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2706:. Beginning at the one's place, 4 is not less than 2 so the difference 2 is written down in the result's one's place. In the ten's place, 0 is less than 1, so the 0 is increased by 10, and the difference with 1, which is 9, is written down in the ten's place. The American method corrects for the increase of ten by reducing the digit in the minuend's hundreds place by one. That is, the 7 is struck through and replaced by a 6. The subtraction then proceeds in the hundreds place, where 6 is not less than 5, so the difference is written down in the result's hundred's place. We are now done, the result is 192.
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2637:{\displaystyle {\begin{array}{rrrr}&\color {Red}-1\\&C&D&U\\&7&0&4\\&5&1&2\\\hline &1&9&2\\\end{array}}{\begin{array}{l}{\color {Red}\longleftarrow {\rm {carry}}}\\\\\longleftarrow \;{\rm {Minuend}}\\\longleftarrow \;{\rm {Subtrahend}}\\\longleftarrow {\rm {Rest\;or\;Difference}}\\\end{array}}}
193:
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Almost all
American schools currently teach a method of subtraction using borrowing or regrouping (the decomposition algorithm) and a system of markings called crutches. Although a method of borrowing had been known and published in textbooks previously, the use of crutches in American schools spread
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The partial differences method is different from other vertical subtraction methods because no borrowing or carrying takes place. In their place, one places plus or minus signs depending on whether the minuend is greater or smaller than the subtrahend. The sum of the partial differences is the total
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There is an additional subtlety in that the student always employs a mental subtraction table in the
American method. The Austrian method often encourages the student to mentally use the addition table in reverse. In the example above, rather than adding 1 to 5, getting 6, and subtracting that from
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The
Austrian method does not reduce the 7 to 6. Rather it increases the subtrahend hundreds digit by one. A small mark is made near or below this digit (depending on the school). Then the subtraction proceeds by asking what number when increased by 1, and 5 is added to it, makes 7. The answer is 1,
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In this method, each digit of the subtrahend is subtracted from the digit above it starting from right to left. If the top number is too small to subtract the bottom number from it, we add 10 to it; this 10 is "borrowed" from the top digit to the left, which we subtract 1 from. Then we move on to
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The method of complements is especially useful in binary (radix 2) since the ones' complement is very easily obtained by inverting each bit (changing "0" to "1" and vice versa). And adding 1 to get the two's complement can be done by simulating a carry into the least significant bit. For example:
302:{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,+\,{\text{term}}\\\scriptstyle {\text{summand}}\,+\,{\text{summand}}\\\scriptstyle {\text{addend}}\,+\,{\text{addend}}\\\scriptstyle {\text{augend}}\,+\,{\text{addend}}\end{matrix}}\right\}\,=\,}
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655:{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\frac {\scriptstyle {\text{dividend}}}{\scriptstyle {\text{divisor}}}}\\\scriptstyle {\frac {\scriptstyle {\text{numerator}}}{\scriptstyle {\text{denominator}}}}\end{matrix}}\right\}\,=\,}
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Some
European schools employ a method of subtraction called the Austrian method, also known as the additions method. There is no borrowing in this method. There are also crutches (markings to aid memory), which vary by country.
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1765:: the difference is not a natural number unless the minuend is greater than or equal to the subtrahend. For example, 26 cannot be subtracted from 11 to give a natural number. Such a case uses one of two approaches:
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2163:, a specific process is taught to students at the end of the 1st year (or during the 2nd year) for use with multi-digit whole numbers, and is extended in either the fourth or fifth grade to include
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The same change method uses the fact that adding or subtracting the same number from the minuend and subtrahend does not change the answer. One simply adds the amount needed to get zeros in the subtrahend.
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A column of two numbers, with the lower number in red, usually indicates that the lower number in the column is to be subtracted, with the difference written below, under a line. This is most common in
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532:{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{factor}}\,\times \,{\text{factor}}\\\scriptstyle {\text{multiplier}}\,\times \,{\text{multiplicand}}\end{matrix}}\right\}\,=\,}
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published a studyβclaiming that crutches were beneficial to students using this method. This system caught on rapidly, displacing the other methods of subtraction in use in
America at that time.
1809:
inverses. The subtraction of a real number (the subtrahend) from another (the minuend) can then be defined as the addition of the minuend and the additive inverse of the subtrahend. For example,
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8(1):4β11. p. 8: "This new version of the decomposition algorithm has so completely dominated the field that it is rare to see any other algorithm used to teach subtraction today ."
417:{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,-\,{\text{term}}\\\scriptstyle {\text{minuend}}\,-\,{\text{subtrahend}}\end{matrix}}\right\}\,=\,}
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Both these methods break up the subtraction as a process of one digit subtractions by place value. Starting with a least significant digit, a subtraction of the subtrahend:
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1716:. This picture is inadequate to describe what would happen after going 3 steps to the left of position 3. To represent such an operation, the line must be extended.
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805:{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{base}}^{\text{exponent}}\\\scriptstyle {\text{base}}^{\text{power}}\end{matrix}}\right\}\,=\,}
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by one (or continuing the borrow leftwards until there is a non-zero digit from which to borrow). The
European method corrects by increasing the subtrahend digit
720:{\displaystyle \scriptstyle \left\{{\begin{matrix}\scriptstyle {\text{fraction}}\\\scriptstyle {\text{quotient}}\\\scriptstyle {\text{ratio}}\end{matrix}}\right.}
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is increased by 10 and some other digit is modified to correct for this increase. The
American method corrects by attempting to decrease the minuend digit
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As an example, suppose that 30% of widgets made in a factory are defective. Six months later, 20% of widgets are defective. The percentage change is
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Because the next digit of the minuend is smaller than the subtrahend, we subtract one from our penciled-in-number and mentally add ten to the next.
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vary from country to country, and within a country, different methods are adopted at different times. In what is known in the United States as
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1842:, meaning that if one reverses the terms in a difference left-to-right, the result is the negative of the original result. Symbolically, if
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is a technique used to subtract one number from another using only the addition of positive numbers. This method was commonly used in
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1096:, subtraction can also represent removing or decreasing physical and abstract quantities using different kinds of objects including
1080:. Subtraction is an operation that represents removal of objects from a collection. For example, in the adjacent picture, there are
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1164:, subtraction of it does not change a number. Subtraction also obeys predictable rules concerning related operations, such as
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Brownell, W.A. (1939). Learning as reorganization: An experimental study in third-grade arithmetic, Duke
University Press.
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Instead of finding the difference digit by digit, one can count up the numbers between the subtrahend and the minuend.
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01100100 (x) + 11101001 (ones' complement of y) + 1 (to get the two's complement) ββββββββββ 101001110
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of real numbers can be defined specifying only two binary operations, addition and multiplication, together with
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is still invalid, since it again leaves the line. The natural numbers are not a useful context for subtraction.
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1978:, they must have the same unit. In most cases, the difference will have the same unit as the original numbers.
1145:. In words: the difference of two numbers is the number that gives the first one when added to the second one.
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Susan Ross and Mary Pratt-Cotter. 2000. "Subtraction in the United States: An
Historical Perspective,"
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1926:), but these two possibilities lead to different answers. To resolve this issue, one must establish an
1821:. Alternatively, instead of requiring these unary operations, the binary operations of subtraction and
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7, the student is asked to consider what number, when increased by 1, and 5 is added to it, makes 7.
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subtracting the next digit and borrowing as needed, until every digit has been subtracted. Example:
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3348:"Subtrahend" is shortened by the inflectional Latin suffix -us, e.g. remaining un-declined as in
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1741:(..., β3, β2, β1, 0, 1, 2, 3, ...). This way, it takes 4 steps to the left from 3 to get to β1:
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peachesβmeaning 5 peaches with 2 taken away, resulting in a total of 3 peaches. Therefore, the
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Because the next digit in the minuend is not smaller than the subtrahend, we keep this number.
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Subtraction of numbers 0β10. Line labels = minuend. X axis = subtrahend. Y axis = difference.
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3826:
Subtraction in the United States: An
Historical Perspective, Susan Ross, Mary Pratt-Cotter,
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We add a 10 to the 1. Because the 10 is "borrowed" from the nearby 5, the 5 is lowered by 1.
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There are also situations where subtraction is "understood", even though no symbol appears:
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So, we add 10 to it. The 10 is "borrowed" from the digit on the left, which goes down by 1.
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Ross, Susan C.; Pratt-Cotter, Mary (1999). "Subtraction From a Historical Perspective".
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Because the minuend is smaller than the subtrahend, this difference has a minus sign.
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A variant of the American method where all borrowing is done before all subtraction.
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Because the minuend is greater than the subtrahend, this difference has a plus sign.
3119:
Because the minuend is greater than the subtrahend, this difference has a plus sign.
2131:
and one is added to the sum. The leading digit "1" of the result is then discarded.
1593:
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1975:
1601:
1580:"to reduce or diminish", one gets "minuend", which means "thing to be diminished".
1207:, operations between numbers are actually defined using "truncated subtraction" or
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3574:
Teachers Versus the Public: What Americans Think about Schools and How to Fix Them
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1152:, meaning that changing the order changes the sign of the answer. It is also not
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So, we add 10 to it and put a 1 under the next higher place in the subtrahend.
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advertising subtraction of 20% from the price of the second perfume purchased.
972:{\displaystyle \scriptstyle \log _{\text{base}}({\text{anti-logarithm}})\,=\,}
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3494:(1 (reprint) ed.). Malabar, FL: Robert E. Krieger Publishing Company.
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change is simply the number obtained by subtracting the two percentages.
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886:{\displaystyle \scriptstyle {\sqrt{\scriptstyle {\text{radicand}}}}\,=\,}
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Dropping the initial "1" gives the answer: 01001110 (equals decimal 78)
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1735:
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Now the subtraction works, and we write the difference under the line.
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01100100 (x, equals decimal 100) - 00010110 (y, equals decimal 22)
1769:
Conclude that 26 cannot be subtracted from 11; subtraction becomes a
1605:
1516:{\displaystyle {\rm {minuend}}-{\rm {subtrahend}}={\rm {difference}}}
3830:, Vol. 8, No. 1 (original publication) and Vol. 10, No. 1 (reprint.)
3587:
Convergence Journalism: Writing and Reporting across the News Media
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The minuend is 704, the subtrahend is 512. The minuend digits are
1708:. From position 3, it takes no steps to the left to stay at 3, so
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results in "subtrahend", "thing to be subtracted". Likewise, from
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2060:%, while the percentage point change is β10 percentage points.
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When subtracting two numbers with units of measurement such as
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25:
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Add up the value from each step to get the total difference:
1115:
In a sense, subtraction is the inverse of addition. That is,
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1592:
3572:
Paul E. Peterson, Michael Henderson, Martin R. West (2014)
3428:
Computability: an introduction to recursive function theory
3181:
Example: 1234 β 567 = can be found by the following steps:
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199:
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Example: 1234 β 567 = can be solved in the following way:
1631:. This movement to the right is modeled mathematically by
3807:
The Many Ways of Arithmetic in UCSMP Everyday Mathematics
3788:
The Many Ways of Arithmetic in UCSMP Everyday Mathematics
3769:
The Many Ways of Arithmetic in UCSMP Everyday Mathematics
3750:
The Many Ways of Arithmetic in UCSMP Everyday Mathematics
3720:
The Many Ways of Arithmetic in UCSMP Everyday Mathematics
1712:. It takes 2 steps to the left to get to position 1, so
1670:. This movement to the left is modeled by subtraction:
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Formally, the number being subtracted is known as the
1148:
Subtraction follows several important patterns. It is
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and is written down in the result's hundreds place.
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56:. Unsourced material may be challenged and removed.
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3147:The smaller number is subtracted from the greater:
3131:The smaller number is subtracted from the greater:
3115:The smaller number is subtracted from the greater:
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2002:between the two quantities as a percentage, while
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3618:The Teaching of Arithmetic: A Manual for Teachers
3238:is to split up the subtraction into small steps.
1784:, so the result of subtracting 26 from 11 is β15.
3600:Racism and Education: Coincidence Or Conspiracy?
1396:, while the number it is subtracted from is the
442:{\displaystyle \scriptstyle {\text{difference}}}
997:{\displaystyle \scriptstyle {\text{logarithm}}}
19:"Subtract" redirects here. For other uses, see
1342:(pronounced as "six minus three equals three")
136:= 3" (verbally, "five minus two equals three")
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1696:Now, a line segment labeled with the numbers
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558:{\displaystyle \scriptstyle {\text{product}}}
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3589:Rowman & Littlefield Publishers, p. 180
1942:also plays a special role: for any integer
1938:In the context of integers, subtraction of
1308:(pronounced as "four minus two equals two")
830:{\displaystyle \scriptstyle {\text{power}}}
16:One of the four basic arithmetic operations
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911:{\displaystyle \scriptstyle {\text{root}}}
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3279:"1234 β 567 =" can be solved as follows:
2815:The difference is written under the line.
2791:Now we can find the difference as before.
2751:The difference is written under the line.
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327:{\displaystyle \scriptstyle {\text{sum}}}
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116:Learn how and when to remove this message
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2998:We write the difference under the line.
2934:We write the difference under the line.
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2123:(the minuend), the ones' complement of
1990:can be reported in at least two forms,
3797:Subtraction: Left to Right Subtraction
2151:The teaching of subtraction in schools
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2377:
2278:is a digit, proceeds by writing down
2155:Methods used to teach subtraction to
2119:(the subtrahend) from another number
1527:All of this terminology derives from
1112:, decimals, functions, and matrices.
7:
3560:participating institution membership
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3394:
54:adding citations to reliable sources
3576:Brookings Institution Press, p. 163
1958:, also known as the predecessor of
1558:"to pull". Thus, to subtract is to
1231:"β" between the terms; that is, in
1176:, starting with the subtraction of
3670:10.1111/j.1949-8594.1999.tb17499.x
3234:Another method that is useful for
2765:The required sum (5) is too small.
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1235:. The result is expressed with an
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1954:is the largest integer less than
1199:, considering subtraction is not
3894:Subtraction on a Japanese abacus
3759:Subtraction: Partial Differences
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1734:The solution is to consider the
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3739:Partial-Differences Subtraction
1902:can be defined to mean either (
41:needs additional citations for
3658:School Science and Mathematics
3457:(1 ed.). Binghamton, NY:
3352:"the number to be subtracted".
2833:Subtraction from left to right
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2193:Comparing the two main methods
2076:, and is still used in modern
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1:
4221:Conway chained arrow notation
3809:Subtraction: Same Change Rule
2948:The minuend (5) is too small!
2676:. The subtrahend digits are
2115:To subtract a binary number
2104:
2096:
1627:steps to the right to reach
1584:Of integers and real numbers
1539:word derived from the Latin
1172:. All of these rules can be
21:Subtraction (disambiguation)
3898:Abacus: Mystery of the Bead
3880:More Subtraction Worksheets
3851:Encyclopedia of Mathematics
3692:David Eugene Smith (1913).
3230:Breaking up the subtraction
3063:Working from right to left:
3051:So we proceed as in step 1.
2310:, and so forth, as long as
1057:(which is signified by the
4278:
4173:Inverse for right argument
3695:The Teaching of Arithmetic
1850:are any two numbers, then
1615:and the right end labeled
1611:with the left end labeled
1088:of 5 and 2 is 3; that is,
1046:Placard outside a shop in
18:
4231:Knuth's up-arrow notation
3940:
3547:Oxford English Dictionary
3284:1234 β 567 = 1237 β 570 =
4236:SteinhausβMoser notation
3828:The Mathematics Educator
3778:Subtraction: Counting Up
3729:Subtraction: Trade First
3644:The Mathematics Educator
3224:3 + 30 + 400 + 234 = 667
1215:Notation and terminology
3682:Klapper 1916, pp. 177β.
3552:Oxford University Press
3331:(truncated subtraction)
2167:of fractional numbers.
2165:decimal representations
2161:traditional mathematics
1825:can be taken as basic.
3876:Four Digit Subtraction
3864:Subtraction Worksheets
3862:Printable Worksheets:
3598:David Gillborn (2008)
3374:"What is to Subtract?"
2638:
2074:mechanical calculators
1776:Give the answer as an
1719:To subtract arbitrary
1693:
1597:
1517:
1373:
1372:{\displaystyle 4-6=-2}
1336:
1302:
1268:
1224:
1051:
998:
973:
912:
887:
831:
806:
721:
656:
559:
533:
443:
418:
328:
303:
137:
4262:Elementary arithmetic
4226:Grzegorczyk hierarchy
3934:Elementary arithmetic
3872:Two Digit Subtraction
3868:One Digit Subtraction
3585:Janet Kolodzy (2006)
3459:John Wiley & Sons
3407:mathworld.wolfram.com
3314:Method of complements
3309:Elementary arithmetic
3049:4 β 9 = not possible.
3035:1 β 3 = not possible.
3010:The total difference.
2827:The total difference.
2639:
2070:method of complements
1692:
1596:
1546:, which in turn is a
1518:
1374:
1337:
1335:{\displaystyle 6-3=3}
1303:
1301:{\displaystyle 4-2=2}
1269:
1267:{\displaystyle 2-1=1}
1222:
1066:arithmetic operations
1064:) is one of the four
1045:
999:
974:
913:
888:
832:
807:
722:
657:
560:
534:
444:
419:
329:
304:
145:Arithmetic operations
131:
3350:numerus subtrahendus
3324:Plus and minus signs
2369:
1966:Units of measurement
1411:
1400:. The result is the
1348:
1314:
1280:
1246:
1197:computability theory
1184:and beyond. General
984:
935:
898:
857:
817:
745:
668:
583:
545:
466:
429:
351:
314:
194:
50:improve this article
4200:Super-logarithm (4)
4159:Root extraction (3)
3550:(Online ed.).
3492:Decimal Computation
3453:Decimal Computation
3401:Weisstein, Eric W.
3299:Absolute difference
3169:Nonvertical methods
3163:+300 β 40 + 2 = 262
3095:Partial differences
2718:Subtraction by hand
2178:William A. Brownell
1928:order of operations
4216:Ackermann function
4110:Exponentiation (3)
4105:Multiplication (2)
3946:
3793:2014-02-25 at the
3774:2014-02-25 at the
3755:2014-02-25 at the
3744:2014-06-23 at the
3725:2014-02-25 at the
3520:www.mathsisfun.com
2634:
2632:
2489:
2459:
2384:
1834:Anti-commutativity
1694:
1598:
1513:
1369:
1332:
1298:
1264:
1225:
1106:irrational numbers
1052:
994:
993:
969:
968:
908:
907:
883:
882:
869:
827:
826:
802:
801:
790:
787:
769:
717:
716:
711:
708:
697:
686:
652:
651:
640:
637:
634:
627:
613:
610:
603:
555:
554:
529:
528:
517:
514:
493:
439:
438:
414:
413:
402:
399:
378:
324:
323:
299:
298:
287:
284:
263:
242:
221:
138:
4244:
4243:
4137:for left argument
4047:
4046:
4042:
4041:
3698:. Ginn. pp.
3558:(Subscription or
3501:978-0-89874-318-0
3468:978-0-471-76180-8
3236:mental arithmetic
2803:(4 + 1) + ... = 7
2228:from the minuend
2157:elementary school
2141:becomes the sum:
2113:
2112:
1992:percentage change
1874:Non-associativity
1554:"from under" and
1186:binary operations
1162:additive identity
1040:
1039:
1007:
1006:
991:
958:
946:
905:
875:
873:
867:
824:
784:
779:
766:
761:
706:
695:
684:
635:
632:
625:
611:
608:
601:
552:
512:
502:
491:
481:
436:
397:
387:
376:
366:
321:
282:
272:
261:
251:
240:
230:
219:
209:
126:
125:
118:
100:
4269:
4209:Related articles
4074:
4067:
4060:
4051:
4022:
3997:
3976:
3955:
3943:
3927:
3920:
3913:
3904:
3885:Subtraction Game
3859:
3810:
3804:
3798:
3785:
3779:
3766:
3760:
3736:
3730:
3717:
3711:
3710:
3708:
3707:
3689:
3683:
3680:
3674:
3673:
3653:
3647:
3640:
3634:
3633:
3631:
3630:
3609:
3603:
3596:
3590:
3583:
3577:
3570:
3564:
3563:
3555:
3543:
3536:
3530:
3529:
3527:
3526:
3512:
3506:
3505:
3484:
3473:
3472:
3456:
3443:
3432:
3431:
3426:Cutland, Nigel.
3423:
3417:
3416:
3414:
3413:
3398:
3389:
3388:
3386:
3385:
3370:
3353:
3346:
3288:
3287:1267 β 600 = 667
3285:
3225:
3218:
3209:
3200:
3191:
3160:
3144:
3128:
3112:
3086:
3074:
3060:
3046:
3032:
3007:
2995:
2983:
2969:
2957:
2943:
2931:
2919:
2896:
2884:
2872:
2860:
2846:
2824:
2812:
2800:
2786:
2774:
2760:
2748:
2736:
2705:
2695:
2685:
2675:
2665:
2655:
2643:
2641:
2640:
2635:
2633:
2629:
2628:
2568:
2567:
2526:
2525:
2494:
2490:
2488:
2487:
2460:
2442:
2424:
2406:
2388:
2375:
2319:does not exceed
2309:
2293:
2083:
2059:
2057:
2056:
2053:
2050:
2046:
2040:
2038:
2037:
2034:
2031:
2024:
2022:
2021:
2018:
2015:
2004:percentage point
1996:percentage point
1953:
1840:anti-commutative
1820:
1799:unary operations
1771:partial function
1747:
1730:
1726:
1715:
1711:
1619:. Starting from
1522:
1520:
1519:
1514:
1512:
1511:
1475:
1474:
1438:
1437:
1378:
1376:
1375:
1370:
1341:
1339:
1338:
1333:
1307:
1305:
1304:
1299:
1273:
1271:
1270:
1265:
1190:abstract algebra
1144:
1128:
1098:negative numbers
1091:
1083:
1063:
1032:
1025:
1018:
1003:
1001:
1000:
995:
992:
989:
978:
976:
975:
970:
959:
956:
948:
947:
944:
917:
915:
914:
909:
906:
903:
892:
890:
889:
884:
876:
874:
871:
868:
865:
862:
836:
834:
833:
828:
825:
822:
811:
809:
808:
803:
795:
791:
786:
785:
782:
780:
777:
768:
767:
764:
762:
759:
726:
724:
723:
718:
715:
712:
707:
704:
696:
693:
685:
682:
661:
659:
658:
653:
645:
641:
636:
633:
630:
626:
623:
620:
612:
609:
606:
602:
599:
596:
564:
562:
561:
556:
553:
550:
538:
536:
535:
530:
522:
518:
513:
510:
503:
500:
492:
489:
482:
479:
448:
446:
445:
440:
437:
434:
423:
421:
420:
415:
407:
403:
398:
395:
388:
385:
377:
374:
367:
364:
333:
331:
330:
325:
322:
319:
308:
306:
305:
300:
292:
288:
283:
280:
273:
270:
262:
259:
252:
249:
241:
238:
231:
228:
220:
217:
210:
207:
178:
168:
161:
154:
147:
140:
135:
121:
114:
110:
107:
101:
99:
58:
34:
26:
4277:
4276:
4272:
4271:
4270:
4268:
4267:
4266:
4247:
4246:
4245:
4240:
4204:
4185:Subtraction (1)
4180:Predecessor (0)
4168:
4149:Subtraction (1)
4144:Predecessor (0)
4129:
4083:
4081:Hyperoperations
4078:
4048:
4043:
4038:
4027:
4023:
4018:
4013:
4002:
3998:
3993:
3988:
3981:
3977:
3972:
3967:
3960:
3956:
3951:
3936:
3931:
3844:
3841:
3819:
3814:
3813:
3805:
3801:
3795:Wayback Machine
3786:
3782:
3776:Wayback Machine
3767:
3763:
3757:Wayback Machine
3746:Wayback Machine
3737:
3733:
3727:Wayback Machine
3718:
3714:
3705:
3703:
3691:
3690:
3686:
3681:
3677:
3655:
3654:
3650:
3641:
3637:
3628:
3626:
3611:
3610:
3606:
3602:Routledge p. 46
3597:
3593:
3584:
3580:
3571:
3567:
3557:
3538:
3537:
3533:
3524:
3522:
3514:
3513:
3509:
3502:
3488:Schmid, Hermann
3486:
3485:
3476:
3469:
3447:Schmid, Hermann
3445:
3444:
3435:
3425:
3424:
3420:
3411:
3409:
3400:
3399:
3392:
3383:
3381:
3380:. 28 April 2022
3372:
3371:
3367:
3362:
3357:
3356:
3347:
3343:
3338:
3319:Negative number
3295:
3286:
3283:
3270:
3232:
3223:
3212:
3203:
3194:
3185:
3176:
3171:
3164:
3161:
3152:
3150:
3148:
3145:
3136:
3134:
3132:
3129:
3120:
3118:
3117:700 β 400 = 300
3116:
3113:
3097:
3090:
3087:
3078:
3075:
3066:
3064:
3061:
3052:
3050:
3047:
3038:
3036:
3033:
3018:
3011:
3008:
2999:
2996:
2987:
2984:
2975:
2973:
2970:
2961:
2958:
2949:
2947:
2944:
2935:
2932:
2923:
2920:
2907:
2905:American method
2900:
2897:
2888:
2885:
2876:
2873:
2864:
2861:
2852:
2850:
2847:
2835:
2828:
2825:
2816:
2813:
2804:
2801:
2792:
2790:
2787:
2778:
2775:
2766:
2764:
2761:
2752:
2749:
2740:
2737:
2725:
2723:Austrian method
2720:
2703:
2697:
2693:
2687:
2683:
2677:
2673:
2667:
2663:
2657:
2653:
2647:
2631:
2630:
2570:
2569:
2528:
2527:
2495:
2492:
2491:
2458:
2457:
2452:
2447:
2440:
2439:
2434:
2429:
2422:
2421:
2416:
2411:
2404:
2403:
2398:
2393:
2386:
2385:
2367:
2366:
2356:
2346:
2336:
2327:
2318:
2308:
2301:
2295:
2292:
2285:
2279:
2277:
2268:
2255:
2248:
2239:
2224:
2217:
2208:
2195:
2186:
2173:
2153:
2145:
2139:
2092:
2087:
2066:
2054:
2051:
2048:
2047:
2044:
2042:
2035:
2032:
2029:
2028:
2026:
2019:
2016:
2013:
2012:
2010:
2000:relative change
1984:
1968:
1947:
1936:
1880:non-associative
1878:Subtraction is
1876:
1838:Subtraction is
1836:
1831:
1810:
1791:
1782:negative number
1780:representing a
1759:natural numbers
1757:Subtraction of
1755:
1753:Natural numbers
1745:
1728:
1724:
1721:natural numbers
1713:
1709:
1666:to get back to
1591:
1586:
1560:draw from below
1409:
1408:
1346:
1345:
1312:
1311:
1278:
1277:
1244:
1243:
1239:. For example,
1217:
1205:natural numbers
1150:anticommutative
1132:
1116:
1089:
1081:
1061:
1036:
982:
981:
939:
933:
932:
896:
895:
855:
854:
815:
814:
789:
788:
775:
771:
770:
757:
749:
743:
742:
710:
709:
699:
698:
688:
687:
672:
666:
665:
639:
638:
615:
614:
587:
581:
580:
543:
542:
516:
515:
495:
494:
470:
464:
463:
427:
426:
401:
400:
380:
379:
355:
349:
348:
312:
311:
286:
285:
265:
264:
244:
243:
223:
222:
198:
192:
191:
172:
143:
133:
122:
111:
105:
102:
59:
57:
47:
35:
24:
17:
12:
11:
5:
4275:
4273:
4265:
4264:
4259:
4249:
4248:
4242:
4241:
4239:
4238:
4233:
4228:
4223:
4218:
4212:
4210:
4206:
4205:
4203:
4202:
4197:
4192:
4187:
4182:
4176:
4174:
4170:
4169:
4167:
4166:
4164:Super-root (4)
4161:
4156:
4151:
4146:
4140:
4138:
4131:
4130:
4128:
4127:
4122:
4117:
4112:
4107:
4102:
4097:
4091:
4089:
4085:
4084:
4079:
4077:
4076:
4069:
4062:
4054:
4045:
4044:
4040:
4039:
4016:
4014:
4000:Multiplication
3991:
3989:
3970:
3968:
3949:
3947:
3941:
3938:
3937:
3932:
3930:
3929:
3922:
3915:
3907:
3901:
3900:
3896:selected from
3891:
3882:
3860:
3840:
3839:External links
3837:
3836:
3835:
3823:
3818:
3815:
3812:
3811:
3799:
3780:
3761:
3731:
3712:
3684:
3675:
3648:
3635:
3604:
3591:
3578:
3565:
3531:
3507:
3500:
3474:
3467:
3433:
3418:
3390:
3364:
3363:
3361:
3358:
3355:
3354:
3340:
3339:
3337:
3334:
3333:
3332:
3326:
3321:
3316:
3311:
3306:
3301:
3294:
3291:
3290:
3289:
3269:
3266:
3265:
3264:
3257:
3250:
3231:
3228:
3220:
3219:
3210:
3201:
3192:
3175:
3172:
3170:
3167:
3166:
3165:
3162:
3155:
3153:
3146:
3139:
3137:
3130:
3123:
3121:
3114:
3107:
3096:
3093:
3092:
3091:
3088:
3081:
3079:
3076:
3069:
3067:
3062:
3055:
3053:
3048:
3041:
3039:
3034:
3027:
3017:
3014:
3013:
3012:
3009:
3002:
3000:
2997:
2990:
2988:
2985:
2978:
2976:
2971:
2964:
2962:
2959:
2952:
2950:
2945:
2938:
2936:
2933:
2926:
2924:
2921:
2914:
2906:
2903:
2902:
2901:
2898:
2891:
2889:
2886:
2879:
2877:
2874:
2867:
2865:
2862:
2855:
2853:
2848:
2841:
2834:
2831:
2830:
2829:
2826:
2819:
2817:
2814:
2807:
2805:
2802:
2795:
2793:
2788:
2781:
2779:
2776:
2769:
2767:
2762:
2755:
2753:
2750:
2743:
2741:
2738:
2731:
2724:
2721:
2719:
2716:
2701:
2691:
2681:
2671:
2661:
2651:
2627:
2624:
2621:
2618:
2615:
2612:
2609:
2606:
2603:
2600:
2596:
2593:
2589:
2586:
2583:
2580:
2575:
2572:
2571:
2566:
2563:
2560:
2557:
2554:
2551:
2548:
2545:
2542:
2539:
2533:
2530:
2529:
2524:
2521:
2518:
2515:
2512:
2509:
2506:
2500:
2497:
2496:
2493:
2486:
2483:
2480:
2477:
2474:
2469:
2464:
2463:
2456:
2453:
2451:
2448:
2446:
2443:
2441:
2438:
2435:
2433:
2430:
2428:
2425:
2423:
2420:
2417:
2415:
2412:
2410:
2407:
2405:
2402:
2399:
2397:
2394:
2392:
2389:
2387:
2383:
2380:
2376:
2374:
2351:
2341:
2332:
2323:
2314:
2306:
2299:
2290:
2283:
2273:
2264:
2258:
2257:
2253:
2243:
2235:
2226:
2225:
2222:
2212:
2204:
2194:
2191:
2185:
2182:
2172:
2169:
2152:
2149:
2143:
2137:
2111:
2110:
2107:
2103:
2102:
2099:
2095:
2094:
2089:
2065:
2062:
1983:
1980:
1967:
1964:
1946:, the integer
1935:
1932:
1900:
1899:
1875:
1872:
1871:
1870:
1835:
1832:
1830:
1827:
1807:multiplicative
1790:
1787:
1786:
1785:
1774:
1754:
1751:
1750:
1749:
1687:
1686:
1652:
1651:
1590:
1587:
1585:
1582:
1525:
1524:
1510:
1507:
1504:
1501:
1498:
1495:
1492:
1489:
1486:
1483:
1478:
1473:
1470:
1467:
1464:
1461:
1458:
1455:
1452:
1449:
1446:
1441:
1436:
1433:
1430:
1427:
1424:
1421:
1418:
1390:
1389:
1381:
1380:
1368:
1365:
1362:
1359:
1356:
1353:
1343:
1331:
1328:
1325:
1322:
1319:
1309:
1297:
1294:
1291:
1288:
1285:
1275:
1263:
1260:
1257:
1254:
1251:
1233:infix notation
1216:
1213:
1170:multiplication
1130:if and only if
1074:multiplication
1038:
1037:
1035:
1034:
1027:
1020:
1012:
1009:
1008:
1005:
1004:
979:
966:
962:
957:anti-logarithm
954:
951:
942:
930:
927:
926:
919:
918:
893:
880:
852:
849:
848:
838:
837:
812:
799:
794:
773:
772:
755:
754:
751:
740:
737:
736:
733:Exponentiation
729:
728:
714:
701:
700:
690:
689:
679:
678:
675:
662:
649:
644:
617:
616:
593:
592:
589:
578:
575:
574:
567:
566:
539:
526:
521:
507:
497:
496:
486:
476:
475:
472:
461:
458:
457:
454:Multiplication
450:
449:
424:
411:
406:
392:
382:
381:
371:
361:
360:
357:
346:
343:
342:
335:
334:
309:
296:
291:
277:
267:
266:
256:
246:
245:
235:
225:
224:
214:
204:
203:
200:
189:
186:
185:
174:
173:
171:
170:
163:
156:
148:
124:
123:
38:
36:
29:
15:
13:
10:
9:
6:
4:
3:
2:
4274:
4263:
4260:
4258:
4255:
4254:
4252:
4237:
4234:
4232:
4229:
4227:
4224:
4222:
4219:
4217:
4214:
4213:
4211:
4207:
4201:
4198:
4196:
4195:Logarithm (3)
4193:
4191:
4188:
4186:
4183:
4181:
4178:
4177:
4175:
4171:
4165:
4162:
4160:
4157:
4155:
4152:
4150:
4147:
4145:
4142:
4141:
4139:
4136:
4132:
4126:
4123:
4121:
4120:Pentation (5)
4118:
4116:
4115:Tetration (4)
4113:
4111:
4108:
4106:
4103:
4101:
4098:
4096:
4095:Successor (0)
4093:
4092:
4090:
4086:
4082:
4075:
4070:
4068:
4063:
4061:
4056:
4055:
4052:
4037:
4035:
4031:
4026:
4021:
4015:
4012:
4010:
4006:
4001:
3996:
3990:
3987:
3985:
3980:
3975:
3969:
3966:
3964:
3959:
3954:
3948:
3945:
3944:
3939:
3935:
3928:
3923:
3921:
3916:
3914:
3909:
3908:
3905:
3899:
3895:
3892:
3890:
3886:
3883:
3881:
3877:
3873:
3869:
3865:
3861:
3857:
3853:
3852:
3847:
3846:"Subtraction"
3843:
3842:
3838:
3834:
3831:
3829:
3824:
3821:
3820:
3816:
3808:
3803:
3800:
3796:
3792:
3789:
3784:
3781:
3777:
3773:
3770:
3765:
3762:
3758:
3754:
3751:
3747:
3743:
3740:
3735:
3732:
3728:
3724:
3721:
3716:
3713:
3701:
3697:
3696:
3688:
3685:
3679:
3676:
3671:
3667:
3664:(7): 389β93.
3663:
3659:
3652:
3649:
3645:
3639:
3636:
3624:
3620:
3619:
3614:
3613:Klapper, Paul
3608:
3605:
3601:
3595:
3592:
3588:
3582:
3579:
3575:
3569:
3566:
3561:
3553:
3549:
3548:
3542:
3541:"Subtraction"
3535:
3532:
3521:
3517:
3516:"Subtraction"
3511:
3508:
3503:
3497:
3493:
3489:
3483:
3481:
3479:
3475:
3470:
3464:
3460:
3455:
3454:
3448:
3442:
3440:
3438:
3434:
3429:
3422:
3419:
3408:
3404:
3403:"Subtraction"
3397:
3395:
3391:
3379:
3375:
3369:
3366:
3359:
3351:
3345:
3342:
3335:
3330:
3327:
3325:
3322:
3320:
3317:
3315:
3312:
3310:
3307:
3305:
3302:
3300:
3297:
3296:
3292:
3282:
3281:
3280:
3277:
3274:
3267:
3262:
3258:
3255:
3251:
3248:
3244:
3243:
3242:
3239:
3237:
3229:
3227:
3216:
3211:
3207:
3202:
3198:
3193:
3189:
3184:
3183:
3182:
3179:
3173:
3168:
3159:
3154:
3143:
3138:
3127:
3122:
3111:
3106:
3104:
3101:
3094:
3085:
3080:
3073:
3068:
3059:
3054:
3045:
3040:
3031:
3026:
3024:
3021:
3015:
3006:
3001:
2994:
2989:
2982:
2977:
2968:
2963:
2956:
2951:
2942:
2937:
2930:
2925:
2918:
2913:
2911:
2904:
2895:
2890:
2883:
2878:
2871:
2866:
2859:
2854:
2845:
2840:
2838:
2832:
2823:
2818:
2811:
2806:
2799:
2794:
2785:
2780:
2773:
2768:
2759:
2754:
2747:
2742:
2735:
2730:
2728:
2722:
2717:
2715:
2711:
2707:
2700:
2690:
2680:
2670:
2660:
2650:
2644:
2454:
2449:
2444:
2436:
2431:
2426:
2418:
2413:
2408:
2400:
2395:
2390:
2381:
2378:
2364:
2362:
2358:
2354:
2350:
2344:
2340:
2335:
2331:
2328:. Otherwise,
2326:
2322:
2317:
2313:
2305:
2298:
2289:
2282:
2276:
2272:
2267:
2263:
2252:
2246:
2242:
2238:
2234:
2231:
2230:
2229:
2221:
2215:
2211:
2207:
2203:
2200:
2199:
2198:
2192:
2190:
2183:
2181:
2179:
2170:
2168:
2166:
2162:
2158:
2150:
2148:
2142:
2136:
2132:
2130:
2126:
2122:
2118:
2108:
2105:
2100:
2097:
2090:
2085:
2084:
2081:
2079:
2075:
2071:
2063:
2061:
2007:
2005:
2001:
1997:
1993:
1989:
1981:
1979:
1977:
1973:
1965:
1963:
1961:
1957:
1951:
1945:
1941:
1933:
1931:
1929:
1925:
1921:
1917:
1913:
1909:
1905:
1897:
1893:
1889:
1885:
1884:
1883:
1881:
1873:
1868:
1864:
1860:
1856:
1853:
1852:
1851:
1849:
1845:
1841:
1833:
1828:
1826:
1824:
1818:
1814:
1808:
1804:
1800:
1796:
1788:
1783:
1779:
1775:
1772:
1768:
1767:
1766:
1764:
1760:
1752:
1744:
1743:
1742:
1740:
1737:
1732:
1722:
1717:
1707:
1703:
1699:
1691:
1684:
1680:
1676:
1673:
1672:
1671:
1669:
1665:
1662:steps to the
1661:
1657:
1649:
1645:
1641:
1638:
1637:
1636:
1634:
1630:
1626:
1622:
1618:
1614:
1610:
1607:
1603:
1595:
1588:
1583:
1581:
1579:
1575:
1572:
1569:
1565:
1561:
1557:
1553:
1549:
1545:
1542:
1538:
1534:
1530:
1476:
1439:
1407:
1406:
1405:
1403:
1399:
1395:
1386:
1385:
1384:
1366:
1363:
1360:
1357:
1354:
1351:
1344:
1329:
1326:
1323:
1320:
1317:
1310:
1295:
1292:
1289:
1286:
1283:
1276:
1261:
1258:
1255:
1252:
1249:
1242:
1241:
1240:
1238:
1234:
1230:
1221:
1214:
1212:
1210:
1206:
1202:
1198:
1193:
1191:
1187:
1183:
1179:
1175:
1171:
1167:
1163:
1159:
1155:
1151:
1146:
1143:
1139:
1135:
1131:
1127:
1123:
1119:
1113:
1111:
1107:
1103:
1099:
1095:
1087:
1079:
1075:
1071:
1067:
1060:
1056:
1049:
1044:
1033:
1028:
1026:
1021:
1019:
1014:
1013:
1011:
1010:
980:
964:
949:
940:
931:
928:
924:
920:
894:
878:
853:
850:
846:
844:
839:
813:
797:
792:
741:
738:
734:
730:
727:
673:
663:
647:
642:
579:
576:
572:
568:
565:
540:
524:
519:
505:
484:
462:
459:
455:
451:
425:
409:
404:
390:
369:
347:
344:
340:
336:
310:
294:
289:
275:
254:
233:
212:
190:
187:
183:
179:
176:
175:
169:
164:
162:
157:
155:
150:
149:
146:
142:
141:
130:
120:
117:
109:
98:
95:
91:
88:
84:
81:
77:
74:
70:
67: β
66:
65:"Subtraction"
62:
61:Find sources:
55:
51:
45:
44:
39:This article
37:
33:
28:
27:
22:
4190:Division (2)
4184:
4154:Division (2)
4148:
4125:Hexation (6)
4100:Addition (1)
4017:
3992:
3978:
3973:
3971:
3950:
3889:cut-the-knot
3849:
3827:
3817:Bibliography
3802:
3783:
3764:
3734:
3715:
3704:. Retrieved
3694:
3687:
3678:
3661:
3657:
3651:
3643:
3638:
3627:. Retrieved
3617:
3607:
3599:
3594:
3586:
3581:
3573:
3568:
3545:
3534:
3523:. Retrieved
3519:
3510:
3491:
3452:
3427:
3421:
3410:. Retrieved
3406:
3382:. Retrieved
3377:
3368:
3349:
3344:
3278:
3275:
3271:
3260:
3253:
3246:
3240:
3233:
3221:
3214:
3205:
3196:
3187:
3180:
3177:
3133:90 β 50 = 40
3102:
3100:difference.
3098:
3022:
3019:
2972:15 β 9 = ...
2908:
2836:
2789:9 + ... = 15
2726:
2712:
2708:
2698:
2688:
2678:
2668:
2658:
2648:
2645:
2365:
2360:
2359:
2352:
2348:
2342:
2338:
2333:
2329:
2324:
2320:
2315:
2311:
2303:
2296:
2287:
2280:
2274:
2270:
2265:
2261:
2259:
2250:
2244:
2240:
2236:
2232:
2227:
2219:
2213:
2209:
2205:
2201:
2196:
2187:
2174:
2154:
2146:
2140:
2133:
2128:
2127:is added to
2124:
2120:
2116:
2114:
2067:
2064:In computing
2008:
1985:
1969:
1959:
1955:
1949:
1943:
1937:
1923:
1919:
1915:
1911:
1907:
1903:
1901:
1895:
1891:
1887:
1877:
1866:
1862:
1858:
1854:
1847:
1843:
1837:
1816:
1812:
1792:
1789:Real numbers
1756:
1733:
1718:
1695:
1682:
1678:
1674:
1667:
1663:
1659:
1655:
1653:
1647:
1643:
1639:
1628:
1624:
1620:
1616:
1612:
1608:
1602:line segment
1599:
1577:
1573:
1566:. Using the
1563:
1559:
1555:
1551:
1543:
1526:
1404:. That is,
1401:
1397:
1393:
1391:
1382:
1226:
1201:well-defined
1194:
1182:real numbers
1147:
1141:
1137:
1133:
1125:
1121:
1117:
1114:
1085:
1054:
1053:
842:
511:multiplicand
338:
112:
103:
93:
86:
79:
72:
60:
48:Please help
43:verification
40:
4257:Subtraction
3979:Subtraction
3621:. pp.
3378:SplashLearn
3268:Same change
3174:Counting up
3016:Trade first
2986:6 β 4 = ...
2946:5 β 9 = ...
2922:3 β 1 = ...
2763:9 + ... = 5
2739:1 + ... = 3
2363:704 β 512.
2260:where each
2093:complement
1988:percentages
1986:Changes in
1982:Percentages
1934:Predecessor
1739:number line
1658:, it takes
1623:, it takes
1533:Subtraction
1388:accounting.
1237:equals sign
1154:associative
1068:along with
1055:Subtraction
631:denominator
339:Subtraction
4251:Categories
3706:2016-03-11
3629:2016-03-11
3562:required.)
3525:2020-08-26
3412:2020-08-26
3384:2022-12-13
3360:References
3077:14 β 9 = 5
3065:11 β 3 = 8
2875:15 β 9 = 6
2171:In America
2043:β33
1829:Properties
1746:3 β 4 = β1
1600:Imagine a
1544:subtrahere
1402:difference
1394:subtrahend
1229:minus sign
1094:arithmetic
1086:difference
1059:minus sign
501:multiplier
435:difference
396:subtrahend
76:newspapers
3856:EMS Press
3490:(1983) .
3304:Decrement
3276:Example:
3149:3 β 1 = 2
3103:Example:
3089:6 β 4 = 2
3023:Example:
2899:3 β 1 = 2
2849:7 β 4 = 3
2837:Example:
2727:Example:
2574:⟵
2532:⟵
2499:⟵
2468:⟵
2379:−
2184:In Europe
2078:computers
2014:20% β 30%
1972:kilograms
1801:yielding
1725:3 β 3 = 0
1714:3 β 2 = 1
1710:3 β 0 = 3
1568:gerundive
1564:take away
1440:−
1364:−
1355:−
1321:−
1287:−
1253:−
1102:fractions
1090:5 β 2 = 3
990:logarithm
950:
923:Logarithm
624:numerator
506:×
485:×
391:−
370:−
4025:Division
3958:Addition
3791:Archived
3772:Archived
3753:Archived
3742:Archived
3723:Archived
3615:(1916).
3449:(1974).
3293:See also
2361:Example:
2357:by one.
1823:division
1815:= 3 + (β
1803:additive
1633:addition
1589:Integers
1562:, or to
1548:compound
1535:" is an
1178:integers
1166:addition
1078:division
1070:addition
1048:Bordeaux
866:radicand
765:exponent
694:quotient
683:fraction
600:dividend
571:Division
182:Addition
106:May 2018
4135:Inverse
4088:Primary
4034:∕
3984:−
3974:−
3858:, 2001
3245:1234 β
3213:1000 +
2058:
2039:
2027:
2023:
2011:
1778:integer
1761:is not
1736:integer
1578:minuere
1556:trahere
1537:English
1398:minuend
1160:is the
1110:vectors
845:th root
607:divisor
551:product
386:minuend
239:summand
229:summand
90:scholar
4030:÷
4020:÷
4009:·
4005:×
3995:×
3878:, and
3498:
3465:
3259:674 β
3252:734 β
3217:= 1234
3208:= 1000
3204:600 +
3195:570 +
3186:567 +
2176:after
2091:Ones'
2088:digit
2086:Binary
1976:pounds
1763:closed
1727:. But
1704:, and
1606:length
1571:suffix
1174:proven
872:degree
490:factor
480:factor
281:addend
271:augend
260:addend
250:addend
92:
85:
78:
71:
63:
3963:+
3953:+
3556:
3336:Notes
3329:Monus
3263:= 667
3256:= 674
3249:= 734
3199:= 600
3190:= 570
1795:field
1729:3 β 4
1654:From
1529:Latin
1209:monus
1203:over
1082:5 β 2
925:(log)
823:power
783:power
705:ratio
134:5 β 2
97:JSTOR
83:books
3496:ISBN
3463:ISBN
2696:and
2666:and
2269:and
2249:...
2218:...
2068:The
1994:and
1952:β 1)
1910:) β
1861:= β(
1846:and
1811:3 β
1805:and
1793:The
1664:left
1541:verb
1168:and
1076:and
945:base
904:root
778:base
760:base
375:term
365:term
218:term
208:term
69:news
4032:or
4007:or
3887:at
3833:PDF
3666:doi
3247:500
3215:234
3206:400
2704:= 2
2694:= 1
2684:= 5
2674:= 4
2664:= 0
2654:= 7
2025:= β
2020:30%
1974:or
1940:one
1918:β (
1914:or
1604:of
1574:-nd
1552:sub
1550:of
1531:. "
1195:In
941:log
847:(β)
735:(^)
573:(Γ·)
456:(Γ)
341:(β)
320:sum
184:(+)
52:by
4253::
4036:)
4011:)
3986:)
3965:)
3874:,
3870:,
3866:,
3854:,
3848:,
3748:;
3700:77
3662:99
3660:.
3623:80
3544:.
3518:.
3477:^
3461:.
3436:^
3405:.
3393:^
3376:.
3254:60
3226:.
3197:30
2686:,
2656:,
2355:+1
2345:+1
2302:β
2294:,
2286:β
2247:β1
2216:β1
2109:0
2106:1
2101:1
2098:0
2080:.
2041:=
1962:.
1922:β
1906:β
1894:β
1890:β
1867:a)
1865:β
1857:β
1700:,
1681:=
1677:β
1646:=
1642:+
1635::
1211:.
1192:.
1140:=
1136:+
1124:β
1120:=
1108:,
1104:,
1100:,
1072:,
4073:e
4066:t
4059:v
4028:(
4003:(
3982:(
3961:(
3926:e
3919:t
3912:v
3709:.
3702:β
3672:.
3668::
3632:.
3625:β
3554:.
3528:.
3504:.
3471:.
3430:.
3415:.
3387:.
3261:7
3188:3
2702:1
2699:s
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2649:m
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2579:R
2565:d
2562:n
2559:e
2556:h
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2505:M
2485:y
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2476:a
2473:c
2455:2
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2233:m
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2202:s
2129:x
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2121:x
2117:y
2055:3
2052:/
2049:1
2045:+
2036:3
2033:/
2030:1
2017:/
1960:a
1956:a
1950:a
1948:(
1944:a
1924:c
1920:b
1916:a
1912:c
1908:b
1904:a
1898:"
1896:c
1892:b
1888:a
1886:"
1869:.
1863:b
1859:b
1855:a
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1844:a
1819:)
1817:Ο
1813:Ο
1773:.
1748:.
1706:3
1702:2
1698:1
1685:.
1683:a
1679:b
1675:c
1668:a
1660:b
1656:c
1650:.
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1644:b
1640:a
1629:c
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1617:c
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1609:b
1523:.
1509:e
1506:c
1503:n
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1497:r
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1477:=
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1448:u
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1367:2
1361:=
1358:6
1352:4
1330:3
1327:=
1324:3
1318:6
1296:2
1293:=
1290:2
1284:4
1262:1
1259:=
1256:1
1250:2
1158:0
1142:a
1138:b
1134:c
1126:b
1122:a
1118:c
1062:β
1031:e
1024:t
1017:v
965:=
961:)
953:(
879:=
843:n
798:=
793:}
674:{
648:=
643:}
525:=
520:}
410:=
405:}
295:=
290:}
276:+
255:+
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213:+
167:e
160:t
153:v
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119:)
113:(
108:)
104:(
94:Β·
87:Β·
80:Β·
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46:.
23:.
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