CN1624651A - Q increasing scale carry line digital engineering method and written calculation engineering - Google Patents
Q increasing scale carry line digital engineering method and written calculation engineering Download PDFInfo
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Abstract
The invention relates to digital engineering method and manual computation engineering method field. The invention uses 'adding Q scale', 'carry row method ', which is exchanging K numbers of Q scale in the calculation into number of adding Q scale , then summation of adding Q scale to the K numbers, from the lowest order or each order 'pulsing according to each order' simultaneously, the summation is recorded to the next calculation hierarchy, meanwhile, the 'adding Q carry' is recorded to the highest position adjacent to one willing carry row of the next calculation hierarchy adjacent to the; repeat the calculations until no 'adding Q carry' is generated, then in the last time, the summation acquired by 'pulsing according to each order' is the addition result of Q scale. Meanwhile, the invention has supplied digital engineering technological proposal of adding Q scale and carry row manual computation of digital engineering field.
Description
Technical field
The present invention relates to digital engineering method and written calculation engineering field
Background technology
Digital engineering comprises numerically-controlled machine, big-and-middle-sized digitizer and digital display circuit engineering or the like." digital engineering " is special finger " digital computing system engineering " among the present invention.It is not to solve concrete one by one arithmetic problem or theorem proving or geometrical issues or certain mathematical thought, but solves the digital engineering realization technical scheme of computing systems such as arithmetic rule itself.It and concrete computational tool are closely related.As everyone knows, " calculating " has multiple, outside (mental arithmetic, finger counting, mental arithmetic comprise pithy formula, calculate quickly, estimate), then is " digital computation that adopts instrument " except that " approximate treatment ", " analog computation " reach " no instrument calculates "." adopt the digital computation of instrument " and comprise that in history written calculation, rechoning by the abacus, machinery are calculated, zooming, and calculate etc.Only remaining three kinds of modern times, Here it is digital zooming, rechoning by the abacus, written calculation.Corresponding digital computing system engineering also just only has three kinds therewith: digital machine; Abacus; The digital computing system engineering that adopts pen and paper to carry out written calculation abbreviates " written calculation engineering " as.
Arithmetic is the fundamental operation of number.Said as Engels: " four fundamental rules (key elements of all mathematics)." addition is again the most basic computing of arithmetic.Therefore, we naturally should especially give special concern to additive operation to arithmetic.Arithmetic in the current number digit engineering method at first is an addition, and many parts not fully up to expectations are arranged.It is slow mainly to show as arithmetic speed; In subtraction, fail to make full use of the effect of negative, and, can not " connect and subtract ".Especially in the plus-minus join operation, can not settle at one go; In multiplication, the shortcoming of addition enlarges seriously more; In division, above-mentioned shortcoming still.In a word, at several bodies of minimum---in the rational number body, the arithmetic situation is dissatisfied.
In the written calculation digital engineering,, show to have some implicit operation journeys to the dissection of computing
The formula same form two prefaces are so that produce " hidden danger ".With the addition is example, example one " two number additions ", and formula is suc as formula one.[Wen Zhongfan does not indicate the number of numeral system, all refers to common decimal number.Down together.] wherein, on ten with several 3, dissect, its microprogram operation is:
The carry of coming up in individual position (seeing sign)
Ten last 5,7 liang of numerals and the addition of low level carry, i.e. (5+7+1).Get itself and the position.
Above-listed (5+7+1) and carry deliver to a high position (seeing sign).All the other every situations are similar.And for example example two, establish three number summations, and formula is suc as formula two 78+297+259=634.As can be seen, above-mentioned situation more increases the weight of.
Obviously, there is following shortcoming:
A. carry indicates difficulty.If word table is bright decimally, then easily obscure and literal limited.Particularly just more annoying during table 456789; If write between numeral with ". " word, then easy and radix point is obscured and is represented that 456789 is also inconvenient; If with finger number number, then speed is slow and inconvenient; If mental arithmetic then takes mentality and fallibility.In a word, more disagreeable, easily make mistakes.
B. general two numbers will have three number additions to sue for peace during additions each on.So, need triple computings.When three and three above number additions are sued for peace, then more inconvenient.
C. checking computations are difficult.The general employing reformed one time, wastes time and energy.
Subtraction bothers than addition.And can not same vertical in " connect subtract ", must disconnect.When the plus-minus hybrid operation, can not settle at one go especially.In the multiplication and division, this class situation is even more serious.And addition subtraction multiplication and division computing form disunity is made a fresh start during division.
On the other hand, in the digital engineering of robot calculator, a large amount of numerical operations is arranged equally.These numbers generally all adopt the ordinary binary numeral system to represent.Its negative is often represented with true form, radix-minus-one complement, complement code, frameshit and so on.Computing is all with two number computings in active computer, and can't realize " multiple arithmetic ".So-called " multiple arithmetic " is meant more than two numbers and adds and subtracts simultaneously.
In the robot calculator of common numeral systems such as other common Q systems of employing, there are corresponding many complicacy.[Q is a natural number.]
Summary of the invention
The present invention proposes another new digital engineering method, significantly improves arithmetic speed;
Another object of the present invention provides another new " written calculation engineering " technical scheme, significantly improves the arithmetic speed of " written calculation engineering ".Strengthen the guarantee of computing correctness simultaneously, in " written calculation engineering ", reduce the error rate of written calculation greatly.
According to an aspect of the present invention, provide a kind of Q of increasing system, carry line number digit engineering method, adopt " increasing the Q system " " carry row method ".May further comprise the steps:
The 1st step, establish K common Q system number and participate in computing, K is 〉=2 positive integer, Q is a natural number; These number conversions are become to increase Q system number;
The 2nd step, the K number is increased simultaneously the summation operation of Q system, begin or every addition of step-by-step simultaneously from lowest order, promptly on a certain position, get two number step-by-step additions in the K number, obtain " step-by-step and " for this this two number additions and number, this and number scale are gone into next operation layer, as " partly and " number; Simultaneously gained " increases the Q carry ", then is stored in the high level adjacent with this in arbitrary carry row of next operation layer;
The 3rd step, on this, get two numbers in addition in the K number, carry out the computing in the 2nd step, so repeatedly, till K number average got; When in the K number during an only remaining number, go up as " partly and " number the same position that then directly moves to next operation layer;
The 4th step on an adjacent high position of above-mentioned certain, repeated the computing in the 2nd step and the 3rd step, until each all operation of K operand; When everybody of K number carried out the 2nd step and the 3rd step computing simultaneously, then this step can jump over over;
In the 5th step, in next operation layer, " carry digit " in above-mentioned " step-by-step and " number and the carry row carried out aforementioned the 2nd step, the 3rd step, the 4th step summation operation;
The 6th step repeated the computing in the 2nd step to the 5th step, and till not producing " increasing the Q carry ", then last " addition without carry " gained and number are institute and ask and increase Q system additive operation result.
The above-mentioned Q of increasing system number can not encoded; Or with the ordinary binary number encoder; Or wait with positive and negative sign indicating number and to encode; Or encode with complete one yard, be about to each and increase each figure place S of Q system number, all to arrange corresponding from the lowest order order to high-order with | S| 1, all the other high positions are 0, and total bit then is the Q/2 position; Simultaneously, will increase the number symbol of this position in the Q system number, promptly represent this position for just for negative, accord with as counting on each in corresponding complete a yard.Complete one yard coding increases the Q system when counting, and two number additions only are 1 not repeated arrangement in two numbers, are called " row 1 ".
Above-mentioned operand is to increase Q system number, and Q is a natural number; Or common symmetrical Q system number, Q is＞1 integer; Or mix and count the numeral system number.
According to another aspect of the present invention, provide a kind of Q of increasing system, carry row " written calculation engineering " technical scheme, comprising: adopt that to increase Q system, carry row method, the number conversion of common Q system be common Q system number for increasing Q system number and increasing the number conversion of Q system.Scheme also comprises the complete one yard coding of employing; Perhaps adopt the binary number coding; Perhaps do not encode.In calculating process, at first with the number conversion of common Q system for increasing Q system number, adopt " carry row method " to carry out computing then.Under the situation of plus and minus calculation, increase the summation operation of Q system, carry row " enhancement method ZJF "." increasing the Q number " that operation result can " increase the Q system ".When ultimate demand, will " increase the Q number " again and be converted to common Q system number; Perhaps common decimal number.In the new written calculation engineering technical scheme, generally adopt " multiple arithmetic ".That is, the plus-minus of a plurality of numbers is finished in disposable computing.Like this, just thoroughly solved the difficulty that " even subtracting " reaches " connect and add and subtract ".Simultaneously, multiplication is exactly " connect and add " in essence, and division is exactly " connect and subtract " in essence.Therefore, in multiplication and division, also can use " multiple arithmetic " to handle.
In the new written calculation engineering technical scheme, extensively utilization " liquidating " (mixed approximately) and " drawing ten " computing is in order to improve arithmetic speed and to simplify the computing picture.So-called " liquidating ", i.e. two opposite number additions, itself and be 0.When so-called " drawing ten ", promptly decimal two number additions, its addition without carry and be zero, but produce carry (its symbol is consistent with two numerical symbols).？？？？
Operand is to increase Q system number in " written calculation engineering ", and Q is a natural number; Or common symmetrical Q system number, Q is＞1 integer; Or mix and count the numeral system number.
Embodiment
First increases Q system, carry line number digit engineering method
1. " carry row method "
1.1 carry and " carry row method "
In robot calculator, one of key that arithmetic speed improves just is " carry ".The acquisition of carry, the computing of participating in of the storage of carry and carry all is vital." carry " is exactly to strive " speed ".In written calculation engineering, also directly have influence on " error rate ".
So-called " carry row method " be exactly, in calculating process, the carry that produces left in participate on the equal position of computing and " step-by-step and " number, carries out computing with " step-by-step with " then.Usually will be with two numbers to be during additions in the operation layer, the carry on everybody is arranged in delegation, is called " carry row ".(notion of operation layer sees next section)
Be exemplified below, establish two common decimal number summations, formula is with vertical summation.Suc as formula three:
For simplicity, here will be anyhow formula close and write.Individual bit arithmetic (6+8)=14, its carry 1
123456+345678＝469134
Formula three
Be written on the Gao Yiwei of next line.The rest may be inferred.Two numbers are disregarded the summation of carry during additions in the formula on everybody, are called " addition without carry ".Itself and be called " step-by-step and ".Step-by-step and the computing row, be called " is capable ".
The row that each carry is lined up is called " carry row ".Form " operation layer " by is capable with the carry row.
Some "+" number save in the formula.Can know that later in increasing Q system, carry line number digit engineering method " enhancement method ZJF ", only there is a kind of computing in each " operation layer ", Here it is "+".So can in operation layer, write out "+" number.
1.2 " carry row method " analyzed
1.2.1 the analysis of two number summations
The additive operation of adopting " carry row method " is by last joint as can be known:
1. two numbers have only two number additions during additions on each;
Formula four formulas five
2. in the carry row, directly indicate carry, do not have hell and high water;
3. check very convenient.
During [lemma one] two number additions, the position is gone up or is had carry to be designated as 1 arbitrarily, or no-carry is designated as 0;
During the additions of [lemma two] two number, the on the position and can be one of 0～9 arbitrarily.But, when oriented high-order carry is gone up in this position, the on this and can only be one of 0～8, and can not be 9.
Can get by [lemma one] and [lemma two]:
During the additions of [theorem one] two number, and if only if, and go up not when high-order carry certain position, the on this and just may occur 9.
1.2.2 level notion and operation layer
If two number summations.Formula is formula four, formula five
By formula four as seen, computing is carried out by different level.Operation layer is dissected into little computing, sub-computing with a computing.Each operation layer is only finished a simple operation." level " notion of computing that Here it is." level " notion is the key concept in the mathematics, and " carry row method " set up on this basis just.Additive operation method in the past, also implicit in essence " level " notion.Therefore, " level " in " carry row method " do not increase the complicacy of computing in general.Otherwise method has in the past further increased the complicacy of computing on the contrary owing to implied " level ".This point also further causes arithmetic speed to be lowered.Both contrasts will be perfectly clear.
In " carry row method ", each operation layer of two number additions can be merged into an operation layer.Suc as formula five.Ask for an interview further analysis.
1.2.3 unique operation layer
During two number additions, under the particular case repeatedly operation layer can appear.Each layer is tied to form upright just like ShiShimonoseki.[lemma three] two number additions, when on certain the last operation layer carry being arranged, each computing thereafter
Formula six formulas seven
On the layer all carry can not appear.(getting) by lemma one, two
[lemma four] two number additions, when on the operation layer after certain carry being arranged, must no-carry on each operation layer before it.(getting) by lemma one, two
During the additions of [theorem two] two number, on each operation layer of same position, or no-carrys all, or a carry can only be arranged.(getting) by lemma three, four
[inference] can merge into a carry row with whole each layer carry row, and each operation layer is merged into an operation layer.
1.2.4 three numbers and the above summation of three numbers are analyzed
If three number summations, formula is 231+786+989=2006 (seeing formula six)
Key points for operation: the 1. utilization of " drawing Q ";
So-called " drawing Q ", promptly two of Q carry numbers on certain position during addition, its addition without carry and be zero, but the last generation carry (consistent) in this position with two numerical symbols.Carry is put into the carry row; Simultaneously, on certain position, this two number average is no longer participated in computing.When the decimal system, be " drawing ten ".
A, same position last two are several and when being " ten ", can in formula two numerals be scratched with oblique line, mend 1 then on a high position.
B, same position are gone up several numbers and are 20,30,40 ... Deng the time, several numerals all can be scratched, on a high position, mend 2,3,4 then ... Deng.
Establish six number summations again.Formula is 786+666+575+321+699+999=2046 (seeing formula seven).
Operation layer more than two and two can appear in 2. a plurality of several additions.In order to reduce the computing number of plies, in the same operation layer room on the same position, the occupy-place arbitrarily of carry and and number.
3. reduce operation layer as far as possible.A, less number directly merge and calculate; B, the carry in " pairing " of trying one's best; The number of c, the minimizing addition number on first operation layer of trying one's best makes second and two above operation layer not occur as far as possible.
4. on the same position, " identical number ", " consecutive numbers " etc. can directly obtain " part with ".
2. mix number and mix the number numeral system
" 2.1 numeral system theory "
2.1.1 by with a kind of regular record number, be convenient to be used in a number system, carrying out the system of the number of computing, be called " system of number representation system ".Abbreviate " numeral system " as.The matter of one number is at first decided by the numeral system under it.En Gesi points out: " single number has obtained certain germplasm in number scale, and matter decides according to this number scale." " law of all numbers all depends on the number scale that is adopted, and is determined by this number scale.”
" numeral system theory " science that to be exactly the generation of studying numeral system, classification, analysis, comparison, conversion etc. and numeral system use in each contiguous subject and practice.It is one of basic theory of mathematics.Science of mathematics, the i.e. science of " number ".The basic of " number " is " numeral system ".Therefore, " numeral system theory " is the basis of " number theory ", is one of " core " of " core mathematics ".
Numeral system is the attribute of number.There is not the number that does not have affiliated numeral system, do not have the numeral system that does not have affiliated number yet.
2.1.2 place value system numeral system
If, construct a number system, number wherein locational to have nothing in common with each other " number symbol " is represented." number symbol " claims " numeral " again.Numeral is horizontal from right to left usually, and for all given unit value of the whole numerals on each numerical digit (claiming " place value " again), it is worth by low (little) to high (greatly).With this numeral system of representing each number in the whole number system, be called " place value system numeral system ".
Our numeral system discussed below all is " a place value system numeral system ".Abbreviate " numeral system " as.The number of being discussed all is decided to be integer approximately except that indicating especially.
2.1.3 three big key element: numerical digit I of numeral system, several collection Zi of unit and power Li.
A, numerical digit I, the position of each bit digital of number in the expression numeral system.Represent from a left side from the right side with I (ordinal number).That is, i=1,2,3 ... represent the 1st, 2,3 of this number ... the position.
B, several collection Zi of unit represent the set that " several unit " on the I position forms.In the same number system, all of distinct symbols gone up in the same position of each number, forms the number symbol collection on this position.The element that this number symbol is concentrated is called " element of number ".Abbreviate " several unit " as.Therefore, this number symbol collection is called " several units collection ".Several collection Zi of unit can be different and different along with the value of i, also can be identical.When the Zi on everybody was identical Z, corresponding numeral system was called " single numeral system "; Zi on everybody is incomplete when identical, and corresponding numeral system is called " mixing numeral system ".When single numeral system is the Q system, be called " single system "; When the mixing numeral system all belongs to the Q system, be called " mixed scale ".(this section aftermentioned is seen in the definition of Q system.)
Several units among several collection Zi of unit can be plural number or other varied symbols.In " numeral system theory ", with a
jRepresent several (a of unit
1, a
2, a
3...), j is a natural number.With ia
jRepresent several first a on the i position
jAgreement, a
jDuring=-A (A is a real number), can be expressed as a
j=A.Several collection Zi of unit are with set { a
1..., a
j... represent i.e. Zi={a
1..., a
j....Perhaps Zi shows its feature with literal.
The radix Pi (Pi is a natural number) of several collection Zi of unit has represented the element sum of collection.En Gesi points out: it " not only determines its matter, and determines the matter of other all numbers." the value difference of Pi, indicated the variation of several collection Zi of unit.Pi on everybody is identical P, then is called " single radix "; Otherwise, be called " mixed radix ".
In " the place value system numeral system " of " numeral system theory ", the room in the definition number represents 0, has implicit " room 0 "; Concentrate in several units, " room " is a kind of special several units, is called " room unit ".Abbreviate " empty unit " as." empty unit " is several units that each " place value system numeral system " several units collection all has, and it is " room " in the expression that several units concentrate.On the other hand, " empty unit " is that several units concentrate unique several first a that are not counted in usually
j, also disregard number, promptly number is several units of 0; Under particular case, then " empty unit " to be indicated it is counted several units, its number counts 1.
C, power Li represent the place value size on the i position.Special this place value that claims is " power Li ".
Li is real number (owing to the non-orderly body of set of complex numbers, so do not adopt).Different Li has just determined different place values.In " coding theory ", the principal character of " coding " just is to weigh Li.
Power Li common in the reality adopts so-called " power power ".That is, make Li=Q
i (i-1), Q
iBe real number.For ease of calculating, get Q usually
iBe natural number.Common every Li is power power, and becomes the numeral system of geometric ratio Q.Q is called " truth of a matter " or " truth of a matter " of numeral system of numeral system power power.The difference of truth of a matter Q has determined different Li, thereby has determined different place values.This numeral system is called " Q system ".Abbreviate " system " as.Q
iCan represent by arabic numeral, also can Chinese small letter numeral represent.Work as Q=2, during 3,10 grades, corresponding system just is called as " scale-of-two ", " three-shift ", " decimal system " etc.
Another kind of power Li commonly used adopts " waiting power ", and promptly the power L on everybody is identical.
In any Q system numeral system, when P=Q, the form that natural number can be unique continuously in this numeral system is expressed, and is called " numeral system continuously ", claims again " common numeral system ";
When P＞Q, natural number can be continuous in this numeral system, but express with variform sometimes, is called " repetition numeral system ";
When P＜Q, the form that natural number can only be interrupted in this numeral system is expressed, and is called " interrupted numeral system ".
According to three big key elements of above-mentioned numeral system, numeral system can have inexhaustible kind.
2.2 mix number and mix the number numeral system
In several unit collection Zi, when containing several unit 0, this corresponding numeral system is called as " containing 0 numeral system "; In several unit collection Zi, when not containing several unit 0, this corresponding numeral system is called as " not containing 0 numeral system ".
In several unit collection Zi, existing positive number unit, when negative unit was arranged again, corresponding numeral system was called as " mixing the number numeral system "; Mix the number in the number numeral system, be called " mixing number ".Existing positive number unit has the number of negative unit again in " mix number ", claims " pure mixed number ".At { Q
△In the number, existing positive number unit has the number of negative unit again, is called " pure { Q
△Number ".({ Q
△Define as follows and one save.)
In several unit collection Zi, whole several units are continuous integral number when becoming " integer section ", and this corresponding numeral system is called as " integer hop count system "; Engels points out: " zero all numbers all have more abundant content than other." in view of this special significance of " 0 ", in " numeral system theory ", contain 0 integer section and remove at 0 o'clock, still as a kind of special integer section.
In several collection Zi of unit, when positive negative unit was opposite number, corresponding numeral system was called " symmetrical numeral system "; Obviously, " symmetrical numeral system " is a kind of of " mixing the number numeral system ".
2.3 increase Q system { Q
△}
In " numeral system theory ", the title of a numeral system adopts " Zi Li ".To the Q system, then be ZiQi; During single system, then be ZQ.Wherein, Q
iRepresent with Chinese small letter number.For example 0,1, the 2} three-shift.
For the common Q system that contains 0, Z={0,1 ..., (Q-1) }.So ZQ={0,1 ..., (Q-1) } and Q, Q is＞1 integer, is called " containing 0 common Q system ".Symbolic representation is for { to contain 0, Q}; For do not contain 01,2 ..., Q}Q, Q is a natural number, is called " not containing 0 common Q system ".Symbolic representation is not for { to contain 0, Q}.
Contain 0 and do not contain 0 common Q system, be referred to as " common Q system " altogether, Q is a natural number.Symbolic representation is { Q}.When unlikely misunderstanding, " containing 0 common Q system " also can be described as " common Q system ", and also { Q} represents with symbol.So can symbol { two } and { ten } represent the ordinary binary and the common decimal system.
Mixed numeral system herein is mainly following a few class.
For contain 00, ± 1 ..., ± Q/2}Q system, Q is a positive even numbers, is called " contain 0 and increase the Q system ".Symbolic representation is for { to contain 0, Q
△; For do not contain 0 ± 1, ± 2 ..., ± (Q+1)/2}Q, Q is a positive odd number, is called " do not contain 0 and increase the Q system ".Symbolic representation is not for { to contain 0, Q
△.
Contain 0 and do not contain 0 increase the Q system, be referred to as " increasing the Q system " altogether, Q is a natural number.Symbolic representation is { Q
△.When unlikely misunderstanding, " contain 0 and increase the Q system " also can be described as " increasing the Q system ", also with symbol { Q
△Represent.So can symbol { ten
△And { two
△Represent that " increasing the decimal system " reaches " increasing scale-of-two ".In " numeral system theory ", { ten
△Title be: " single radix P=11 contains 0, the integer section, the symmetry the decimal system ".Can be written as { 11, contain 0, integer section, symmetry } decimal system, perhaps be written as 0, ± 1, ± 2 ..., ± 5} the decimal system.Generally speaking, further symbolic representation is { ten
△, be called " increasing the decimal system "; { two
△Title be: " single radix P=3 contains 0, the integer section, the symmetry scale-of-two ".Can be written as { three, contain 0, integer section, symmetry } scale-of-two, perhaps be written as 0, ± 1} scale-of-two.Generally speaking, further symbolic representation is { two
△, be called " increasing scale-of-two ".
In mixing the number numeral systems, another kind of for common symmetry contain 00, ± 1 ..., ± (Q-1)/2}Q system, Q is＞1 odd number, is called " containing 0 common symmetrical Q system ".Symbolic representation is for { containing 0, claim Q}; To do not contain 0 ± 1 ..., ± Q/2}Q system, Q is a positive even numbers, is called " not containing 0 common symmetrical Q system ".Symbolic representation is not for { containing 0, claim Q}.
Contain 0 and do not contain 0 common symmetrical Q system, be referred to as " common symmetrical Q system " altogether.Q is＞1 integer.Symbolic representation is { to claim Q}.When unlikely misunderstanding, " containing 0 common symmetrical Q system ", also can be described as " common symmetrical Q system ", also { claim that Q} represents with symbol.
3. " enhancement method ZJF " and increase the decimal system { ten
△Arithmetic.
Employing increases the method that Q system and " carry row method " carry out rational number operation, is called " increasing Q system, carry row method ", abbreviates " enhancement method ZJF " as.When being used for abacus or written calculation digital engineering, employing be { ten
△Increase " the enhancement method ZJF " of decimal system etc.In the time of among being used for robot calculator etc., employing be { two
△Increase scale-of-two and { ten
△Increase " the enhancement method ZJF " of decimal system etc.
3.1{ ten
△Add rule: 1 23+,344,=43 3 (seeing formula eight)
Formula eight
Try to achieve in the formula and be 43 3.When needs are converted into the common decimal system { ten } when number and are 427.In general, sue for peace and 43 3 needn't transform (particularly as computation process intermediate result time).When really needing to transform, method is seen 4.1 conversion rules.
3.2{ ten
△Subtraction
3.2.1 example 1 23-344=1 23+ 34 4=34 1
Example 11,2+1 4 4-32-1 25+1,3 3-5 4=1 32 (seeing formula nine)
At first subtraction turns to addition and comes computing, and this is owing to mix the characteristic of number and determine.This comes, and in the actual computation, plus-minus has just been merged into addition.This has just eliminated the difficulty that connects plus-minus usually.
3.2.2 it is mixed approximately.This was meant for 2 whens summation number, and the opposite number on the same position can cancellation, also can be described as " offseting " or " liquidating ".In formula, can scratch by oblique line.That is to say, so-called " liquidating ", i.e. two opposite numbers, itself and be zero.On this two number is no longer participated in later computing.In actual operation, adopt elder generation " liquidating " back " stroke Q " to obtain to increase the result of Q number.
Formula nine formulas ten
3.3{ ten
△Multiplication example 2 42 * 1 31=11502 (seeing formula ten)
3.4{ ten
△Division
Formula ten same form 12 formulas 13
Example 14 33 2 ÷ 23,=25 1 ... 1
Main points: 1. formula 11 adopts former common division, now adopts four fundamental rules to unify formula suc as formula 12.
2. can make " subtracting " process in the division become the process of " adding " owing to adopt to mix number in the formula 12.All the other herewith.
We can make the dividend reversion in order to remove the thinking of " subtracting " process, and then, whole " subtracting " process becomes " adding " process fully.This can make the complicacy of whole computing further reduce.After, our division just carries out with this.Should be noted that then will with this remainder reversion after be only the remainder of final operation result if remainder occurs this moment.
4. " increase the decimal system " { ten
△With the relation of " the common decimal system " { ten }.
4.1{ ten
△The transformation approach several with { ten }
Here the situation that refers to integer, for example { ten
△222 32 4={ ten } 221716 (formulas 13).
4.1.1{ ten } number needs to be converted into { ten through table one
△Several, need only sign symbol with these common Q system numbers, be assigned to corresponding these several each and get on.
4.1.2{ ten
△Number conversion one-tenth { ten }.Method has several: a kind of is with { ten
△Number becomes the summation of one positive one negative two { ten } number.This has good multimode.Wherein, be typically with this { ten
△In the number each positive number word bit and 0 as { a ten } number just, and with each negative word bit as one negative { ten } number.Example { ten
△3 82 2 96={ ten } 302006-80290=221716
Another is on everybody of this number, makes positive number constant; Negative becomes its absolute value and gets " benefit " number to 10, subtracts for 1 (promptly adding 1) in an adjacent high position simultaneously.Another kind method is: formula ten four { ten
△In the number, the digital section of positive digital (or 0) is constant according to writing continuously.As 222 * 2 *.But, when its not { ten
△When counting end (individual position), then lowest order adds 1; The digital section of negative word then makes the negative word become its absolute value and gets " benefit " number to 9 continuously, as * * * 6 * 5.Then, add 1 at its lowest order.
| { ten } → { ten △{ ten △} → { ten } | |
| ??0?????0 ??1?????1 ??2?????2 ??3?????3 ??4?????4 ??5?????5，1 5 ??6?????1 4 ??7?????1 3 ??8?????1 2 ??9?????1 1 ??10????10 ??11????11 ?????? | ????0????????0 ????1????????1 ????2????????2 ????3????????3 ????4????????4 ????5????????5> ????1 5?????5 ????1 4?????6 ????1 3?????7 ????1 2?????8 ????1 1?????9 ????10???????10 ????11???????11 ?????????? |
Table one
Like this, trying to achieve the result is 221716, is corresponding { ten } number.
4.2{ ten
△And { ten } table of comparisons and explanation (table of comparisons sees Table) thereof
Illustrate:
1. in { ten } number numeral 5 is arranged, they corresponding { ten
△Number can have repeat number, can not have yet;
Numeral 5 (plus or minus) are arranged when occurring in 2. all { ten } number, then corresponding { ten } number has { ten of repetition
△Number.At this moment, in this corresponding { ten } number numeral 5 can be arranged, also can not have.Because concentrating, several units of { ten } not only contain 5 but also contain 5 and just produce corresponding repeat number in fact.In other words, as long as { ten
△Several units concentrate and to remove 5 or 5, then can not produce repeat number.At this moment, the numeral system of corresponding this no repeat number is called inclined to one side Q system { Q ' }, the situation of Q=10.
3. { ten
△Several repeat numbers to { ten } number, be " the main repetition " with 5=1 5 and 5=15, promptly all the other repeat all can release thus.
4.3{ ten
△And { ten } relationship analysis
4.3.1{ ten } number and { ten
△The relation of number partly is " more than one corresponding " relation, rather than " corresponding one by one " concerns.Just because this, { ten
△Just obtained the dirigibility of the various processing of part.This is { ten
△The reason of part diversity, rapidity in the computing.From this point, { ten
△Has a stronger function.
4.3.2{ ten
△Number conversion be { ten } number, can only turn to a corresponding unique number.This be because, { ten
△Number can directly obtain through { ten } number plus-minus, and the result behind { ten } number plus and minus calculation is unique.Otherwise { ten } number also can only turn to corresponding unique one group { ten
△Number.So, " " and { ten of this { ten } number
△To organize both are " one by one corresponding " relations for " " of number.
Thus, can set up a kind of { ten
△Several mapping relations each other of counting with { ten }.For arithmetic system, { ten } and { ten
△Number system " isomorphism ".The various fundamental operation character of corresponding { ten } number are also { ten
△Set up in the number system.
4.3.3{ ten
△Middle P＞Q, thereby the variform expression appears in natural number sometimes in this numeral system.This is this numeral system dirigibility place just, and it is simple and efficient that it is able to computing.Also we can say { ten
△Be to have exchanged the part dirigibility for the part diversity.{ ten } P=Q in, thereby in this numeral system, natural number is that continuous unique form is expressed.It does not have this species diversity, has lacked this corresponding dirigibility yet.
It has been arranged, " enhancement method ZJF " just arranged, another new solution of " written calculation engineering " has just been arranged.It has been arranged, another new solution of processor and respective electronic computing machine thereof has also just been arranged.
4.3.4 should be pointed out that obviously, above-mentioned to { ten } and { ten
△Analysis, fully corresponding to { Q} and { Q
△Analysis because { ten } with { Q} is an isomorphism.Hence one can see that, 1. { Q} number and { Q
△Several relations is part " corresponding more than one ", rather than " corresponding one by one ".2. simultaneously, { " one " number among the Q} and corresponding { Q
△In " one " group number, be " one by one corresponding " relation between the two.3. { Q} and { Q
△Number system " isomorphism ".It is corresponding that { the various fundamental operation character of Q} number system are also at { Q
△Set up in the number system.
5. comprehensively above-mentioned, following terse conclusion can be arranged:
Increase Q system { Q
△Reach " enhancement method ZJF " in digital engineering, can significantly improve arithmetic speed, and reduce the error rate of written calculation greatly.It is the mathematics tri-layer " direct applied engineering " pointed out of Qian Xuesen just.The method that this " engineering " and digital computation engineering are combined closely is called " increasing Q system, carry line number digit engineering method ".
Second portion increases Q system, carry row written calculation engineering scheme
(1) in the written calculation engineering, numerical operation is under the correct prerequisite of principle, and most important have 2 points: a bit being not make mistakes as far as possible, a bit is to wish that arithmetic speed is fast as far as possible.Yet in practice, these 2 usually are in opposition contradiction state again.Because otherwise make mistakes, usually have to reduce arithmetic speed.Otherwise, usually make mistakes again fast.
Where restrict above-mentioned key at 2? the key just is " advance and retreat position ".Use the aforementioned written calculation engineering that increases Q system, carry line number digit engineering method, can be in the numerical operation process, make that the notion on each hierarchy of 00operation is simpler, more basic, more clear.Simultaneously, corresponding operation can be more convenient.This just makes the fallibility of numerical operation obviously reduce, and arithmetic speed is able to obvious raising.
Because human the most frequently used number is common decimal number, therefore, background mathematics all is to use common decimal number.New written calculation engineering technical scheme has adopted and has increased Q system { Q
△.That is, to increase Q system { Q
△In increase the decimal system, replace the common decimal system { ten }.
(2) increase the written calculation engineering technical scheme of Q system, carry line number digit engineering method, further adopt again to increase the decimal system { ten
△" enhancement method ZJF ".Mixed counting method combines with the carry row method, makes both just in time complementary, promotes mutually.Therefore, in " enhancement method ZJF ", arithmetic speed improves greatly; Simultaneously, in written calculation engineering, error rate is reduced greatly.
(3) in the new written calculation engineering technical scheme, generally adopt " multiple arithmetic ".That is, the plus-minus of a plurality of numbers is finished in disposable computing.Like this, just thoroughly solved the difficulty that " even subtracting " reaches " connect and add and subtract ".Simultaneously, multiplication is exactly " connect and add " in essence, and division is exactly " connect and subtract " in essence.Therefore, in multiplication and division, also can use " multiple arithmetic " to handle.
(4) in the new written calculation engineering technical scheme, extensively utilization " liquidating " (mixed approximately) and " drawing ten " computing is in order to improve arithmetic speed and to simplify the computing picture.So-called " liquidating ", i.e. two opposite number additions, itself and be 0.When so-called " drawing ten ", promptly decimal two number additions, its addition without carry and be zero, but produce carry (its symbol is consistent with two numerical symbols).
Increase the complete one yard coding of normal employing in Q system, the carry line number digit engineering method.But, in the application of written calculation engineering because the word length of complete one yard coded number is longer, so though available complete one yard encode and increase decimal number, also can not encode separately.
Theory and practice proves, increases the written calculation engineering technical scheme that Q system, carry line number digit engineering method and written calculation engineering are a kind of excellences.Essentially, it makes ten-* the ÷ arithmetic, rational number operation just, comprehensively, systematically take on a new look.It is convenient and easy, even for the beginner, operand also can expand any multidigit to quickly, need not to be limited especially at all.Its low error rate has successfully realized the pleasure principle of mathematics and education thereof with quick.Its birth helps mathematics throughout the ages and education family estate.
(5) brief summary:
Increase the Q system, carry line number digit engineering method is used for written calculation engineering, is practicable.The written calculation engineering new solution can improve arithmetic speed greatly, reduces error rate simultaneously greatly.Increase the decimal system { ten
△Application in written calculation engineering, be a revolution with respect to the common decimal system { ten }.
This written calculation engineering new solution particularly has the important meaning in the science and education in the human brain written calculation in textbook.Consider today and following background mathematics and widespread use and the important meaning of education in human lives, production, teaching or the like field thereof, so, the purposes of written calculation engineering new solution and value are exactly self-evident.
Third part increases Q system { Q
△And complete one yard
1. increase Q system { Q
△}
1.1 definition and symbol
In a Q system numeral system, the system of all P=Q+1＞Q is called " enhancement mode Q system ".Q is a natural number.Abbreviate " increasing the Q system " as, with symbol { Q
△Represent.Increase Q system { Q
△Have a variety of.Wherein symmetry be the aforementioned Q of increasing system, in addition, also have the asymmetric Q of increasing system { Q
△.
1.2 increase a system {
△And computing
Increase Q system { Q
△In, when Q=1, be and increase a system {
△.Increase a system {
△In, mainly contain two kinds.{ 0,1} one system, its components and parts are the two condition device to the first.It two is that { 1,1} one system, its components and parts also are the two condition device, and it also can represent whole integers.Allegedly below this paper " increase a system {
△", except that indicating especially, all refer to 0,1} one system.
Increase a system {
△Computing.Here list additive operation, for example { ten } 4+3+2=9={ one
△110101+1011+101=11001100010101011.
1.3 increase a system {
△And { the relation of Q}.
1.3.1{ one
△Number and the { transformation approach of Q} number.
{ one
△{ the Q} number can be with { one for number conversion one-tenth
△Each bit digital 1 in the number, so that { the Q} counting gets final product.Gained Q} counting and, be corresponding { Q} number.In other words, { one
△Have in the number severally 1, it is then corresponding that { the Q} number is several.Obviously, this is foolproof rule.(seeing Table two)
{ the Q} number conversion becomes { one
△Number, can { Q} counts everybody and all multiply by power on everybody, then these is amassed with 1 of same number, respectively to be expressed { one
△Numerical digit puts, and lists getting final product in unduplicated mode.In other words, and the Q} number is several, then { one
△Just have several 1 in the number.Obviously, this also is foolproof rule.(seeing Table three)
1.3.2{ one
△Count with { Q} counts the table of comparisons and explanation thereof
{ one
△{ two } { ten } { ten } { two } { one
△}
000?????0?????0????0????000??0…00000000＝
001?????1?????1????1????001??0…00000001＝1＝
010?????1?????1????2????010??0…00000011＝11＝
011?????10????2????3????011??0…00000111＝111＝
100?????1?????1????4????100??0…00001111＝1111＝
101?????10????2????5????101??0…00011111＝11111＝
110?????10????2????6????110??0…00111111＝111111＝
111?????11????3????7????111??0…01111111＝1111111＝
?????????????????????????????????
Table two table three
1
11 poplars
121 brightness
1331 three
14641 jiaos
shape
Table four
Illustrate: 1. { one
△Number can represent all { Q} numbers
2. more repeat number is arranged, with 4 { one
△Number is for example, except that 0 and 4 unique, all the other all have repeat number.Wherein, 1 has 4; 2 have 6; 3 have 4.So, be respectively 1,4 from 0～4 repeat number, 6,4,1.This and binomial expansion coefficient C
K nBe consistent.(figure place n is a natural number, and K is 0～n.) (see Table four and raise the brightness triangle.)
3. in the table
Be expressed as any nonnegative integer position continuous 0.Be called " infinitely prolonging number "
{ one
△In the number, infinitely prolonging number has and only has one, is
1.3.3{ one
△And { Q} relationship analysis.
(1) Q 1, and Q is a natural number; 1 is minimum natural number, also is the most basic natural number unit.Q really comprises 1, and this makes that { Q} reaches { one accordingly
△Between have the contact of nature.
(2) { Q} number and {
△The relation of number is " more than one corresponding " relation, rather than " corresponding one by one " concerns.Just because this, { one
△Just obtained the dirigibility of various processing.This is { one
△One of the reason of rapidity in the computing.From this point, { one
△Has a stronger function.
(3) { one
△Number conversion is that { the Q} number can only turn to a corresponding unique number.This be because, { one
△Number can be through { the Q} plus-minus directly obtains, and { result that Q} counts behind the plus and minus calculation is unique.Otherwise { Q} also can only turn to corresponding unique one group { one
△Infinitely prolong number.So, this { " " and { of Q} number
△Infinitely prolonging several " one ", to organize both are " one by one corresponding " relations.Thus, can set up a kind of { one
△Number and the { mapping relations each other of Q} number.For arithmetic system, { Q} and {
△Number system is " isomorphism ".It is corresponding that { the various fundamental operation character of Q} number are also { one
△Set up in the number system.
(4) { one
△Middle P=Q+1＞Q, thereby in this numeral system, variform appears in natural number sometimes expresses, and this is this numeral system dirigibility place just, and it is simple and efficient that it makes that computing is able to.Also we can say { one
△Be to have exchanged dirigibility for diversity.
P=Q among the Q}, thereby in such number, natural number is that continuous unique form is expressed.It does not have this species diversity, has lacked this corresponding dirigibility yet.
(5) above-mentioned { one
△And { Q
△Combine, make function strengthen more.Consider { one
△} → { Q} → { Q
△, this wherein has inherent contact.Obviously, everything is all in the contemplation.
1.4 increase a system {
△Application
1.4.1 increase a system {
△Computing be a kind of computing of excellence.Since it with the unit 1 be equipped with 0 the structure number, and power be 1, so its " computing " often realizes with " transmission ".This is { one
△One of rapidity reason in the number computings.{ one
△" carry " of number in the computings, also with the addition without carry of two number present bit be 0, and carry is Q's " drawing Q " logic realization.This " transmission " reaches the logic realization of " drawing Q ", and structure is simple especially, and speed is special fast.This is { one
△Rapidity former therefore two in the number computings.
When { one
△Number and { Q
△Number is during associative operation, replenished again that " liquidating " this structure is more simple, speed logic more fast.This is { one
△Number computing rapidities former therefore three.
1.4.2{ one
△And { Q
△In conjunction with the technical scheme that can be used as multiple hypervelocity robot calculator of new generation.
2. a full system, complete is counted and complete one yard
2.1 a full system and a full number
Increase a system {
△The number diversity be { one
△Number computings one of reason fast.But, because { one
△Number has extreme variously, can to occur once in same number above infinitely prolongs number, often causes several expression-forms to be difficult to assurance.Cause operand too to disperse thus, be not easy to control, certainly will increase equipment and influence arithmetic speed.Therefore, in the ordinary course of things, be necessary { one
△Number certain constraint condition in addition.This has just produced " a full system ".
Increasing a system {
△Positive integer in, limit each group and infinitely prolong number.Only choose from a position beginning, arrange unique a kind of form of unit 1 from right to left continuously and express; Be 0 on the high position, represent with the room.For example: { ten } number 3={ one
△Several 111/1110/1101/ ... ("/" table " perhaps ") is defined as { ten } 3={--
Δ}
Like this, the repetition number average that each group is infinitely prolonged in the number is deleted, and the only surplus next one is unique form of 1 entirely, and we are called " a full number ".The system of expressing " a full number " is referred to as " a full system ".In the table three, { one
△The leftmost form of number, be " a full system " number.When considering positive negative integer, can be with the sign symbol of this full system number, be assigned to this number everybody get on, thereby construct a signed full system.Following " a full system " all is the signed full system of this kind.
Therefore, " a full system " can be the system { that increases that adds the particular constraints condition
△.
In " the place value system numeral system " of " numeral system theory ", the room in the definition number represents 0, has implicit " room 0 "; Concentrate in several units, " room " is a kind of special several units, is called " room unit ".Abbreviate " empty unit " as.Therefore, " a full system " can never contain 0 common Q system do not contain 0, and among the Q} { the 1} one system position of putting in marks obtains; " a full system " can also never contain 0 and increase a system and { do not contain 0, one
△In " 1,1} one system " and the acquisition of addition of constraints condition, constraint condition is that necessary everybody two symbols of this system number are all identical; In addition, also have several different methods to obtain.
2.2 complete one yard
A full system obviously has following relative merits.Advantage: 1. fast operation." transmission " replaced " upset ".2. during multiple arithmetic, do not need two, two summations, only need elder generation " to liquidate " and the back " draw Q " and get final product the result.This has just accelerated overall arithmetic speed greatly.3. with { Q} changes convenient.Shortcoming: 1. " word length " is oversize, and figure place is many.(when getting variable word length, its average word length only is half.) 2. the load quantity of information is less.Therefore,, maximize favourable factors and minimize unfavourable ones, with a full system { Q that encodes according to the relative merits of a full system
△Be suitable.Encode with " a full system ", be called " a full coding "." a full number " that adopts in " a full coding " is called " complete one yard ".By an above-mentioned full system is as can be known signed, and complete one yard also is signed.Table five demonstrates complete one yard one, the situation of coding { two } several units.By table five as seen, { two } of complete one yard coding number is { two } number itself.Table six demonstrates with complete one yard nine, the situation of coding { ten } several units.By table six as seen, { ten } of complete one yard nine coding, word length increases to 9 times.(when getting variable word length, its average word length only is 5 times.) for example: the complete one yard=≡ of { ten } 23=.
Complete one yard { two } nine complete one yard { ten }
0????????????0???????00??…???0???????0
1????????????1???????00??…???1???????1
00??…??11???????2
???????????
Table five
111111111????????9
Table six
For increasing Q system { Q
△, also can complete one yard encode.It is pointed out that { two of complete one yard coding here
△Number, be { two
△Number itself; Here { ten
△Number, then encode with five complete one yard.
2.3 complete one yard calculating.
Complete one yard calculating is very simple.With two number additions is example, only is 1 not repeated arrangement in two numbers, is called " row 1 ".As 11+111=11111.Particularly, at { Q
△In the digital engineering, only need earlier " liquidating " back " to draw Q " and just can obtain { Q
△The number operation result.When net result need be exported, just will be with { the Q of complete one yard coding
△The output of number conversion one-tenth { ten } number.
2.4 complete one yard application.
Complete one yard is mainly used in { Q} and { Q
△Count and encode.Particularly,
1. adopt complete one yard nine codings { ten } number, can realize the common decimal system { ten }, complete one yard, carry row processor.
2. adopt complete one yard five codings { ten
△Number, can realize increasing the decimal system { ten
△, complete one yard, carry row processor.
3. adopt a full coding { Q
△Number, can realize increasing Q system { Q
△, complete one yard, carry row processor.
4. adopt complete one yard five codings { ten } or { ten
△Number, come secondary coding (that is, with corresponding positive negative to encoding) with " positive and negative sign indicating number " again, can realize the new solution of another abacus.
5. adopt complete one yard five codings { ten } or { ten
△Number, come secondary coding (that is, with corresponding positive negative to encoding) with " positive and negative sign indicating number " again, can realize the new solution of another written calculation engineering.
Claims (7)
1. one kind increases Q system, carry line number digit engineering method, may further comprise the steps:
The 1st step, establish K common Q system number and participate in computing, K is 〉=2 positive integer, Q is a natural number; These number conversions are become to increase Q system number;
The 2nd step, the K number is increased simultaneously the summation operation of Q system, begin or every addition of step-by-step simultaneously from lowest order, promptly on a certain position, get two number step-by-step additions in the K number, obtain " step-by-step and " for this this two number additions and number, this and number scale are gone into next operation layer, as " partly and " number; Simultaneously gained " increases the Q carry ", then is stored in the high level adjacent with this in arbitrary carry row of next operation layer;
The 3rd step, on this, get two numbers in addition in the K number, carry out the computing in the 2nd step, so repeatedly, till K number average got; When in the K number during an only remaining number, go up as " partly and " number the same position that then directly moves to next operation layer;
The 4th step on an adjacent high position of above-mentioned certain, repeated the computing in the 2nd step and the 3rd step, until each all operation of K operand; When everybody of K number carried out the 2nd step and the 3rd step computing simultaneously, then this step can jump over over;
In the 5th step, in next operation layer, " carry digit " in above-mentioned " step-by-step and " number and the carry row carried out aforementioned the 2nd step, the 3rd step, the 4th step summation operation;
The 6th step repeated the computing in the 2nd step to the 5th step, and till not producing " increasing the Q carry ", then last " addition without carry " gained and number are institute and ask and increase Q system additive operation result.
2. Q system, the carry line number digit engineering method of increasing as claimed in claim 1 is characterized in that computing is " increasing the Q system " computing; Wherein, 0, ± 1 ..., ± Q/2}Q system, Q is a positive even numbers, is called " contain 0 symmetry and increase the Q system "; ± 1 ..., ± (Q+1)/2}Q system, Q is a positive odd number, is called " do not contain 0 symmetry and increase the Q system "; When unlikely misunderstanding, " increasing the Q system " promptly refers to " contain 0 symmetry and increase the Q system ".
3. Q system, the carry line number digit engineering method of increasing as claimed in claim 1 is characterized in that computing employing " carry row method "; Promptly in calculating process, the carry that produces is left in the adjacent high position " carry row ", carry out computing with " step-by-step and " then.
As claim 1-3 any increase Q system, carry line number digit engineering method, it is characterized in that on a certain position, when two numbers in the K number are carried out summation operation, if wherein this position of two operands is an opposite number, this position and be zero then, with certain all reset logically of position of these two operands, no longer participate in later computing then, this is called " liquidating "; If on a certain position, when two numbers in the K number are carried out summation operation, the addition without carry of two operands and be zero wherein, but generation carry, then its carry is put into the adjacent high position of arbitrary carry row, with certain all reset logically of position of these two operands, no longer participate in later computing then, this is called " drawing Q "; Perhaps, do not adopt " liquidating " to reach " drawing Q ".
As claim 1-4 any increase Q system, carry line number digit engineering method, it is characterized in that: increasing Q system number can not encode; Or with the ordinary binary number encoder; Or wait with positive and negative sign indicating number and to encode; Or encode with complete one yard, be about to each and increase each figure place S of Q system number, all to arrange corresponding from the lowest order order to high-order with | S| 1, all the other high positions are 0, and total bit then is the Q/2 position; Simultaneously, will increase the number symbol of this position in the Q system number, the number of promptly representing this is a plus or minus, as the number symbol on each in corresponding complete a yard.
As claim 1-5 any increase Q system, carry line number digit engineering method, it is characterized in that when adopting complete one yard to encode when increasing the Q system and counting two number additions only are 1 not repeated arrangement in two numbers, are called " row 1 ".
Claim 1-4 any increase Q system, carry line number digit engineering method, wherein said operand is to increase Q system number, Q is a positive even numbers; Or common symmetrical Q system number, Q is＞1 integer; Or mix and count the numeral system number.
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|---|---|---|---|
| CN 200410084836 CN1624651A (en) | 2004-09-30 | 2004-09-30 | Q increasing scale carry line digital engineering method and written calculation engineering |
| US11/664,397 US20070271326A1 (en) | 2004-09-30 | 2005-09-29 | Technical Solution to Written Calculations Engineering of the Digital Engineering Method for Hybrid Numeral Carry System and Carry Line |
| PCT/CN2005/001598 WO2006034649A2 (en) | 2004-09-30 | 2005-09-29 | Manual computation engineering technical proposal of mixed digital scale and carry line digital engineering method |
| CN 200510107786 CN1760825A (en) | 2004-09-30 | 2005-09-30 | Written calculation engineering scheme of carry line digital engineering method in increasing Q scale |
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