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2717:. 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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2648:{\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}}}
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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:
313:{\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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666:{\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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1776:: 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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2174:, 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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543:{\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.
1820:
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 ."
428:{\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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1727:. 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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816:{\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
731:{\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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2020:
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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1853:, 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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1107:, subtraction can also represent removing or decreasing physical and abstract quantities using different kinds of objects including
1091:. Subtraction is an operation that represents removal of objects from a collection. For example, in the adjacent picture, there are
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1175:, 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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1167:, meaning that when one subtracts more than two numbers, the order in which subtraction is performed matters. Because
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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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1989:, they must have the same unit. In most cases, the difference will have the same unit as the original numbers.
1156:. 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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1937:), but these two possibilities lead to different answers. To resolve this issue, one must establish an
1832:. 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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3359:"Subtrahend" is shortened by the inflectional Latin suffix -us, e.g. remaining un-declined as in
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1752:(..., 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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1234:
Subtraction of numbers 0β10. Line labels = minuend. X axis = subtrahend. Y axis = difference.
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3837:
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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1164:
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Because the minuend is greater than the subtrahend, this difference has a plus sign.
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Because the minuend is greater than the subtrahend, this difference has a plus sign.
2142:
and one is added to the sum. The leading digit "1" of the result is then discarded.
1604:
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1986:
1612:
1591:"to reduce or diminish", one gets "minuend", which means "thing to be diminished".
1218:, operations between numbers are actually defined using "truncated subtraction" or
3704:
3585:
Teachers Versus the Public: What Americans Think about Schools and How to Fix Them
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1163:, 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.
983:{\displaystyle \scriptstyle \log _{\text{base}}({\text{anti-logarithm}})\,=\,}
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3505:(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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897:{\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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1746:
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Now the subtraction works, and we write the difference under the line.
2149:
01100100 (x, equals decimal 100) - 00010110 (y, equals decimal 22)
1780:
Conclude that 26 cannot be subtracted from 11; subtraction becomes a
1616:
1527:{\displaystyle {\rm {minuend}}-{\rm {subtrahend}}={\rm {difference}}}
3841:, Vol. 8, No. 1 (original publication) and Vol. 10, No. 1 (reprint.)
3598:
Convergence Journalism: Writing and Reporting across the News Media
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3339:
2657:
The minuend is 704, the subtrahend is 512. The minuend digits are
1719:. 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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2071:%, while the percentage point change is β10 percentage points.
1981:
When subtracting two numbers with units of measurement such as
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36:
3233:
Add up the value from each step to get the total difference:
1126:
In a sense, subtraction is the inverse of addition. That is,
1699:
1603:
3583:
Paul E. Peterson, Michael Henderson, Martin R. West (2014)
3439:
Computability: an introduction to recursive function theory
3192:
Example: 1234 β 567 = can be found by the following steps:
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Example: 1234 β 567 = can be solved in the following way:
1642:. This movement to the right is modeled mathematically by
3818:
The Many Ways of Arithmetic in UCSMP Everyday Mathematics
3799:
The Many Ways of Arithmetic in UCSMP Everyday Mathematics
3780:
The Many Ways of Arithmetic in UCSMP Everyday Mathematics
3761:
The Many Ways of Arithmetic in UCSMP Everyday Mathematics
3731:
The Many Ways of Arithmetic in UCSMP Everyday Mathematics
1723:. It takes 2 steps to the left to get to position 1, so
1681:. This movement to the left is modeled by subtraction:
1403:
Formally, the number being subtracted is known as the
1159:
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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67:. Unsourced material may be challenged and removed.
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3158:The smaller number is subtracted from the greater:
3142:The smaller number is subtracted from the greater:
3126:The smaller number is subtracted from the greater:
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2013:between the two quantities as a percentage, while
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3629:The Teaching of Arithmetic: A Manual for Teachers
3249:is to split up the subtraction into small steps.
1795:, so the result of subtracting 26 from 11 is β15.
3611:Racism and Education: Coincidence Or Conspiracy?
1407:, while the number it is subtracted from is the
453:{\displaystyle \scriptstyle {\text{difference}}}
1008:{\displaystyle \scriptstyle {\text{logarithm}}}
30:"Subtract" redirects here. For other uses, see
1353:(pronounced as "six minus three equals three")
147:= 3" (verbally, "five minus two equals three")
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1707:Now, a line segment labeled with the numbers
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569:{\displaystyle \scriptstyle {\text{product}}}
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3600:Rowman & Littlefield Publishers, p. 180
1953:also plays a special role: for any integer
1949:In the context of integers, subtraction of
1319:(pronounced as "four minus two equals two")
841:{\displaystyle \scriptstyle {\text{power}}}
27:One of the four basic arithmetic operations
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922:{\displaystyle \scriptstyle {\text{root}}}
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3290:"1234 β 567 =" can be solved as follows:
2826:The difference is written under the line.
2802:Now we can find the difference as before.
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1238:Subtraction is usually written using the
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127:Learn how and when to remove this message
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3009:We write the difference under the line.
2945:We write the difference under the line.
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2134:(the minuend), the ones' complement of
2001:can be reported in at least two forms,
3808:Subtraction: Left to Right Subtraction
2162:The teaching of subtraction in schools
2477:
2388:
2289:is a digit, proceeds by writing down
2166:Methods used to teach subtraction to
2130:(the subtrahend) from another number
1538:All of this terminology derives from
1123:, decimals, functions, and matrices.
7:
3571:participating institution membership
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3405:
65:adding citations to reliable sources
3587:Brookings Institution Press, p. 163
1969:, also known as the predecessor of
1569:"to pull". Thus, to subtract is to
1242:"β" between the terms; that is, in
1187:, starting with the subtraction of
3681:10.1111/j.1949-8594.1999.tb17499.x
3245:Another method that is useful for
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1246:. The result is expressed with an
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1965:is the largest integer less than
1210:, considering subtraction is not
3905:Subtraction on a Japanese abacus
3770:Subtraction: Partial Differences
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1745:The solution is to consider the
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3750:Partial-Differences Subtraction
1913:can be defined to mean either (
52:needs additional citations for
3669:School Science and Mathematics
3468:(1 ed.). Binghamton, NY:
3363:"the number to be subtracted".
2844:Subtraction from left to right
2584:
2542:
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2478:
2204:Comparing the two main methods
2087:, and is still used in modern
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1:
4232:Conway chained arrow notation
3820:Subtraction: Same Change Rule
2959:The minuend (5) is too small!
2687:. The subtrahend digits are
2126:To subtract a binary number
2115:
2107:
1638:steps to the right to reach
1595:Of integers and real numbers
1550:word derived from the Latin
1183:. All of these rules can be
32:Subtraction (disambiguation)
3909:Abacus: Mystery of the Bead
3891:More Subtraction Worksheets
3862:Encyclopedia of Mathematics
3703:David Eugene Smith (1913).
3241:Breaking up the subtraction
3074:Working from right to left:
3062:So we proceed as in step 1.
2321:, and so forth, as long as
1068:(which is signified by the
4289:
4184:Inverse for right argument
3706:The Teaching of Arithmetic
1861:are any two numbers, then
1626:and the right end labeled
1622:with the left end labeled
1099:of 5 and 2 is 3; that is,
1057:Placard outside a shop in
29:
4242:Knuth's up-arrow notation
3951:
3558:Oxford English Dictionary
3295:1234 β 567 = 1237 β 570 =
4247:SteinhausβMoser notation
3839:The Mathematics Educator
3789:Subtraction: Counting Up
3740:Subtraction: Trade First
3655:The Mathematics Educator
3235:3 + 30 + 400 + 234 = 667
1226:Notation and terminology
3693:Klapper 1916, pp. 177β.
3563:Oxford University Press
3342:(truncated subtraction)
2178:of fractional numbers.
2176:decimal representations
2172:traditional mathematics
1836:can be taken as basic.
3887:Four Digit Subtraction
3875:Subtraction Worksheets
3873:Printable Worksheets:
3609:David Gillborn (2008)
3385:"What is to Subtract?"
2649:
2085:mechanical calculators
1787:Give the answer as an
1730:To subtract arbitrary
1704:
1608:
1528:
1384:
1383:{\displaystyle 4-6=-2}
1347:
1313:
1279:
1235:
1062:
1009:
984:
923:
898:
842:
817:
732:
667:
570:
544:
454:
429:
339:
314:
148:
4273:Elementary arithmetic
4237:Grzegorczyk hierarchy
3945:Elementary arithmetic
3883:Two Digit Subtraction
3879:One Digit Subtraction
3596:Janet Kolodzy (2006)
3470:John Wiley & Sons
3418:mathworld.wolfram.com
3325:Method of complements
3320:Elementary arithmetic
3060:4 β 9 = not possible.
3046:1 β 3 = not possible.
3021:The total difference.
2838:The total difference.
2650:
2081:method of complements
1703:
1607:
1557:, which in turn is a
1529:
1385:
1348:
1346:{\displaystyle 6-3=3}
1314:
1312:{\displaystyle 4-2=2}
1280:
1278:{\displaystyle 2-1=1}
1233:
1077:arithmetic operations
1075:) is one of the four
1056:
1010:
985:
924:
899:
843:
818:
733:
668:
571:
545:
455:
430:
340:
315:
156:Arithmetic operations
142:
3361:numerus subtrahendus
3335:Plus and minus signs
2380:
1977:Units of measurement
1422:
1411:. The result is the
1359:
1325:
1291:
1257:
1208:computability theory
1195:and beyond. General
995:
946:
909:
868:
828:
756:
679:
594:
556:
477:
440:
362:
325:
205:
61:improve this article
4211:Super-logarithm (4)
4170:Root extraction (3)
3561:(Online ed.).
3503:Decimal Computation
3464:Decimal Computation
3412:Weisstein, Eric W.
3310:Absolute difference
3180:Nonvertical methods
3174:+300 β 40 + 2 = 262
3106:Partial differences
2729:Subtraction by hand
2189:William A. Brownell
1939:order of operations
4227:Ackermann function
4121:Exponentiation (3)
4116:Multiplication (2)
3957:
3804:2014-02-25 at the
3785:2014-02-25 at the
3766:2014-02-25 at the
3755:2014-06-23 at the
3736:2014-02-25 at the
3531:www.mathsisfun.com
2645:
2643:
2500:
2470:
2395:
1845:Anti-commutativity
1705:
1609:
1524:
1380:
1343:
1309:
1275:
1236:
1117:irrational numbers
1063:
1005:
1004:
980:
979:
919:
918:
894:
893:
880:
838:
837:
813:
812:
801:
798:
780:
728:
727:
722:
719:
708:
697:
663:
662:
651:
648:
645:
638:
624:
621:
614:
566:
565:
540:
539:
528:
525:
504:
450:
449:
425:
424:
413:
410:
389:
335:
334:
310:
309:
298:
295:
274:
253:
232:
149:
4255:
4254:
4148:for left argument
4058:
4057:
4053:
4052:
3709:. Ginn. pp.
3569:(Subscription or
3512:978-0-89874-318-0
3479:978-0-471-76180-8
3247:mental arithmetic
2814:(4 + 1) + ... = 7
2239:from the minuend
2168:elementary school
2152:becomes the sum:
2124:
2123:
2003:percentage change
1885:Non-associativity
1565:"from under" and
1197:binary operations
1173:additive identity
1051:
1050:
1018:
1017:
1002:
969:
957:
916:
886:
884:
878:
835:
795:
790:
777:
772:
717:
706:
695:
646:
643:
636:
622:
619:
612:
563:
523:
513:
502:
492:
447:
408:
398:
387:
377:
332:
293:
283:
272:
262:
251:
241:
230:
220:
137:
136:
129:
111:
16:(Redirected from
4280:
4220:Related articles
4085:
4078:
4071:
4062:
4033:
4008:
3987:
3966:
3954:
3938:
3931:
3924:
3915:
3896:Subtraction Game
3870:
3821:
3815:
3809:
3796:
3790:
3777:
3771:
3747:
3741:
3728:
3722:
3721:
3719:
3718:
3700:
3694:
3691:
3685:
3684:
3664:
3658:
3651:
3645:
3644:
3642:
3641:
3620:
3614:
3607:
3601:
3594:
3588:
3581:
3575:
3574:
3566:
3554:
3547:
3541:
3540:
3538:
3537:
3523:
3517:
3516:
3495:
3484:
3483:
3467:
3454:
3443:
3442:
3437:Cutland, Nigel.
3434:
3428:
3427:
3425:
3424:
3409:
3400:
3399:
3397:
3396:
3381:
3364:
3357:
3299:
3298:1267 β 600 = 667
3296:
3236:
3229:
3220:
3211:
3202:
3171:
3155:
3139:
3123:
3097:
3085:
3071:
3057:
3043:
3018:
3006:
2994:
2980:
2968:
2954:
2942:
2930:
2907:
2895:
2883:
2871:
2857:
2835:
2823:
2811:
2797:
2785:
2771:
2759:
2747:
2716:
2706:
2696:
2686:
2676:
2666:
2654:
2652:
2651:
2646:
2644:
2640:
2639:
2579:
2578:
2537:
2536:
2505:
2501:
2499:
2498:
2471:
2453:
2435:
2417:
2399:
2386:
2330:does not exceed
2320:
2304:
2094:
2070:
2068:
2067:
2064:
2061:
2057:
2051:
2049:
2048:
2045:
2042:
2035:
2033:
2032:
2029:
2026:
2015:percentage point
2007:percentage point
1964:
1851:anti-commutative
1831:
1810:unary operations
1782:partial function
1758:
1741:
1737:
1726:
1722:
1630:. Starting from
1533:
1531:
1530:
1525:
1523:
1522:
1486:
1485:
1449:
1448:
1389:
1387:
1386:
1381:
1352:
1350:
1349:
1344:
1318:
1316:
1315:
1310:
1284:
1282:
1281:
1276:
1201:abstract algebra
1155:
1139:
1109:negative numbers
1102:
1094:
1074:
1043:
1036:
1029:
1014:
1012:
1011:
1006:
1003:
1000:
989:
987:
986:
981:
970:
967:
959:
958:
955:
928:
926:
925:
920:
917:
914:
903:
901:
900:
895:
887:
885:
882:
879:
876:
873:
847:
845:
844:
839:
836:
833:
822:
820:
819:
814:
806:
802:
797:
796:
793:
791:
788:
779:
778:
775:
773:
770:
737:
735:
734:
729:
726:
723:
718:
715:
707:
704:
696:
693:
672:
670:
669:
664:
656:
652:
647:
644:
641:
637:
634:
631:
623:
620:
617:
613:
610:
607:
575:
573:
572:
567:
564:
561:
549:
547:
546:
541:
533:
529:
524:
521:
514:
511:
503:
500:
493:
490:
459:
457:
456:
451:
448:
445:
434:
432:
431:
426:
418:
414:
409:
406:
399:
396:
388:
385:
378:
375:
344:
342:
341:
336:
333:
330:
319:
317:
316:
311:
303:
299:
294:
291:
284:
281:
273:
270:
263:
260:
252:
249:
242:
239:
231:
228:
221:
218:
189:
179:
172:
165:
158:
151:
146:
132:
125:
121:
118:
112:
110:
69:
45:
37:
21:
4288:
4287:
4283:
4282:
4281:
4279:
4278:
4277:
4258:
4257:
4256:
4251:
4215:
4196:Subtraction (1)
4191:Predecessor (0)
4179:
4160:Subtraction (1)
4155:Predecessor (0)
4140:
4094:
4092:Hyperoperations
4089:
4059:
4054:
4049:
4038:
4034:
4029:
4024:
4013:
4009:
4004:
3999:
3992:
3988:
3983:
3978:
3971:
3967:
3962:
3947:
3942:
3855:
3852:
3830:
3825:
3824:
3816:
3812:
3806:Wayback Machine
3797:
3793:
3787:Wayback Machine
3778:
3774:
3768:Wayback Machine
3757:Wayback Machine
3748:
3744:
3738:Wayback Machine
3729:
3725:
3716:
3714:
3702:
3701:
3697:
3692:
3688:
3666:
3665:
3661:
3652:
3648:
3639:
3637:
3622:
3621:
3617:
3613:Routledge p. 46
3608:
3604:
3595:
3591:
3582:
3578:
3568:
3549:
3548:
3544:
3535:
3533:
3525:
3524:
3520:
3513:
3499:Schmid, Hermann
3497:
3496:
3487:
3480:
3458:Schmid, Hermann
3456:
3455:
3446:
3436:
3435:
3431:
3422:
3420:
3411:
3410:
3403:
3394:
3392:
3391:. 28 April 2022
3383:
3382:
3378:
3373:
3368:
3367:
3358:
3354:
3349:
3330:Negative number
3306:
3297:
3294:
3281:
3243:
3234:
3223:
3214:
3205:
3196:
3187:
3182:
3175:
3172:
3163:
3161:
3159:
3156:
3147:
3145:
3143:
3140:
3131:
3129:
3128:700 β 400 = 300
3127:
3124:
3108:
3101:
3098:
3089:
3086:
3077:
3075:
3072:
3063:
3061:
3058:
3049:
3047:
3044:
3029:
3022:
3019:
3010:
3007:
2998:
2995:
2986:
2984:
2981:
2972:
2969:
2960:
2958:
2955:
2946:
2943:
2934:
2931:
2918:
2916:American method
2911:
2908:
2899:
2896:
2887:
2884:
2875:
2872:
2863:
2861:
2858:
2846:
2839:
2836:
2827:
2824:
2815:
2812:
2803:
2801:
2798:
2789:
2786:
2777:
2775:
2772:
2763:
2760:
2751:
2748:
2736:
2734:Austrian method
2731:
2714:
2708:
2704:
2698:
2694:
2688:
2684:
2678:
2674:
2668:
2664:
2658:
2642:
2641:
2581:
2580:
2539:
2538:
2506:
2503:
2502:
2469:
2468:
2463:
2458:
2451:
2450:
2445:
2440:
2433:
2432:
2427:
2422:
2415:
2414:
2409:
2404:
2397:
2396:
2378:
2377:
2367:
2357:
2347:
2338:
2329:
2319:
2312:
2306:
2303:
2296:
2290:
2288:
2279:
2266:
2259:
2250:
2235:
2228:
2219:
2206:
2197:
2184:
2164:
2156:
2150:
2103:
2098:
2077:
2065:
2062:
2059:
2058:
2055:
2053:
2046:
2043:
2040:
2039:
2037:
2030:
2027:
2024:
2023:
2021:
2011:relative change
1995:
1979:
1958:
1947:
1891:non-associative
1889:Subtraction is
1887:
1849:Subtraction is
1847:
1842:
1821:
1802:
1793:negative number
1791:representing a
1770:natural numbers
1768:Subtraction of
1766:
1764:Natural numbers
1756:
1739:
1735:
1732:natural numbers
1724:
1720:
1677:to get back to
1602:
1597:
1571:draw from below
1420:
1419:
1357:
1356:
1323:
1322:
1289:
1288:
1255:
1254:
1250:. For example,
1228:
1216:natural numbers
1161:anticommutative
1143:
1127:
1100:
1092:
1072:
1047:
993:
992:
950:
944:
943:
907:
906:
866:
865:
826:
825:
800:
799:
786:
782:
781:
768:
760:
754:
753:
721:
720:
710:
709:
699:
698:
683:
677:
676:
650:
649:
626:
625:
598:
592:
591:
554:
553:
527:
526:
506:
505:
481:
475:
474:
438:
437:
412:
411:
391:
390:
366:
360:
359:
323:
322:
297:
296:
276:
275:
255:
254:
234:
233:
209:
203:
202:
183:
154:
144:
133:
122:
116:
113:
70:
68:
58:
46:
35:
28:
23:
22:
15:
12:
11:
5:
4286:
4284:
4276:
4275:
4270:
4260:
4259:
4253:
4252:
4250:
4249:
4244:
4239:
4234:
4229:
4223:
4221:
4217:
4216:
4214:
4213:
4208:
4203:
4198:
4193:
4187:
4185:
4181:
4180:
4178:
4177:
4175:Super-root (4)
4172:
4167:
4162:
4157:
4151:
4149:
4142:
4141:
4139:
4138:
4133:
4128:
4123:
4118:
4113:
4108:
4102:
4100:
4096:
4095:
4090:
4088:
4087:
4080:
4073:
4065:
4056:
4055:
4051:
4050:
4027:
4025:
4011:Multiplication
4002:
4000:
3981:
3979:
3960:
3958:
3952:
3949:
3948:
3943:
3941:
3940:
3933:
3926:
3918:
3912:
3911:
3907:selected from
3902:
3893:
3871:
3851:
3850:External links
3848:
3847:
3846:
3834:
3829:
3826:
3823:
3822:
3810:
3791:
3772:
3742:
3723:
3695:
3686:
3659:
3646:
3615:
3602:
3589:
3576:
3542:
3518:
3511:
3485:
3478:
3444:
3429:
3401:
3375:
3374:
3372:
3369:
3366:
3365:
3351:
3350:
3348:
3345:
3344:
3343:
3337:
3332:
3327:
3322:
3317:
3312:
3305:
3302:
3301:
3300:
3280:
3277:
3276:
3275:
3268:
3261:
3242:
3239:
3231:
3230:
3221:
3212:
3203:
3186:
3183:
3181:
3178:
3177:
3176:
3173:
3166:
3164:
3157:
3150:
3148:
3141:
3134:
3132:
3125:
3118:
3107:
3104:
3103:
3102:
3099:
3092:
3090:
3087:
3080:
3078:
3073:
3066:
3064:
3059:
3052:
3050:
3045:
3038:
3028:
3025:
3024:
3023:
3020:
3013:
3011:
3008:
3001:
2999:
2996:
2989:
2987:
2982:
2975:
2973:
2970:
2963:
2961:
2956:
2949:
2947:
2944:
2937:
2935:
2932:
2925:
2917:
2914:
2913:
2912:
2909:
2902:
2900:
2897:
2890:
2888:
2885:
2878:
2876:
2873:
2866:
2864:
2859:
2852:
2845:
2842:
2841:
2840:
2837:
2830:
2828:
2825:
2818:
2816:
2813:
2806:
2804:
2799:
2792:
2790:
2787:
2780:
2778:
2773:
2766:
2764:
2761:
2754:
2752:
2749:
2742:
2735:
2732:
2730:
2727:
2712:
2702:
2692:
2682:
2672:
2662:
2638:
2635:
2632:
2629:
2626:
2623:
2620:
2617:
2614:
2611:
2607:
2604:
2600:
2597:
2594:
2591:
2586:
2583:
2582:
2577:
2574:
2571:
2568:
2565:
2562:
2559:
2556:
2553:
2550:
2544:
2541:
2540:
2535:
2532:
2529:
2526:
2523:
2520:
2517:
2511:
2508:
2507:
2504:
2497:
2494:
2491:
2488:
2485:
2480:
2475:
2474:
2467:
2464:
2462:
2459:
2457:
2454:
2452:
2449:
2446:
2444:
2441:
2439:
2436:
2434:
2431:
2428:
2426:
2423:
2421:
2418:
2416:
2413:
2410:
2408:
2405:
2403:
2400:
2398:
2394:
2391:
2387:
2385:
2362:
2352:
2343:
2334:
2325:
2317:
2310:
2301:
2294:
2284:
2275:
2269:
2268:
2264:
2254:
2246:
2237:
2236:
2233:
2223:
2215:
2205:
2202:
2196:
2193:
2183:
2180:
2163:
2160:
2154:
2148:
2122:
2121:
2118:
2114:
2113:
2110:
2106:
2105:
2100:
2076:
2073:
1994:
1991:
1978:
1975:
1957:, the integer
1946:
1943:
1911:
1910:
1886:
1883:
1882:
1881:
1846:
1843:
1841:
1838:
1818:multiplicative
1801:
1798:
1797:
1796:
1785:
1765:
1762:
1761:
1760:
1698:
1697:
1663:
1662:
1601:
1598:
1596:
1593:
1536:
1535:
1521:
1518:
1515:
1512:
1509:
1506:
1503:
1500:
1497:
1494:
1489:
1484:
1481:
1478:
1475:
1472:
1469:
1466:
1463:
1460:
1457:
1452:
1447:
1444:
1441:
1438:
1435:
1432:
1429:
1401:
1400:
1392:
1391:
1379:
1376:
1373:
1370:
1367:
1364:
1354:
1342:
1339:
1336:
1333:
1330:
1320:
1308:
1305:
1302:
1299:
1296:
1286:
1274:
1271:
1268:
1265:
1262:
1244:infix notation
1227:
1224:
1181:multiplication
1141:if and only if
1085:multiplication
1049:
1048:
1046:
1045:
1038:
1031:
1023:
1020:
1019:
1016:
1015:
990:
977:
973:
968:anti-logarithm
965:
962:
953:
941:
938:
937:
930:
929:
904:
891:
863:
860:
859:
849:
848:
823:
810:
805:
784:
783:
766:
765:
762:
751:
748:
747:
744:Exponentiation
740:
739:
725:
712:
711:
701:
700:
690:
689:
686:
673:
660:
655:
628:
627:
604:
603:
600:
589:
586:
585:
578:
577:
550:
537:
532:
518:
508:
507:
497:
487:
486:
483:
472:
469:
468:
465:Multiplication
461:
460:
435:
422:
417:
403:
393:
392:
382:
372:
371:
368:
357:
354:
353:
346:
345:
320:
307:
302:
288:
278:
277:
267:
257:
256:
246:
236:
235:
225:
215:
214:
211:
200:
197:
196:
185:
184:
182:
181:
174:
167:
159:
135:
134:
49:
47:
40:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4285:
4274:
4271:
4269:
4266:
4265:
4263:
4248:
4245:
4243:
4240:
4238:
4235:
4233:
4230:
4228:
4225:
4224:
4222:
4218:
4212:
4209:
4207:
4206:Logarithm (3)
4204:
4202:
4199:
4197:
4194:
4192:
4189:
4188:
4186:
4182:
4176:
4173:
4171:
4168:
4166:
4163:
4161:
4158:
4156:
4153:
4152:
4150:
4147:
4143:
4137:
4134:
4132:
4131:Pentation (5)
4129:
4127:
4126:Tetration (4)
4124:
4122:
4119:
4117:
4114:
4112:
4109:
4107:
4106:Successor (0)
4104:
4103:
4101:
4097:
4093:
4086:
4081:
4079:
4074:
4072:
4067:
4066:
4063:
4048:
4046:
4042:
4037:
4032:
4026:
4023:
4021:
4017:
4012:
4007:
4001:
3998:
3996:
3991:
3986:
3980:
3977:
3975:
3970:
3965:
3959:
3956:
3955:
3950:
3946:
3939:
3934:
3932:
3927:
3925:
3920:
3919:
3916:
3910:
3906:
3903:
3901:
3897:
3894:
3892:
3888:
3884:
3880:
3876:
3872:
3868:
3864:
3863:
3858:
3857:"Subtraction"
3854:
3853:
3849:
3845:
3842:
3840:
3835:
3832:
3831:
3827:
3819:
3814:
3811:
3807:
3803:
3800:
3795:
3792:
3788:
3784:
3781:
3776:
3773:
3769:
3765:
3762:
3758:
3754:
3751:
3746:
3743:
3739:
3735:
3732:
3727:
3724:
3712:
3708:
3707:
3699:
3696:
3690:
3687:
3682:
3678:
3675:(7): 389β93.
3674:
3670:
3663:
3660:
3656:
3650:
3647:
3635:
3631:
3630:
3625:
3624:Klapper, Paul
3619:
3616:
3612:
3606:
3603:
3599:
3593:
3590:
3586:
3580:
3577:
3572:
3564:
3560:
3559:
3553:
3552:"Subtraction"
3546:
3543:
3532:
3528:
3527:"Subtraction"
3522:
3519:
3514:
3508:
3504:
3500:
3494:
3492:
3490:
3486:
3481:
3475:
3471:
3466:
3465:
3459:
3453:
3451:
3449:
3445:
3440:
3433:
3430:
3419:
3415:
3414:"Subtraction"
3408:
3406:
3402:
3390:
3386:
3380:
3377:
3370:
3362:
3356:
3353:
3346:
3341:
3338:
3336:
3333:
3331:
3328:
3326:
3323:
3321:
3318:
3316:
3313:
3311:
3308:
3307:
3303:
3293:
3292:
3291:
3288:
3285:
3278:
3273:
3269:
3266:
3262:
3259:
3255:
3254:
3253:
3250:
3248:
3240:
3238:
3227:
3222:
3218:
3213:
3209:
3204:
3200:
3195:
3194:
3193:
3190:
3184:
3179:
3170:
3165:
3154:
3149:
3138:
3133:
3122:
3117:
3115:
3112:
3105:
3096:
3091:
3084:
3079:
3070:
3065:
3056:
3051:
3042:
3037:
3035:
3032:
3026:
3017:
3012:
3005:
3000:
2993:
2988:
2979:
2974:
2967:
2962:
2953:
2948:
2941:
2936:
2929:
2924:
2922:
2915:
2906:
2901:
2894:
2889:
2882:
2877:
2870:
2865:
2856:
2851:
2849:
2843:
2834:
2829:
2822:
2817:
2810:
2805:
2796:
2791:
2784:
2779:
2770:
2765:
2758:
2753:
2746:
2741:
2739:
2733:
2728:
2726:
2722:
2718:
2711:
2701:
2691:
2681:
2671:
2661:
2655:
2465:
2460:
2455:
2447:
2442:
2437:
2429:
2424:
2419:
2411:
2406:
2401:
2392:
2389:
2375:
2373:
2369:
2365:
2361:
2355:
2351:
2346:
2342:
2339:. Otherwise,
2337:
2333:
2328:
2324:
2316:
2309:
2300:
2293:
2287:
2283:
2278:
2274:
2263:
2257:
2253:
2249:
2245:
2242:
2241:
2240:
2232:
2226:
2222:
2218:
2214:
2211:
2210:
2209:
2203:
2201:
2194:
2192:
2190:
2181:
2179:
2177:
2173:
2169:
2161:
2159:
2153:
2147:
2143:
2141:
2137:
2133:
2129:
2119:
2116:
2111:
2108:
2101:
2096:
2095:
2092:
2090:
2086:
2082:
2074:
2072:
2018:
2016:
2012:
2008:
2004:
2000:
1992:
1990:
1988:
1984:
1976:
1974:
1972:
1968:
1962:
1956:
1952:
1944:
1942:
1940:
1936:
1932:
1928:
1924:
1920:
1916:
1908:
1904:
1900:
1896:
1895:
1894:
1892:
1884:
1879:
1875:
1871:
1867:
1864:
1863:
1862:
1860:
1856:
1852:
1844:
1839:
1837:
1835:
1829:
1825:
1819:
1815:
1811:
1807:
1799:
1794:
1790:
1786:
1783:
1779:
1778:
1777:
1775:
1771:
1763:
1755:
1754:
1753:
1751:
1748:
1743:
1733:
1728:
1718:
1714:
1710:
1702:
1695:
1691:
1687:
1684:
1683:
1682:
1680:
1676:
1673:steps to the
1672:
1668:
1660:
1656:
1652:
1649:
1648:
1647:
1645:
1641:
1637:
1633:
1629:
1625:
1621:
1618:
1614:
1606:
1599:
1594:
1592:
1590:
1586:
1583:
1580:
1576:
1572:
1568:
1564:
1560:
1556:
1553:
1549:
1545:
1541:
1487:
1450:
1418:
1417:
1416:
1414:
1410:
1406:
1397:
1396:
1395:
1377:
1374:
1371:
1368:
1365:
1362:
1355:
1340:
1337:
1334:
1331:
1328:
1321:
1306:
1303:
1300:
1297:
1294:
1287:
1272:
1269:
1266:
1263:
1260:
1253:
1252:
1251:
1249:
1245:
1241:
1232:
1225:
1223:
1221:
1217:
1213:
1209:
1204:
1202:
1198:
1194:
1190:
1186:
1182:
1178:
1174:
1170:
1166:
1162:
1157:
1154:
1150:
1146:
1142:
1138:
1134:
1130:
1124:
1122:
1118:
1114:
1110:
1106:
1098:
1090:
1086:
1082:
1078:
1071:
1067:
1060:
1055:
1044:
1039:
1037:
1032:
1030:
1025:
1024:
1022:
1021:
991:
975:
960:
951:
942:
939:
935:
931:
905:
889:
864:
861:
857:
855:
850:
824:
808:
803:
752:
749:
745:
741:
738:
684:
674:
658:
653:
590:
587:
583:
579:
576:
551:
535:
530:
516:
495:
473:
470:
466:
462:
436:
420:
415:
401:
380:
358:
355:
351:
347:
321:
305:
300:
286:
265:
244:
223:
201:
198:
194:
190:
187:
186:
180:
175:
173:
168:
166:
161:
160:
157:
153:
152:
141:
131:
128:
120:
109:
106:
102:
99:
95:
92:
88:
85:
81:
78: β
77:
76:"Subtraction"
73:
72:Find sources:
66:
62:
56:
55:
50:This article
48:
44:
39:
38:
33:
19:
4201:Division (2)
4195:
4165:Division (2)
4159:
4136:Hexation (6)
4111:Addition (1)
4028:
4003:
3989:
3984:
3982:
3961:
3900:cut-the-knot
3860:
3838:
3828:Bibliography
3813:
3794:
3775:
3745:
3726:
3715:. Retrieved
3705:
3698:
3689:
3672:
3668:
3662:
3654:
3649:
3638:. Retrieved
3628:
3618:
3610:
3605:
3597:
3592:
3584:
3579:
3556:
3545:
3534:. Retrieved
3530:
3521:
3502:
3463:
3438:
3432:
3421:. Retrieved
3417:
3393:. Retrieved
3388:
3379:
3360:
3355:
3289:
3286:
3282:
3271:
3264:
3257:
3251:
3244:
3232:
3225:
3216:
3207:
3198:
3191:
3188:
3144:90 β 50 = 40
3113:
3111:difference.
3109:
3033:
3030:
2983:15 β 9 = ...
2919:
2847:
2800:9 + ... = 15
2737:
2723:
2719:
2709:
2699:
2689:
2679:
2669:
2659:
2656:
2376:
2371:
2370:
2363:
2359:
2353:
2349:
2344:
2340:
2335:
2331:
2326:
2322:
2314:
2307:
2298:
2291:
2285:
2281:
2276:
2272:
2270:
2261:
2255:
2251:
2247:
2243:
2238:
2230:
2224:
2220:
2216:
2212:
2207:
2198:
2185:
2165:
2157:
2151:
2144:
2139:
2138:is added to
2135:
2131:
2127:
2125:
2078:
2075:In computing
2019:
1996:
1980:
1970:
1966:
1960:
1954:
1948:
1934:
1930:
1926:
1922:
1918:
1914:
1912:
1906:
1902:
1898:
1888:
1877:
1873:
1869:
1865:
1858:
1854:
1848:
1827:
1823:
1803:
1800:Real numbers
1767:
1744:
1729:
1706:
1693:
1689:
1685:
1678:
1674:
1670:
1666:
1664:
1658:
1654:
1650:
1639:
1635:
1631:
1627:
1623:
1619:
1613:line segment
1610:
1588:
1584:
1577:. Using the
1574:
1570:
1566:
1562:
1554:
1537:
1415:. That is,
1412:
1408:
1404:
1402:
1393:
1237:
1212:well-defined
1205:
1193:real numbers
1158:
1152:
1148:
1144:
1136:
1132:
1128:
1125:
1096:
1065:
1064:
853:
522:multiplicand
349:
123:
114:
104:
97:
90:
83:
71:
59:Please help
54:verification
51:
4268:Subtraction
3990:Subtraction
3632:. pp.
3389:SplashLearn
3279:Same change
3185:Counting up
3027:Trade first
2997:6 β 4 = ...
2957:5 β 9 = ...
2933:3 β 1 = ...
2774:9 + ... = 5
2750:1 + ... = 3
2374:704 β 512.
2271:where each
2104:complement
1999:percentages
1997:Changes in
1993:Percentages
1945:Predecessor
1750:number line
1669:, it takes
1634:, it takes
1544:Subtraction
1399:accounting.
1248:equals sign
1165:associative
1079:along with
1066:Subtraction
642:denominator
350:Subtraction
4262:Categories
3717:2016-03-11
3640:2016-03-11
3573:required.)
3536:2020-08-26
3423:2020-08-26
3395:2022-12-13
3371:References
3088:14 β 9 = 5
3076:11 β 3 = 8
2886:15 β 9 = 6
2182:In America
2054:β33
1840:Properties
1757:3 β 4 = β1
1611:Imagine a
1555:subtrahere
1413:difference
1405:subtrahend
1240:minus sign
1105:arithmetic
1097:difference
1070:minus sign
512:multiplier
446:difference
407:subtrahend
87:newspapers
18:Subtrahend
3867:EMS Press
3501:(1983) .
3315:Decrement
3287:Example:
3160:3 β 1 = 2
3114:Example:
3100:6 β 4 = 2
3034:Example:
2910:3 β 1 = 2
2860:7 β 4 = 3
2848:Example:
2738:Example:
2585:⟵
2543:⟵
2510:⟵
2479:⟵
2390:−
2195:In Europe
2089:computers
2025:20% β 30%
1983:kilograms
1812:yielding
1736:3 β 3 = 0
1725:3 β 2 = 1
1721:3 β 0 = 3
1579:gerundive
1575:take away
1451:−
1375:−
1366:−
1332:−
1298:−
1264:−
1113:fractions
1101:5 β 2 = 3
1001:logarithm
961:
934:Logarithm
635:numerator
517:×
496:×
402:−
381:−
4036:Division
3969:Addition
3802:Archived
3783:Archived
3764:Archived
3753:Archived
3734:Archived
3626:(1916).
3460:(1974).
3304:See also
2372:Example:
2368:by one.
1834:division
1826:= 3 + (β
1814:additive
1644:addition
1600:Integers
1573:, or to
1559:compound
1546:" is an
1189:integers
1177:addition
1089:division
1081:addition
1059:Bordeaux
877:radicand
776:exponent
705:quotient
694:fraction
611:dividend
582:Division
193:Addition
117:May 2018
4146:Inverse
4099:Primary
4045:∕
3995:−
3985:−
3869:, 2001
3256:1234 β
3224:1000 +
2069:
2050:
2038:
2034:
2022:
1789:integer
1772:is not
1747:integer
1589:minuere
1567:trahere
1548:English
1409:minuend
1171:is the
1121:vectors
856:th root
618:divisor
562:product
397:minuend
250:summand
240:summand
101:scholar
4041:÷
4031:÷
4020:·
4016:×
4006:×
3889:, and
3509:
3476:
3270:674 β
3263:734 β
3228:= 1234
3219:= 1000
3215:600 +
3206:570 +
3197:567 +
2187:after
2102:Ones'
2099:digit
2097:Binary
1987:pounds
1774:closed
1738:. But
1715:, and
1617:length
1582:suffix
1185:proven
883:degree
501:factor
491:factor
292:addend
282:augend
271:addend
261:addend
103:
96:
89:
82:
74:
3974:+
3964:+
3567:
3347:Notes
3340:Monus
3274:= 667
3267:= 674
3260:= 734
3210:= 600
3201:= 570
1806:field
1740:3 β 4
1665:From
1540:Latin
1220:monus
1214:over
1093:5 β 2
936:(log)
834:power
794:power
716:ratio
145:5 β 2
108:JSTOR
94:books
3507:ISBN
3474:ISBN
2707:and
2677:and
2280:and
2260:...
2229:...
2079:The
2005:and
1963:β 1)
1921:) β
1872:= β(
1857:and
1822:3 β
1816:and
1804:The
1675:left
1552:verb
1179:and
1087:and
956:base
915:root
789:base
771:base
386:term
376:term
229:term
219:term
80:news
4043:or
4018:or
3898:at
3844:PDF
3677:doi
3258:500
3226:234
3217:400
2715:= 2
2705:= 1
2695:= 5
2685:= 4
2675:= 0
2665:= 7
2036:= β
2031:30%
1985:or
1951:one
1929:β (
1925:or
1615:of
1585:-nd
1563:sub
1561:of
1542:. "
1206:In
952:log
858:(β)
746:(^)
584:(Γ·)
467:(Γ)
352:(β)
331:sum
195:(+)
63:by
4264::
4047:)
4022:)
3997:)
3976:)
3885:,
3881:,
3877:,
3865:,
3859:,
3759:;
3711:77
3673:99
3671:.
3634:80
3555:.
3529:.
3488:^
3472:.
3447:^
3416:.
3404:^
3387:.
3265:60
3237:.
3208:30
2697:,
2667:,
2366:+1
2356:+1
2313:β
2305:,
2297:β
2258:β1
2227:β1
2120:0
2117:1
2112:1
2109:0
2091:.
2052:=
1973:.
1933:β
1917:β
1905:β
1901:β
1878:a)
1876:β
1868:β
1711:,
1692:=
1688:β
1657:=
1653:+
1646::
1222:.
1203:.
1151:=
1147:+
1135:β
1131:=
1119:,
1115:,
1111:,
1083:,
4084:e
4077:t
4070:v
4039:(
4014:(
3993:(
3972:(
3937:e
3930:t
3923:v
3720:.
3713:β
3683:.
3679::
3643:.
3636:β
3565:.
3539:.
3515:.
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3426:.
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2028:/
1971:a
1967:a
1961:a
1959:(
1955:a
1935:c
1931:b
1927:a
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1880:.
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1824:Ο
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1696:.
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1488:=
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1372:=
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1304:=
1301:2
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1270:=
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1169:0
1153:a
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1073:β
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976:=
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890:=
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809:=
804:}
685:{
659:=
654:}
536:=
531:}
421:=
416:}
306:=
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98:Β·
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