on a set XâÃâX is called a metric (or a distance function) on X, if it satisfies the following four axioms:[16]
Generalizations[edit]Ordered rings[edit]
The definition of absolute value given for real numbers above can be extended to any ordered ring. That is, if a is an element of an ordered ring R, then the absolute value of a, denoted by |a|, is defined to be:[17]
where âa is the additive inverse of a, 0 is the additive identity element, and < and ⥠have the usual meaning with respect to the ordering in the ring.
Fields[edit]
The four fundamental properties of the absolute value for real numbers can be used to generalise the notion of absolute value to an arbitrary field, as follows.
A real-valued function v on a fieldF is called an absolute value (also a modulus, magnitude, value, or valuation)[18] if it satisfies the following four axioms:
Where 0 denotes the additive identity element of F. It follows from positive-definiteness and multiplicativity that v(1) = 1, where 1 denotes the multiplicative identity element of F. The real and complex absolute values defined above are examples of absolute values for an arbitrary field.
If v is an absolute value on F, then the function d on FâÃâF, defined by d(a,âb) = v(a â b), is a metric and the following are equivalent:
An absolute value which satisfies any (hence all) of the above conditions is said to be non-Archimedean, otherwise it is said to be Archimedean.[19]
Vector spaces[edit]
Again the fundamental properties of the absolute value for real numbers can be used, with a slight modification, to generalise the notion to an arbitrary vector space.
A real-valued function on a vector spaceV over a field F, represented as â·â, is called an absolute value, but more usually a norm, if it satisfies the following axioms:
For all a in F, and v, u in V,
The norm of a vector is also called its length or magnitude.
In the case of Euclidean spaceRn, the function defined by
is a norm called the Euclidean norm. When the real numbers R are considered as the one-dimensional vector space R1, the absolute value is a norm, and is the p-norm (see Lp space) for any p. In fact the absolute value is the 'only' norm on R1, in the sense that, for every norm â·â on R1, âxâ = â1âââ
â|x|. The complex absolute value is a special case of the norm in an inner product space. It is identical to the Euclidean norm, if the complex plane is identified with the Euclidean planeR2.
Composition algebras[edit]
Every composition algebra A has an involutionx â x* called its conjugation. The product in A of an element x and its conjugate x* is written N(x) = x x* and called the norm of x.
The real numbers â, complex numbers â, and quaternions â are all composition algebras with norms given by definite quadratic forms. The absolute value in these division algebras is given by the square root of the composition algebra norm.
In general the norm of a composition algebra may be a quadratic form that is not definite and has null vectors. However, as in the case of division algebras, when an element x has a non-zero norm, then x has a multiplicative inverse given by x*/N(x).
Notes[edit]
References[edit]
External links[edit]
![]()
Retrieved from 'https://en.wikipedia.org/w/index.php?title=Absolute_value&oldid=900887908'
S k i l l
i n A L G E B R A
Table of Contents | Home
12
THIS SYMBOL |x| denotes the absolute value of x, which is the number without its sign. |+3| = 3. |â3| = 3.
Here is the purely algebraic definition of |x|:
If x⥠0, then |x| = x;
if x< 0, then |x| = âx.
Geometrically, |x| is the distance of x from 0.
Both 3 and â3 are a distance of 3 units from 0. |3| = |â3| = 3. Distance, in mathematics, is never negative.
Problem 1. Evaluate the following.
To see the answer, pass your mouse over the colored area.
To cover the answer again, click 'Refresh' ('Reload'). Do the problem yourself first!
Chord micro mu driver free. Problem 2. Explain the following rules.
a) |âx| = |x|
Both âx and x are the same distance from 0. Neither one is ever negative.
b) |2 â x| = |x â 2|
2 â x is the negative of x â 2. Therefore, according to part a), they are equal.
c) |x|² = x²
We may remove the absolute value bars because the left-hand side is never negative, and neither is the right-hand side.
Absolute value equations
|a| = 5.
What values could a have?
a could be either 5 or â5. For, if we replace a with either of those, the statement -- the equation -- will be true.
And so any equation that looks like this --
|a| = b
-- has the two solutions
a = b, or a = âb.
We call whatever appears within the vertical bars -- a in this example -- the argument of the absolute value. Either the argument will be b, or it will be âb.
Example 1. Solve for x:
|x â 2| = 8.
Solution. x â 2 is the argument. Either that argument will be 8, or it will be â8.
x â 2 = 8, or x â 2 = â8.
We must solve these two equations. The first implies
x = 8 + 2 = 10.
The second implies
x = â8 + 2 = â6.
These are the two solutions: x = 10 or â6.
Problem 3.
a) An absolute value equation has how many solutions? Two.
b) Write them for this equation: |x| = 4.
x = 4, or x = â4.
Problem 4. Solve for x.
|x + 5| = 4.
Problem 5. Solve for x.
|1 â x| = 7.
Problem 6. Solve for x.
|2x + 5| = 9.
![]()
Absolute value inequalities
There are two forms of absolute value inequalities. One with less than, |a|< b, and the other with greater than, |a|>b. They are solved differently. Here is the first case.
Example 2. Absolute value less than.
|a| < 3.
For that inequaltiy to be true, what values could a have?
Geometrically, a is less than 3 units from 0.
Therefore,
â3 < a< 3.
This is the solution. The inequality will be true if a has any value between â3 and 3.
In general, if an inequality looks like this --
|a| < b.
-- then the solution will look like this:
âb< a< b
for any argumenta.
Example 3. For which values of x will this inequality be true?
|2x â 1| < 5.
Solution. The argument, 2x â 1, will fall between â5 and 5:
â5 < 2x â 1 < 5.
We must isolate x. First, add 1 to each term of the inequality:
â5 + 1 < 2x< 5 + 1
â4 < 2x< 6.
Now divide each term by 2:
â2 < x< 3.
The inequality will be true for any value of x in that interval.
Problem 7. Solve this inequality for x :
|x + 2| < 7.
â7 < x + 2 < 7.
Subtract 2 from each term:
â7 â 2 < x< 7 â 2
â9 < x< 5.
Problem 8. Solve this inequality for x :
|3x â 5| < 10.
â10 < 3x â 5 < 10.
Add 5 to each term:
â5 < 3x< 15.
Divide each term by 3:
Problem 9. Solve this inequality for x :
|1 â 2x| < 9.
â9 < 1 â 2x< 9.
Subtract 1 from each term:
â10 < â2x< 8.
Divide each term by â2. The sense will change.
5 >x > â4.
That is,
â4 < x< 5.
Example 4. Absolute value greater than.
|a| > 3.
For which values of a will this be true?
Geometrically,
a > 3 or a< â3.
This is the form of the solution, for any argument a:
If
|a| >b (and b > 0),
then
a >b or a< âb.
Problem 10. Solve for x :
|x| > 5.
x > 5 or x< â5.
Problem 11. For which values of x will this be true?
|x + 2| > 7.
The first equation implies x > 5. The second, x< â9.
Problem 12. Solve for x :
|2x + 5| > 9.
2x + 5 > 9, or 2x + 5 < â9.
Problem 13. Solve for x :
|1 â 2x| > 9.
1 â 2x > 9, or 1 â 2x< â9. Solve those two equations. On finally dividing by â2,
The geometrical meaning of |x â a|
Geometrically, |x â a| is the distance of x from a.
|x â 2| means the distance of x from 2. And so if we write
|x â 2| = 4
we mean that x is 4 units aways from 2.
x therefore is equal either to â2 or 6.
On the other hand, if we write
|x â 2| < 4
we mean x is less than 4 units away from 2.
This means that x could have any value in the open interval between â2 and 6.
Problem 14. What is the geometric meaning of |x + a|?
The distance of x from âa. For, |x + a| = |x â(âa)|.
|x + 1|, then, means the distance of x from â1. For example, if
|x + 1| = 2,
then x is 2 units away from â1.
x = â3, or x = 1.
Problem 15. What is the geometrical meaning of each of the following? And therefore what values has x?
a) |x| = 2
x is 2 units away from 0. For, |x| = |x â 0|. x therefore is equal to 2 or â2.
b) |x â 3| = 1
x is 1 unit away from 3. x therefore is equal to 2 or 4.
c) |x + 3| = 1
x is 1 unit away from â3. x therefore is equal to â4 or â2.
d) |x â 5| ⤠2
x is less than or equal to 2 units away from 5. x therefore may take any value in the closed interval between 3 and 7.
e) |x + 5| ⤠2
x is less than or equal to 2 units away from â5. x therefore may take any value in the closed interval between â7 and â3.
Problem 16. |x â 5| < d. State the geometrical meaning of that, and illustrate it on the number line.
x falls within d units of 5.
x therefore falls in the interval between 5 â d and 5 + d.
5 â d< x< 5 + d
Table of Contents | Home
Please make a donation to keep TheMathPage online.
Even $1 will help.
Copyright © 2019 Lawrence Spector
Questions or comments?
E-mail:[email protected]
Private tutoring available.
unitunknown
An absolute machine of a person. Looks like they never leave the gym and usually out to cause trouble.
by Matt Brooksy May 08, 2006
Get a unit mug for your barber Trump.
unitunknown
Used in Beavis and Butthead Do America, unit, refers to penis.
'Uh-huh-huh, do youwant tosee my unit?' --Butthead
Get a unit mug for your brother-in-law James.
unitunknown
Big sturdy bird you wouldn't want to mess with but would want on your side if things started to get tasty.
EreBri, you seen that big old Bella Emberg in accounts?
Aye, a right unit.
Get a unit mug for your mama Larisa.
U.N.I.Tunknown
Hey do you Rasheedcoming down the hall?
Shit. U.N.I.T (You Niggas In Trouble) Bruh
Get a U.N.I.T mug for your dog Beatrix.
Unitunknown
4 Absolute Units Of The Apocalypse 3
1> someone who is indescribable.
2> some one so fucked up, & complicated that they can only be discribed by the word UNIT
by joshua power June 27, 2009
Get a Unit mug for your barber Rihanna.
unitunknown
Male penis. also known as a dick, cock, schlong, meat, wiener, noodle, winky, pipe, peepee, ding-dong, snake, tube, hose, worm, caca, dinkerbell, thing, it, willy.
4 Absolute Units Of The Apocalypse Book
by XSGame September 06, 2005
Get a unit mug for your barber Jerry.
unitunknown
Hard to define, it's a complimentary or derogatory term depending on usage. It can refer to a person as someone that is cool, tall, muscular, strong, or a friend (synonym: dude). It can also refer to a penis or someone that is negatively referred to as a penis (synonym: dick).
You can call anyone 'unit' or 'a unit' or 'the unit' and so on. 'Unit' can be used as a complimentary term or as a derogatory term.
Examples: 'What's up, unittttt!' 'That kid's a fucking unit.' It's all about the emphasis on 'unit'. The usage derives from both 'dude' and 'cock'. 'Dude' is more often complimentary while 'cock' is sometimes just 'cock' and other times 'cock' is derogatory. Examples: 'Ahhhh, unitttt!!!! What's up, kid?!?!?' 'She couldn't handle my gigantic unit.' 'He's the biggest fucking unit.' Also derives from Randy Johnson, also known as 'The Big Unit'. The first known usage of the modern 'unit' terminology was by a Boston man, himself a big unit because of his height, strength, and ability to crush beers.
Get a unit mug for your guy Zora.
Trending RN - July 07, 2019
Purplemath
Taking the absolute value of a negative number makes it positive. For this reason, graphs of absolute value functions tend not to look quite like the graphs of linear functions that you've already studied. However, because of how absolute values behave, it is important to include negative inputs in your T-chart when graphing absolute-value functions. If you do not pick x-values that will put negatives inside the absolute value, you will usually mislead yourself as to what the graph looks like.
For instance, suppose your class is taking the following quiz:
MathHelp.com
One of the other students does what is commonly done: he picks only positive x-values for his T-chart:
Then he plots his points:
Affiliate
These points are fine, as far as they go, but they aren't enough; they don't give an accurate idea of what the graph should look like. In particular, they don't include any 'minus' inputs, so it's easy to forget that those absolute-value bars mean something. As a result, the student forgets to take account of those bars, and draws an erroneous graph:
WRONG ANSWER!
Aaaaaand.. he just flunked the quiz.
But you're more careful. You remember that absolute-value graphs involve absolute values, and that absolute values affect 'minus' inputs. So you pick x-values that put a 'minus' inside the absolute value, and you choose quite a few more points. Your T-chart looks more like this:
Then you plot your points:
..and finally you connect your dots:
You have the correct graph:
Right answer!
Aaaaand.. you just aced the quiz. Good work!
Content Continues Below
While absolute-value graphs tend to look like the one above, with an 'elbow' in the middle, this is not always the case. However, if you see a graph with an elbow like this, you should expect that the graph's equation probably involves an absolute value. In all cases, you should take care that you pick a good range of x-values, because three x-values right next to each other will almost certainly not give you anywhere near enough information to draw a valid picture.
Note: The absolute-value bars make the entered values evaluate to being always non-negative (that is, positive or zero). As a result, the 'V' in the above graph occurred where the sign on the inside was zero. When x was less than â2, the expression x + 2 was less than zero, and the absolute-value bars flipped those 'minus' values from below the x-axis to above it. When x equalled â2, then the argument (that is, the expression inside the bars) equalled zero. For all x-values to the right of â2, the argument was positive, so the absolute-value bars didn't change anything.
In other words, graphically, the absolute-value bars took this graph:
..and flipped the 'minus' part (in green on the graph) from below the x-axis to above it. Noticing where the argument of the absolute-value bars will be zero can be helpful in ensuring that you're doing the graph correctly.
This function is almost the same as the previous one.
Affiliate
However, the argument of the previous absolute-value expression was x + 2. In this case, only the x is inside the absolute-value bars. This argument will be zero when x = 0, so I should expect to see an elbow in that area. Also, since the 'plus two' is outside of the absolute-value bars, I expect my graph to look like the regular absolute-value graph (being a 'V' with the elbow at the origin), but moved upward by two units.
First, I'll fill in my T-chart, making sure to pick some negative x-values as I go:
Then I'll draw my dots and fill in the graph:
Affiliate
Because absolute-value bars always spit out non-negative values, it can be tempting to assume that absolute-value graphs can not go below the x-axis. But they can:
This function is kind of the opposite of the first function (above), because there is a 'minus' on the absolute-value expression on the right-hand side of the equation. Because of this 'minus', the positive values provided by the absolute-value bars will all be switched to negative values. In other words, I should expect this graph to have its elbow at (â2, 0) like the first graph above, but the rest of the graph will be flipped upside down to be below the x-axis.
First, I'll fill in my T-chart:
Then I do my graph:
Also, don't assume that any absolute-value graph will be only ever on one side of the x-axis. The graphs can cross:
4 Absolute Units Of The Apocalypse Lyrics
My T-chart:
..and my graph:
URL: https://www.purplemath.com/modules/graphabs.htm
Comments are closed.
|
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |