formulasearchengine - User contributions [en]

Completeness (statistics)

2014-12-01T12:49:01Z

132.66.40.169:

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Quantum cohomology

2014-08-06T08:44:42Z

132.66.40.239:

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2014-07-29T12:46:53Z

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Fresnel diffraction

2014-02-19T09:02:34Z

132.66.51.110: Very well, but, F>>1 is not a sufficient condition for the validity of the Fresnel approximation. See Saleh and Teich pg.119 eq.(4.1-10) and the text on pg.123 below eq.(4.2-4) !

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2014-02-13T07:22:15Z

132.66.50.87: /* Limitations */

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2014-02-06T16:04:05Z

132.66.40.163: /* Lie theory */

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Particle in a box

2014-01-28T16:48:16Z

132.66.201.158: /* One-dimensional solution */

In [[physics]], specifically [[statistical mechanics]], a '''population inversion''' occurs when a system (such as a group of [[atom]]s or [[molecule]]s) exists in a state with more members in an [[excited state]] than in lower energy states. The concept is of fundamental importance in [[laser science]] because the production of a population inversion is a necessary step in the workings of a standard [[laser]].

==Boltzmann distributions and thermal equilibrium==
To understand the concept of a population inversion, it is necessary to understand some [[thermodynamics]] and the way that [[light]] interacts with [[matter]]. To do so, it is useful to consider a very simple assembly of atoms forming a [[active laser medium|laser medium]].

Assume there are a group of ''N'' atoms, each of which is capable of being in one of two [[energy]] states, either
# The ''ground state'', with energy ''E''1; or
# The ''excited state'', with energy ''E''2, with ''E''2 > ''E''1.
The number of these atoms which are in the ground state is given by ''N''1, and the number in the excited state ''N''2. Since there are ''N'' atoms in total,
:<math>N_1+N_2 = N</math>
The energy difference between the two states, given by
:<math>\Delta E_{12} = E_2-E_1,</math>
determines the characteristic [[frequency]] ν12 of light which will interact with the atoms; This is given by the relation
:<math>E_2-E_1 = \Delta E = h\nu_{12},</math>
''h'' being [[Planck's constant]].

If the group of atoms is in [[thermal equilibrium]], it can be shown from [[thermodynamics]] that the ratio of the number of atoms in each state is given by the [[Boltzmann factor]]:
:<math>\frac{N_2}{N_1} = \exp{\frac{-(E_2-E_1)}{kT}},</math>
where ''T'' is the [[thermodynamic temperature]] of the group of atoms, and ''k'' is [[Boltzmanns constant|Boltzmann's constant]].

We may calculate the ratio of the populations of the two states at room temperature (''T'' ≈ 300 [[kelvin|K]]) for an energy difference Δ''E'' that corresponds to light of a frequency corresponding to visible light (ν ≈ 5×1014 Hz). In this case Δ''E'' = ''E''2 - ''E''1 ≈ 2.07 eV, and ''kT'' ≈ 0.026 eV. Since ''E''2 - ''E''1 ≫ ''kT'', it follows that the argument of the exponential in the equation above is a large negative number, and as such ''N''2/''N''1 is vanishingly small; i.e., there are almost no atoms in the excited state. When in thermal equilibrium, then, it is seen that the lower energy state is more populated than the higher energy state, and this is the normal state of the system. As ''T'' increases, the number of electrons in the high-energy state (''N''2) increases, but ''N''2 never exceeds ''N''1 for a system at thermal equilibrium; rather, at infinite temperature, the populations ''N''2 and ''N''1 become equal. In other words, a population inversion (''N''2/''N''1 > 1) can never exist for a system at thermal equilibrium. To achieve population inversion therefore requires pushing the system into a non-equilibrated state.

==The interaction of light with matter==
There are three types of possible interactions between a system of atoms and light that are of interest:

===Absorption===
{{Main|Absorption (optics)}}

If light ([[photon]]s) of [[frequency]] ν12 pass through the group of atoms, there is a possibility of the light being absorbed by atoms which are in the ground state, which will cause them to be excited to the higher energy state. The rate of absorption is [[proportionality (mathematics)|proportional]] to the radiation intensity of the light, and also to the number of atoms currently in the ground state, ''N''1.

===Spontaneous emission===
{{Main|Spontaneous emission}}

If a collection of atoms are in the excited state, spontaneous decay events to the ground state will occur at a rate proportional to ''N''2, the number of atoms in the excited state. The energy difference between the two states Δ''E''21 is emitted from the atom as a photon of frequency ν21 as given by the frequency-energy relation above.

The photons are emitted [[stochastic]]ally, and there is no fixed [[phase (waves)|phase]] relationship between photons emitted from a group of excited atoms; in other words, spontaneous emission is [[coherence (physics)|incoherent]]. In the absence of other processes, the number of atoms in the excited state at time ''t'', is given by
:<math>N_2(t) = N_2(0) \exp{\frac{-t}{\tau_{21}}},</math>
where ''N''2(0) is the number of excited atoms at time ''t'' = 0, and τ21 is the ''[[mean lifetime|lifetime]]'' of the transition between the two states.

===Stimulated emission===
{{Main|Stimulated emission}}
If an atom is already in the excited state, it may be perturbed by the passage of a photon that has a [[frequency]] ν21 corresponding to the energy gap Δ''E'' of the excited state to ground state transition. In this case, the excited atom relaxes to the ground state, and is induced to produce a second photon of frequency ν21. The original photon is not absorbed by the atom, and so the result is two photons of the same frequency. This process is known as ''stimulated emission''.

Specifically, an excited atom will act like a small electric dipole which will oscillate with the external field provided. One of the consequences of this oscillation is that it encourages electrons to decay to the lowest energy state. When this happens due to the presence of the electromagnetic field from a photon, a photon is released in the same phase and direction as the "stimulating" photon, and is called stimulated emission.

[[Image:Stimulated Emission.svg|center|550px]]

The rate at which stimulated emission occurs is proportional to the number of atoms ''N''2 in the excited state, and the radiation density of the light. The base probability of a photon causing stimulated emission in a single excited atom was shown by [[Albert Einstein]] to be exactly equal to the probability of a photon being absorbed by an atom in the ground state. Therefore, when the numbers of atoms in the ground and excited states are equal, the rate of stimulated emission is equal to the rate of absorption for a given radiation density.

The critical detail of stimulated emission is that the induced photon has the same [[frequency]] and [[phase (waves)|phase]] as the incident photon. In other words, the two photons are [[coherence (physics)|coherent]]. It is this property that allows [[optical amplifier|optical amplification]], and the production of a [[laser]] system. During the operation of a laser, all three light-matter interactions described above are taking place. Initially, atoms are energized from the ground state to the excited state by a process called ''[[Laser pumping|pumping]]'', described below. Some of these atoms decay via spontaneous emission, releasing incoherent light as photons of frequency, ν. These photons are fed back into the laser medium, usually by an [[laser construction|optical resonator]]. Some of these photons are absorbed by the atoms in the ground state, and the photons are lost to the laser process. However, some photons cause stimulated emission in excited-state atoms, releasing another coherent photon. In effect, this results in ''optical amplification''.

If the number of photons being amplified per unit time is greater than the number of photons being absorbed, then the net result is a continuously increasing number of photons being produced; the laser medium is said to have a gain of greater than unity.

Recall from the descriptions of absorption and stimulated emission above that the rates of these two processes are proportional to the number of atoms in the ground and excited states, ''N''1 and ''N''2, respectively. If the ground state has a higher population than the excited state (''N''1 > ''N''2), the process of absorption dominates and there is a net attenuation of photons. If the populations of the two states are the same (''N''1 = ''N''2), the rate of absorption of light exactly balances the rate of emission; the medium is then said to be ''optically transparent''.

If the higher energy state has a greater population than the lower energy state (''N''1 < ''N''2), then the emission process dominates, and light in the system undergoes a net increase in intensity. It is thus clear that to produce a faster rate of stimulated emissions than absorptions, it is required that the ratio of the populations of the two states is such that
''N''2/''N''1 > 1; In other words, a population inversion is required for laser operation.

==Selection rules==
{{Main|Selection rule}}
Many transitions involving electromagnetic radiation are strictly forbidden under quantum mechanics. The allowed transitions are described by so-called [[selection rule]]s, which describe the conditions under which a radiative transition is allowed. For instance, transitions are only allowed if Δ''S'' = 0, ''S'' being the total spin angular momentum of the system. In real materials other effects, such as interactions with the crystal lattice, intervene to circumvent the formal rules. In these systems the forbidden transitions can occur, but usually at slower rates than allowed transitions. A classic example is [[phosphorescence]] where a material has a ground state with ''S'' = 0, an excited state with ''S'' = 0, and an intermediate state with ''S'' = 1. The transition from the intermediate state to the ground state by emission of light is slow because of the selection rules. Thus emission may continue after the external illumination is removed. In contrast [[fluorescence]] in materials is characterized by emission which ceases when the external illumination is removed.

Transitions which do not involve the absorption or emission of radiation are not affected by selection rules. Radiationless transition between levels, such as between the excited ''S'' = 0 and ''S'' = 1 states, may proceed quickly enough to siphon off a portion of the ''S'' = 0 population before it spontaneously returns to the ground state.

The existence of intermediate states in materials is essential to the technique of optical pumping of lasers (see below).

==Creating a population inversion==
As described above, a population inversion is required for [[laser]] operation, but cannot be achieved in our theoretical group of atoms with two energy-levels when they are in thermal equilibrium. In fact, any method by which the atoms are directly and continuously excited from the ground state to the excited state (such as optical absorption) will eventually reach equilibrium with the de-exciting processes of spontaneous and stimulated emission. At best, an equal population of the two states, ''N''1 = ''N''2 = ''N''/2, can be achieved, resulting in optical transparency but no net optical gain.

===Three-level lasers===
[[Image:Population-inversion-3level.png|frame|right|A three-level laser energy diagram.]]
To achieve non-equilibrium conditions, an indirect method of populating the excited state must be used. To understand how this is done, we may use a slightly more realistic model, that of a ''three-level laser''. Again consider a group of ''N'' atoms, this time with each atom able to exist in any of three energy states, levels 1, 2 and 3, with energies ''E''1, ''E''2, and ''E''3, and populations ''N''1, ''N''2, and ''N''3, respectively.

Note that ''E''1 < ''E''2 < ''E''3; that is, the energy of level 2 lies between that of the ground state and level 3.

Initially, the system of atoms is at thermal equilibrium, and the majority of the atoms will be in the ground state, i.e., ''N''1 ≈ ''N'', ''N''2 ≈ ''N''3 ≈ 0. If we now subject the atoms to light of a frequency <math>\scriptstyle\nu_{13} \,=\, \frac{1}{h}\left(E_3 - E_1\right)</math>, the process of optical absorption will excite the atoms from the ground state to level 3. This process is called ''[[Laser pumping|pumping]]'' and does not necessarily always directly involve light absorption; other methods of exciting the laser medium, such as electrical discharge or chemical reactions, may be used. The level 3 is sometimes referred to as the ''pump level'' or ''pump band'', and the energy transition ''E''1 → ''E''3 as the ''pump transition'', which is shown as the arrow marked '''P''' in the diagram on the right.

If we continue pumping the atoms, we will excite an appreciable number of them into level 3, such that ''N''3 > 0. In a medium suitable for laser operation, we require these excited atoms to quickly decay to level 2. The energy released in this transition may be emitted as a photon (spontaneous emission), however in practice the 3→2 transition (labeled '''R''' in the diagram) is usually ''radiationless'', with the energy being transferred to vibrational motion ([[heat]]) of the host material surrounding the atoms, without the generation of a photon.

An atom in level 2 may decay by spontaneous emission to the ground state, releasing a photon of frequency ''ν''12 (given by ''E''2 – ''E''1 = ''hν''12), which is shown as the transition '''L''', called the ''laser transition'' in the diagram. If the lifetime of this transition, τ21 is much longer than the lifetime of the radiationless 3 → 2 transition τ32 (if τ21 ≫ τ32, known as a ''favourable lifetime ratio''), the population of the ''E''3 will be essentially zero (''N''3 ≈ 0) and a population of excited state atoms will accumulate in level 2 (''N''2 > 0). If over half the ''N'' atoms can be accumulated in this state, this will exceed the population of the ground state ''N''1. A population inversion (''N''2 > ''N''1 ) has thus been achieved between level 1 and 2, and optical amplification at the frequency ν21 can be obtained.

Because at least half the population of atoms must be excited from the ground state to obtain a population inversion, the laser medium must be very strongly pumped. This makes three-level lasers rather inefficient, despite being the first type of laser to be discovered (based on a [[ruby]] laser medium, by [[Theodore Maiman]] in 1960). A three-level system could also have a radiative transition between level 3 and 2, and a non-radiative transition between 2 and 1. In this case, the pumping requirements are weaker. In practice, most lasers are ''four-level lasers'', described below.

===Four-level lasers===
[[Image:Population-inversion-4level.png|frame|right|A four-level laser energy diagram.]]
Here, there are four energy levels, energies ''E''1, ''E''2, ''E''3, ''E''4, and populations ''N''1, ''N''2, ''N''3, ''N''4, respectively. The energies of each level are such that ''E''1 < ''E''2 < ''E''3 < ''E''4.

In this system, the pumping transition '''P''' excites the atoms in the ground state (level 1) into the pump band (level 4). From level 4, the atoms again decay by a fast, non-radiative transition '''Ra''' into the level 3. Since the lifetime of the laser transition '''L''' is long compared to that of '''Ra''' (τ32 ≫ τ43), a population accumulates in level 3 (the ''upper laser level''), which may relax by spontaneous or stimulated emission into level 2 (the ''lower laser level''). This level likewise has a fast, non-radiative decay '''Rb''' into the ground state.

As before, the presence of a fast, radiationless decay transition results in the population of the pump band being quickly depleted (''N''4 ≈ 0). In a four-level system, any atom in the lower laser level ''E''2 is also quickly de-excited, leading to a negligible population in that state (''N''2 ≈ 0). This is important, since any appreciable population accumulating in level 3, the upper laser level, will form a population inversion with respect to level 2. That is, as long as ''N''3 > 0, then ''N''3 > ''N''2 and a population inversion is achieved. Thus optical amplification, and laser operation, can take place at a frequency of ν32 (''E''3-''E''2 = ''h''ν32).

Since only a few atoms must be excited into the upper laser level to form a population inversion, a four-level laser is much more efficient than a three-level one, and most practical lasers are of this type. In reality, many more than four energy levels may be involved in the laser process, with complex excitation and relaxation processes involved between these levels. In particular, the pump band may consist of several distinct energy levels, or a continuum of levels, which allow optical pumping of the medium over a wide range of wavelengths.

Note that in both three- and four-level lasers, the energy of the pumping transition is greater than that of the laser transition. This means that, if the laser is optically pumped, the frequency of the pumping light must be greater than that of the resulting laser light. In other words, the pump wavelength is shorter than the laser wavelength. It is possible in some media to use multiple photon absorptions between multiple lower-energy transitions to reach the pump level; such lasers are called ''up-conversion'' lasers.

While in many lasers the laser process involves the transition of atoms between different [[electron]]ic energy states, as described in the model above, this is not the only mechanism that can result in laser action. For example, there are many common lasers (e.g., dye lasers, [[carbon dioxide laser]]s) where the laser medium consists of complete molecules, and energy states correspond to vibrational and rotational modes of oscillation of the molecules. This is the case with water [[maser]]s, that occur in nature.

In some media it is possible, by imposing an additional optical or microwave field, to use [[quantum coherence]] effects to reduce the likelihood of an excited-state to ground-state transition. This technique, known as [[lasing without inversion]], allows optical amplification to take place without producing a population inversion between the two states.

==Other methods of creating a population inversion==
Stimulated emission was first observed in the microwave region of the electromagnetic spectrum, giving rise to the acronym [[Maser|MASER]] for Microwave Amplification by Stimulated Emission of Radiation. In the microwave region, the Boltzmann distribution of molecules among energy states is such that, at room temperature all states are populated almost equally.

To create a population inversion under these conditions, it is necessary to selectively remove some atoms or molecules from the system based on differences in properties. For instance, in a [[Maser#Hydrogen maser|hydrogen Maser]], the well-known "[[21 cm line|21cm wave]]" transition in atomic hydrogen, where the lone electron flips its spin state from parallel to the nuclear spin to antiparallel, can be used to create a population inversion because the parallel state has a magnetic moment and the antiparallel state does not. A [[Stern-Gerlach experiment|strong inhomogeneous magnetic field]] will separate out atoms in the higher energy state from a beam of mixed state atoms. The separated population represents a population inversion which can exhibit stimulated emissions.

==See also==
*[[Quantum electronics]]
*[[Negative absolute temperature]]

==References==
*Svelto, Orazio (1998). ''Principles of Lasers'', 4th ed. (trans. David Hanna), Springer. ISBN 0-306-45748-2
{{Use dmy dates|date=September 2010}}

{{DEFAULTSORT:Population Inversion}}
[[Category:Laser science]]
[[Category:Statistical mechanics]]

Variable-order Markov model

2014-01-10T13:01:57Z

132.66.50.87: /* Application areas */

{{numeral systems}}
A '''negative base''' (or negative [[radix]]) may be used to construct a [[non-standard positional numeral system]]. Like other place-value systems, each position holds multiples of the appropriate power of the system's base; but that base is negative—that is to say, the base <math>\scriptstyle b</math> is equal to <math>\scriptstyle -r</math> for some natural number <math>\scriptstyle r</math> (''r ≥ 2'').

Negative-base systems can accommodate all the same numbers as standard place-value systems, but both positive and negative numbers are represented without the use of a [[minus sign]] (or, in computer representation, a [[sign bit]]); this advantage is countered by an increased complexity of arithmetic operations. The need to store the information normally contained by a negative sign often results in a negative-base number being one digit longer than its positive-base equivalent.

The common names for negative-base positional numeral systems are formed by [[prefix (linguistics)|prefixing]] ''nega-'' to the name of the corresponding positive-base system; for example, '''negadecimal''' (base −10) corresponds to [[decimal]] (base 10), '''negaternary''' (base −3) to [[ternary numeral system|ternary]] (base 3), and '''negabinary''' (base −2) to [[binary numeral system|binary]] (base 2).<ref>{{harvnb|Knuth|1998}} and [[#WeissteinNegadecimal|Weisstein]] each refer to the negadecimal system. In the index {{harvnb|Knuth|1998}} refers to the negabinary system, as does [[#WeissteinNegabinary|Weisstein]]. The negaternary system is discussed briefly in {{Citation | last1=Petkovšek | first1=Marko | author1-link=Marko Petkovšek | title=Ambiguous numbers are dense | doi=10.2307/2324393 | mr=1048915 | year=1990 | journal=[[American Mathematical Monthly|The American Mathematical Monthly]] | issn=0002-9890 | volume=97 | issue=5 | pages=408–411}}.</ref>

==Example==
Consider what is meant by the representation ''12,243'' in the negadecimal system, whose base <math>\scriptstyle b</math> is −10:

{| class="wikitable"
|-
|align="center"| multiples of <math>\scriptstyle b^4</math> (i.e., 10,000)
|align="center"| multiples of <math>\scriptstyle b^3</math> (i.e., −1,000)
|align="center"| multiples of <math>\scriptstyle b^2</math> (i.e., 100)
|align="center"| multiples of <math>\scriptstyle b^1</math> (i.e., −10)
|align="center"| multiples of <math>\scriptstyle b^0</math> (i.e., 1)
|-
|align="center"| 1
|align="center"| 2
|align="center"| 2
|align="center"| 4
|align="center"| 3
|-
|}

Since 10,000 + (−2,000) + 200 + (−40) + 3 = 8,163, the representation ''12,243'' in negadecimal notation is equivalent to ''8,163'' in decimal notation.

==History==
Negative numerical bases were first considered by [[Vittorio Grünwald]] in his work ''Giornale di Matematiche di Battaglini'', published in 1885. Grünwald gave algorithms for performing addition, subtraction, multiplication, division, root extraction, divisibility tests, and radix conversion. Negative bases were later independently rediscovered by [[Aubrey J. Kempner|A. J. Kempner]] in 1936 and [[Zdzisław Pawlak]] and A. Wakulicz in 1959{{Citation needed|date=May 2012}}.

Negabinary was implemented in the early [[Poland|Polish]] computer [[BINEG]], built 1957–59, based on ideæ by Z. Pawlak and A. Lazarkiewicz from the [[Mathematical Institute]] in [[Warsaw]].<ref>[http://chc60.fgcu.edu/images/articles/Marczynski.pdf Marczynski, R. W., "The First Seven Years of Polish Computing"], IEEE Annals of the History of Computing, Vol. 2, No 1, January 1980</ref> Implementations since then have been rare.

==Notation and use==
Denoting the base as <math>-r</math>, every [[integer]] <math>a</math> can be written uniquely as
:<math>a = \sum_{i=0}^{n}d_{i}(-r)^{i}</math>
where each digit <math>\scriptstyle d_k</math> is an integer from 0 to <math>\scriptstyle r - 1</math> and the leading digit <math>\scriptstyle d_n</math> is <math>\scriptstyle > 0</math> (unless <math>\scriptstyle n=0</math>). The base <math>\scriptstyle -r</math> expansion of <math>\scriptstyle a</math> is then given by the string <math>\scriptstyle d_n d_{n-1} \ldots d_1 d_0</math>.

Negative-base systems may thus be compared to [[signed-digit representation]]s, such as [[balanced ternary]], where the radix is positive but the digits are taken from a partially negative range.

Some numbers have the same representation in base <math>\scriptstyle -r</math> as in base <math>r</math>. For example, the numbers from 100 to 109 have the same representations in decimal and negadecimal. Similarly,
:<math>17=2^4+2^0=(-2)^4+(-2)^0</math>
and is represented by 10001 in binary and 10001 in negabinary.

Some numbers with their expansions in a number of positive and corresponding negative bases are:

{|class="wikitable" style="margin: 1em auto 1em auto; text-align: right;"
! Decimal !! Negadecimal !! Binary !! Negabinary !! Ternary !! Negaternary
|-
| −15 || 25 || −1111 || 110001 || −120 || 1220
|-
| : || : || : || : || : || :
|-
| −5 || 15 || −101 || 1111 || −12 || 21
|-
| −4 || 16 || −100 || 1100 || −11 || 22
|-
| −3 || 17 || −11 || 1101 || −10 || 10
|-
| −2 || 18 || −10 || 10 || −2 || 11
|-
| −1 || 19 || −1 || 11 || −1 || 12
|-
| 0 || 0 || 0 || 0 || 0 || 0
|-
| 1 || 1 || 1 || 1 || 1 || 1
|-
| 2 || 2 || 10 || 110 || 2 || 2
|-
| 3 || 3 || 11 || 111 || 10 || 120
|-
| 4 || 4 || 100 || 100 || 11 || 121
|-
| 5 || 5 || 101 || 101 || 12 || 122
|-
| 6 || 6 || 110 || 11010 || 20 || 110
|-
| 7 || 7 || 111 || 11011 || 21 || 111
|-
| 8 || 8 || 1000 || 11000 || 22 || 112
|-
| 9 || 9 || 1001 || 11001 || 100 || 100
|-
| 10 || 190 || 1010 || 11110 || 101 || 101
|-
| 11 || 191 || 1011 || 11111 || 102 || 102
|-
| 12 || 192 || 1100 || 11100 || 110 || 220
|-
| 13 || 193 || 1101 || 11101 || 111 || 221
|-
| 14 || 194 || 1110 || 10010 || 112 || 222
|-
| 15 || 195 || 1111 || 10011 || 120 || 210
|-
| 16 || 195 || 10000 || 10000 || 121 || 211
|-
| 17 || 197 || 10001 || 10001 || 122 || 212
|}

Note that the base <math>\scriptstyle -r</math> expansions of negative integers have an [[even number]] of digits, while the base <math>\scriptstyle -r</math> expansions of the non-negative integers have an [[odd number]] of digits.

==Calculation==
The base <math>\scriptstyle -r</math> expansion of a number can be found by repeated division by <math>\scriptstyle -r</math>, recording the non-negative remainders of <math>\scriptstyle 0, 1,\ldots r-1</math>, and concatenating those remainders, starting with the last. Note that if <math>\scriptstyle a / b = c</math>, remainder <math>d</math>, then <math>\scriptstyle bc + d = a</math>. For example, in negaternary:
:<math>\begin{align}
146 & ~/~ -3 = & -48, & ~\mbox{remainder}~ 2 \\
-48 & ~/~ -3 = & 16, & ~\mbox{remainder}~ 0 \\
16 & ~/~ -3 = & -5, & ~\mbox{remainder}~ 1 \\
-5 & ~/~ -3 = & 2, & ~\mbox{remainder}~ 1 \\
2 & ~/~ -3 = & 0, & ~\mbox{remainder}~ 2 \\
\end{align}</math>
Therefore, the negaternary expansion of 146 is 21,102.

Note that in most [[programming languages]], the result (in integer arithmetic) of dividing a negative number by a negative number is rounded towards 0, usually leaving a negative remainder. In such a case we have <math>\scriptstyle a = (-r)c + d = (-r)c + d - r + r = (-r)(c + 1) + (d + r)</math>. Because <math>\scriptstyle |d| < r</math>, <math>\scriptstyle (d + r)</math> is the positive remainder. Therefore, to get the correct result in such case, computer implementations of the above algorithm should add 1 and <math>r</math> to the quotient and remainder respectively (shown below in the [[Python (programming language)|Python]] programming language):
<source lang="python">
def negaternary(i):
digits = ''
if not i:
digits = '0'
else:
while i != 0:
i, remainder = divmod(i, -3)
if remainder < 0:
i, remainder = i + 1, remainder + 3
digits = str(remainder)+ digits
return digits
</source>
C# implementation:
<source lang="csharp">
static string negatenary(int value)
{
string result = string.Empty;

while (value != 0)
{
int remainder = value % -3;
value = value / -3;

if (remainder < 0)
{
remainder += 3;
value += 1;
}

result = remainder.ToString() + result;
}

return result;
}
</source>
Common Lisp implementation:
<source lang="lisp">
(defun negaternary (i)
(if (zerop i)
"0"
(let ((digits "")
(rem 0))
(loop while (not (zerop i)) do
(progn
(multiple-value-setq (i rem) (truncate i -3))
(when (minusp rem)
(incf i)
(incf rem 3))
(setf digits (concatenate 'string (write-to-string rem) digits))))
digits)))
</source>

==Arithmetic operations==
The following describes the arithmetic operations for negabinary; calculations in larger bases are similar.

===Addition===
To add two negabinary numbers, start with a carry of 0, and, starting from the [[least significant bit]]s, add the bits of the two numbers plus the carry. The resulting number is then looked up in the following table to get the bit to write down as result, and the next carry:

{| class="wikitable"
! Number !! Bit !! Carry !! Comment
|-
| −2 || 0 || 1 || −2 occurs only during subtraction.
|-
| −1 || 1 || 1 ||
|-
| 0 || 0 || 0 ||
|-
| 1 || 1 || 0 ||
|-
| 2 || 0 || −1 ||
|-
| 3 || 1 || −1 || 3 occurs only during addition.
|}

The second row of this table, for instance, expresses the fact that '''−1''' = '''1''' + '''1''' × −2; the fifth row says '''2''' = '''0''' + '''−1''' × −2; etc.

As an example, to add 1010101 (1 + 4 + 16 + 64 = 85) and 1110100 (4 + 16 − 32 + 64 = 52),

carry: 1 −1 0 −1 1 −1 0 0 0
first number: 1 0 1 0 1 0 1
second number: 1 1 1 0 1 0 0 +
--------------------------
number: 1 −1 2 0 3 −1 2 0 1
bit (result): 1 1 0 0 1 1 0 0 1
carry: 0 1 −1 0 −1 1 −1 0 0

so the result is 110011001 (1 − 8 + 16 − 128 + 256 = 137).

==== Another Method ====
While adding two negabinary numbers, every time a carry is generated an extra carry should be propagated to next bit.
Consider same example as above
extra carry: 1 1 0 1 0 0 0
carry: 1 0 1 1 0 1 0 0 0
first number: 1 0 1 0 1 0 1
second number: 1 1 1 0 1 0 0 +
--------------------------
Answer: 1 1 0 0 1 1 0 0 1

===Subtraction===
To subtract, multiply each bit of the second number by −1, and add the numbers, using the same table as above.

As an example, to compute 1101001 (1 − 8 − 32 + 64 = 25) minus 1110100 (4 + 16 − 32 + 64 = 52),

carry: 0 1 −1 1 0 0 0
first number: 1 1 0 1 0 0 1
second number: −1 −1 −1 0 −1 0 0 +
--------------------
number: 0 1 −2 2 −1 0 1
bit (result): 0 1 0 0 1 0 1
carry: 0 0 1 −1 1 0 0

so the result is 100101 (1 + 4 −32 = −27).

To negate a number, compute 0 minus the number.

===Multiplication and division===
Shifting to the left multiplies by −2, shifting to the right divides by −2.

To multiply, multiply like normal [[decimal]] or [[binary numeral system|binary]] numbers, but using the negabinary rules for adding the carry, when adding the numbers.

first number: 1 1 1 0 1 1 0
second number: 1 0 1 1 0 1 1 *
-------------------------------------
1 1 1 0 1 1 0
1 1 1 0 1 1 0

1 1 1 0 1 1 0
1 1 1 0 1 1 0

1 1 1 0 1 1 0 +
-------------------------------------
carry: 0 −1 0 −1 −1 −1 −1 −1 0 −1 0 0
number: 1 0 2 1 2 2 2 3 2 0 2 1 0
bit (result): 1 0 0 1 0 0 0 1 0 0 0 1 0
carry: 0 −1 0 −1 −1 −1 −1 −1 0 −1 0 0

For each column, add ''carry'' to ''number'', and divide the sum by −2, to get the new ''carry'', and the resulting bit as the remainder.


==Fractional numbers==
Base <math>\scriptstyle -r</math> representation may of course be carried beyond the [[radix point]], allowing the representation of non-integral numbers.

As with positive-base systems, terminating representations correspond to fractions where the denominator is a power of the base; repeating representations correspond to other rationals, and for the same reason.

===Non-unique representations===
Unlike positive-base systems, where integers and terminating fractions have non-unique representations (for example, in decimal [[0.999… = 1]]) in negative-base systems the integers have only a single representation. However, there do exist rationals with non-unique representations; for example, in negaternary,
: <math>0.(02)\ldots_{(-3)} = \frac{1}{4} = 1.(20)\ldots_{(-3)}</math>.

Such non-unique representations can be found by considering the largest and smallest possible representations with integral parts 0 and 1 respectively, and then noting that they are equal. (Indeed, this works with any integral-base system.) The rationals thus non-uniquely expressible are those of form
: <math>\frac{ar + 1}{b(r + 1)}</math>.

== Imaginary base ==
{{main|Complex base systems}}

Just as using a negative base allows the representation of negative numbers without an explicit negative sign, using an [[imaginary number|imaginary]] base allows the representation of [[Gaussian integer]]s. [[Donald Knuth]] proposed the [[quater-imaginary base]] (base 2i) in 1955.<ref>D. Knuth. The Art of Computer Programming. Volume 2, 3rd Edition. Addison-Wesley. pp. 205, "Positional Number Systems"</ref>

Imaginary-base arithmetic is not much different from negative-base arithmetic, since an imaginary-base number may be considered as the interleave of its real and imaginary parts; using [[INTERCAL]]-72 notation,
: ''x''(2i) + (2''i'')''y''(2i) = ''x''(2i) ¢ ''y''(2i).

== See also ==
* [[Quater-imaginary base]]
* [[Binary numeral system|Binary]]
* [[Balanced ternary]]
* [[Numeral system]]s

== Notes ==
<references/>

==References==
* Vittorio Grünwald. ''Giornale di Matematiche di Battaglini'' (1885), 203-221, 367
* A. J. Kempner. (1936), 610-617
* Z. Pawlak and A. Wakulicz ''Bulletin de l'Academie Polonaise des Scienses'', Classe III, 5 (1957), 233-236; Serie des sciences techniques 7 (1959), 713-721
* L. Wadel ''IRE Transactions EC-6'' 1957, 123
* N. M. Blachman, ''Communications of the ACM'' (1961), 257
* IEEE Transactions 1963, 274-276
* ''Computer Design'' May 1967, 52-63
* R. W. Marczynski, ''Annotated History of Computing'', 1980, 37-48
* {{citation|first=Donald|last=Knuth|title=[[The Art of Computer Programming]], Volume 2|year=1998|edition=3rd|pages=204–205}}.
* {{anchor|WeissteinNegabinary}} {{mathworld|title=Negabinary|urlname=Negabinary}}
* {{anchor|WeissteinNegadecimal}} {{mathworld|title=Negadecimal|urlname=Negadecimal}}

[[Category:Non-standard positional numeral systems]]
[[Category:Computer arithmetic]]

Attributable risk

2012-08-16T05:03:24Z

132.66.7.211:

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Vibration of plates

2011-12-01T10:15:08Z

132.66.52.182: Erased a minus sign

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