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Roman numerals

From Wikipedia, the free encyclopedia

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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

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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

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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).