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Roman numerals
From Wikipedia, the free encyclopedia
(Redirected from Roman numeral system)Jump to: navigation, search
Roman numerals are a numeral system of ancient Rome based on letters of the alphabet,
which are combined to signify the sum (or in some cases, the difference) of their values.The first ten Roman numerals are:
The Roman numeral system is decimal[1] but not directly positional and does not include a
zero. It is a cousin of the Etruscan numerals, and the letters derive from earlier non-
alphabetical symbols; over time the Romans came to identify the symbols with letters of the Latin alphabet. The system was modified slightly during the Middle Ages to produce
the system used today.
Roman numerals are commonly used in numbered lists (such as the outline format of an
article), clock faces, pages preceding the main body of a book, chord triads in musicanalysis, dated notices of copyright, months of the year, successive political leaders or
children with identical names, and the numbering of annual events. See modern usage
below.
For arithmetic involving Roman numerals, see Roman arithmetic and Roman abacus.
Symbols
Roman numerals are based on seven symbols: a stroke (identified with the letter I) for a
unit, a chevron (identified with the letter V) for a five, a cross-stroke (identified with the
letter X) for a ten, a C (identified as an abbreviation of Centum) for a hundred, etc.:
Symbol Value
I 1 (one) (unus)
V 5 (five) (quinque)
X 10 (ten) (decem)
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L 50 (fifty) (quinquaginta)
C 100 (one hundred) (centum)
D 500 (five hundred) (quingenti)
M 1000 (one thousand) (mille)
Symbols are iterated to produce multiples of the decimal (1, 10, 100, 1000) values, withV, L, D substituted for a multiple of five, and the iteration continuing: I "1", II "2", III
"3", V "5", VI "6", VII "7", etc., and the same for other bases: X "10", XX "20", XXX
"30", L "50", LXXX "80"; CC "200", DCC "700", etc. At the fourth iteration, a subtractive principle may be employed, with the base placed before the higher base: IIII
or IV "4", VIIII or IX "9", XXXX or XL "40", LXXXX or XC "90", CCCC or CD "400",
DCCCC or CM "900".
The Romans only used what is called capital (upper case) letters in modern usage. In theMiddle Ages, minuscule (lower case) letters were developed, and these are commonly
used for Roman numerals: i, ii, iii, iv, etc. Also in medieval use was the substitution of j
for a final i to end numbers, such as iij for 3 or vij for 7. This was not a separate letter, butmerely a swash variant of i. It is used today, especially in medical prescriptions, to
prevent tampering with the numbers after they are written.[citation needed ]
For large numbers (4000 and above), a bar can be placed above a base numeral, or
parentheses placed around it, to indicate multiplication by 1000, although the Romansthemselves often just wrote out the "M"s:[2]
Symbol Value
V or (V) five thousand
X or (X) ten thousand
L or (L) fifty thousand
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C or (C) one hundred thousand
D or (D) five hundred thousand
M or (M) one million
The parentheses are more versatile; (II) is synonymous with MM, but II is not found.
The basic multiples of Roman numerals thus follow a pattern:
×1 ×2 ×3 ×4 ×5 ×6 ×7 ×8 ×9
Ones I II III IV V VI VII VIII IX
Tens X XX XXX XL L LX LXX LXXX XC
Hundreds C CC CCC CD D DC DCC DCCC CM
Thousands M MM MMM IV V VI VII VIII IX
Ten thousands X XX XXX XL L LX LXX LXXX XC
Hundred thousands C CC CCC CD D DC DCC DCCC CM
A practical way to write a Roman number is to consider the modern Arabic numeral
system, and separately convert the thousands, hundreds, tens, and ones as given in thechart above. So, for instance, 1234 may be thought of as "one thousand and two hundreds
and three tens and four", obtaining M (one thousand) + CC (two hundreds) + XXX(thirty) + IV (four), for MCCXXXIV. Thus eleven is XI (ten and one), 32 is XXXII
(thirty and two) and 2009 is MMIX (two thousand and nine). Note that the subtractive
principle is not extended beyond the chart, and IL is not used for 49, which can only beforty (XL) and nine (IX), or XLIX.
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Origins
Although the Roman numerals are now written with letters of the Roman alphabet, they
were originally independent symbols. The Etruscans, for example, used I Λ X⋔ 8⊕ for
I V X L C M, of which only I and X happened to be letters in their alphabet. One folk
etymology has it that the V represented a hand, and that the X was made by placing twoVs on top of each other, one inverted. However, the Etrusco-Roman numerals actually
appear to derive from notches on tally sticks, which continued to be used by Italian andDalmatian shepherds into the 19th century.[3]
Thus, 'I' descends not from the letter 'I' but from a notch scored across the stick. Every
fifth notch was double cut (i.e. ⋀, ⋁,⋋,⋌, etc.), and every tenth was cross cut (X),
IIIIΛIIIIXIIIIΛIIIIXII..., much like European tally marks today. This produced a
positional system: Eight on a counting stick was eight tallies, IIIIΛIII, or the eighth of a
longer series of tallies; either way, it could be abbreviated ΛIII (or VIII), as the existenceof a Λ implies four prior notches. By extension, eighteen was the eighth tally after the
first ten, which could be abbreviated X, and so was XΛIII. Likewise, number four on thestick was the I-notch that could be felt just before the cut of the Λ (V), so it could bewritten as either IIII or IΛ (IV). Thus the system was neither additive nor subtractive in
its conception, but ordinal . When the tallies were transferred to writing, the marks were
easily identified with the existing Roman letters I, V, X
The tenth V or X along the stick received an extra stroke. Thus 50 was written variouslyas N, И, K, Ψ, ⋔, etc., but perhaps most often as a chicken-track shape like a
superimposed V and I -ᗐ. This had flattened to⊥ (an inverted T) by the time of
Augustus, and soon thereafter became identified with the graphically similar letter L.
Likewise, 100 was variously Ж,⋉,⋈, H, or as any of the symbols for 50 above plus an
extra stroke. The form Ж (that is, a superimposed X and I) came to predominate. It waswritten variously as >I< or ƆIC, was then abbreviated to Ɔ or C, with C variant finally
winning out because, as a letter, it stood for centum, Latin for "hundred".
The hundredth V or X was marked with a box or circle. Thus 500 was like a Ɔ
superimposed on a⋌ or ⊢— that is, like a Þ with a cross bar,— becoming D or Ð by
the time of Augustus, under the graphic influence of the letter D. It was later identified as
the letter D, perhaps as an abbreviation of demi-mille "half-thousand"; this at least was
the folk etymology given to it later on.
Meanwhile, 1000 was a circled or boxed X:Ⓧ,⊗,⊕, and by Augustinian times was
partially identified with the Greek letter Φ phi. In different traditions it then evolvedalong several different routes. Some variants, such as Ψ and ↀ, were historical dead
ends, although folk etymology later identified D for 500 as graphically half of Φ for 1000
because of the CD variant. A third line,ↀ, survives to this day in two variants:
• One, CIƆ, led to the convention of using parentheses to indicate multiplication by
a thousand: the original CIƆ = (I) 1000, then (III) for 3000, (V) 5000, (IX) 9000,
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(X) 10 000, (L) 50 000, (C) 100 000, (D) 500 000, (M) 1000 000, etc. This was
later extended to double parentheses, as in ↁ ,ↂ, etc. See alternate forms below.
• In the other,ↀ became ∞ and⋈, eventually changing to M under the influence
of the Latin word mille "thousand".
AssociativityFrom Wikipedia, the free encyclopedia
In mathematics, associativity is a property that a binary operation can have. It means
that, within an expression containing two or more occurrences in a row of the sameassociative operator, the order in which the operations are performed does not matter as
long as the sequence of the operands is not changed. That is, rearranging the parentheses
in such an expression will not change its value. Consider for instance the equation
Even though the parentheses were rearranged (the left side requires adding 5 and 2 first,then adding 1 to the result, whereas the right side requires adding 2 and 1 first, then 5),
the value of the expression was not altered. Since this holds true when performing
addition on any real numbers, we say that "addition of real numbers is an associativeoperation."
Associativity is not to be confused with commutativity. Commutativity justifies changing
the order or sequence of the operands within an expression while associativity does not.For example,
is an example of associativity because the parentheses were changed (and consequently
the order of operations during evaluation) while the operands 5, 2, and 1 appeared in the
exact same order from left to right in the expression.
is not an example of associativity because the operand sequence changed when the 2 and5 switched places.
Associative operations are abundant in mathematics; in fact, many algebraic structures
(such as semigroups and categories) explicitly require their binary operations to beassociative.
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However, many important and interesting operations are non-associative; one common
example would be the vector cross product.
Commutativity
From Wikipedia, the free encyclopedia
(Redirected from Commutative law)
Jump to: navigation, search
Example showing the commutativity of addition (3 + 2 = 2 + 3) For other uses, see Commute (disambiguation).
In mathematics, commutativity is the property that changing the order of something does
not change the end result. It is a fundamental property of many binary operations, and
many mathematical proofs depend on it. The commutativity of simple operations, such as
multiplication and addition of numbers, was for many years implicitly assumed and the property was not named until the 19th century when mathematicians began to formalize
the theory of mathematics.
Common uses
The commutative property (or commutative law) is a property associated with binaryoperations and functions. Similarly, if the commutative property holds for a pair of
elements under a certain binary operation then it is said that the two elements commute
under that operation.
In group and set theory, many algebraic structures are called commutative when certainoperands satisfy the commutative property. In higher branches of math, such as analysis
and linear algebra the commutativity of well known operations (such as addition and
multiplication on real and complex numbers) is often used (or implicitly assumed) in proofs.[1][2][3]
Mathematical definitions
The term "commutative" is used in several related senses.[4][5]
1. A binary operation ∗ on a set S is said to be commutative if:
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- An operation that does not satisfy the above property is called noncommutative.
2. One says that x commutes with y under ∗ if:
3. A binary function f: A× A → B is said to be commutative if:
Distributivity
From Wikipedia, the free encyclopedia
(Redirected from Distributive law)Jump to: navigation, search
In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. For example:
2 × (1 + 3) = (2 × 1) + (2 × 3).
In the left-hand side of the above equation, the 2 multiplies the sum of 1 and 3; on the
right-hand side, it multiplies the 1 and the 3 individually, with the results addedafterwards. Because these give the same final answer (8), we say that multiplication by 2
distributes over addition of 1 and 3. Since we could have put any real numbers in place of 2, 1, and 3 above, and still have obtained a true equation, we say that multiplication of
real numbers distributes over addition of real numbers.
Definition
Given a set S and two binary operations · and + on S , we say that the operation ·
• is left-distributive over + if, given any elements x, y, and z of S ,
x · ( y + z ) = ( x · y) + ( x · z );
• is right-distributive over + if, given any elements x, y, and z of S :
( y + z ) · x = ( y · x) + ( z · x);
• is distributive over + if it is both left- and right-distributive.[1]
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Notice that when · is commutative, then the three above conditions are logically
equivalent.
Examples
1. Multiplication of numbers is distributive over addition of numbers, for a broadclass of different kinds of numbers ranging from natural numbers to complex
numbers and cardinal numbers.
2. Multiplication of ordinal numbers, in contrast, is only left-distributive, not right-distributive.
3. Matrix multiplication is distributive over matrix addition, even though it's not
commutative.4. The union of sets is distributive over intersection, and intersection is distributive
over union. Also, intersection is distributive over the symmetric difference.
5. Logical disjunction ("or") is distributive over logical conjunction ("and"), and
conjunction is distributive over disjunction. Also, conjunction is distributive over
exclusive disjunction ("xor").6. For real numbers (or for any totally ordered set), the maximum operation is
distributive over the minimum operation, and vice versa: max(a,min(b,c)) =min(max(a,b),max(a,c)) and min(a,max(b,c)) = max(min(a,b),min(a,c)).
7. For integers, the greatest common divisor is distributive over the least common
multiple, and vice versa: gcd(a,lcm(b,c)) = lcm(gcd(a,b),gcd(a,c)) andlcm(a,gcd(b,c)) = gcd(lcm(a,b),lcm(a,c)).
8. For real numbers, addition distributes over the maximum operation, and also over
the minimum operation: a + max(b,c) = max(a+b,a+c) and a + min(b,c) =
min(a+b,a+c).
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