Review and Preview Math 311 Using the Multiplicative Identity

Natural number

← 0 1 two →
-1 0 1 2 iii 4 5 half dozen 7 8 9 → List of numbers — Integers ← 0 10 twenty 30 xl 50 60 70 lxxx xc →
Key	i
Ordinal	1st (first)
Numeral system	unary
Factorization	∅
Divisors	ane
Greek numeral	Α´
Roman numeral	I, i
Greek prefix	mono-/haplo-
Latin prefix	uni-
Binary	1ii
Ternary	13
Octal	1viii
Duodecimal	112
Hexadecimal	116
Greek numeral	α'
Arabic, Kurdish, Persian, Sindhi, Urdu	١
Assamese & Bengali	১
Chinese numeral	一/弌/壹
Devanāgarī	१
Ge'ez	፩
Georgian	Ⴀ/ⴀ/ა(Ani)
Hebrew	א
Japanese numeral	一/壱
Kannada	೧
Khmer	១
Malayalam	൧
Thai	๑
Tamil	௧
Telugu	೧
Counting rod	𝍠

1 (ane, also called unit, and unity) is a number and a numerical digit used to represent that number in numerals. It represents a single entity, the unit of measurement of counting or measurement. For example, a line segment of unit length is a line segment of length one. In conventions of sign where zero is considered neither positive nor negative, ane is the first and smallest positive integer.^[one] Information technology is also sometimes considered the first of the infinite sequence of natural numbers, followed past 2, although by other definitions i is the 2nd natural number, following 0.

The key mathematical property of one is to be a multiplicative identity, meaning that any number multiplied by one equals the same number. Most if non all backdrop of ane can be deduced from this. In advanced mathematics, a multiplicative identity is often denoted 1, even if it is not a number. 1 is by convention non considered a prime; although universally accepted today, this fact was controversial until the mid-20th century.

The unique mathematical properties of the number have led to its unique uses in other fields, ranging from science to sports. Information technology commonly denotes the first, leading or pinnacle thing in a grouping.

Etymology

The word one can be used as a noun, an describing word, and a pronoun.^[ii]

Information technology comes from the English discussion an,^[two] which comes from the Proto-Germanic root *ainaz.^[ii] The Proto-Germanic root *ainaz comes from the Proto-Indo-European root *oi-no-.^[ii]

Compare the Proto-Germanic root *ainaz to Onetime Frisian an, Gothic ains, Danish en, Dutch een, German eins and One-time Norse einn.

Compare the Proto-Indo-European root *oi-no- (which means "one, single"^[2]) to Greek oinos (which means "ace" on die^[2]), Latin unus (one^[2]), Old Persian aivam, Former Church building Slavonic -inu and ino-, Lithuanian vienas, Old Irish oin and Breton un (one^[2]).

As a number

1, sometimes referred to as unity,^{[3] [ane]} is the offset not-zero natural number. It is thus the integer after cypher.

Any number multiplied by one remains that number, every bit one is the identity for multiplication. As a issue, 1 is its ain factorial, its own square and square root, its ain cube and cube root, and then on. One is also the result of the empty product, as any number multiplied past one is itself. It is also the just natural number that is neither composite nor prime with respect to sectionalisation, but is instead considered a unit (significant of ring theory).

Equally a digit

The glyph used today in the Western world to represent the number 1, a vertical line, ofttimes with a serif at the superlative and sometimes a brusk horizontal line at the bottom, traces its roots back to the Brahmic script of ancient India, where information technology was a unproblematic vertical line. It was transmitted to Europe via the Maghreb and Andalusia during the Middle Ages, through scholarly works written in Arabic.

In some countries, the serif at the top is sometimes extended into a long upstroke, sometimes as long as the vertical line, which can lead to defoliation with the glyph used for seven in other countries. In styles in which the digit 1 is written with a long upstroke, the digit seven is often written with a horizontal stroke through the vertical line, to disambiguate them. Styles that do not use the long upstroke on digit 1 usually do not use the horizontal stroke through the vertical of the digit 7 either.

While the shape of the character for the digit i has an ascender in most modern typefaces, in typefaces with text figures, the glyph unremarkably is of x-elevation, equally, for example, in .

The 24-hour belfry clock in Venice, using J as a symbol for 1

Many older typewriters lack a separate key for one, using the lowercase letter l or uppercase I instead. Information technology is possible to observe cases when the uppercase J is used, though it may be for decorative purposes. In some typefaces, different glyphs are used for I and i, but the numeral 1 resembles a modest caps version of I, with parallel serifs at superlative and bottom, with the uppercase I being full-superlative.

Hoefler Text, a typeface designed in 1991, represents the numeral 1 as similar to a small-caps I.

Mathematics

Definitions

Mathematically, 1 is:

• in arithmetic (algebra) and calculus, the natural number that follows 0 and the multiplicative identity element of the integers, real numbers and complex numbers;
• more generally, in algebra, the multiplicative identity (too called unity), usually of a group or a ring.

Formalizations of the natural numbers have their own representations of ane. In the Peano axioms, 1 is the successor of 0. In Principia Mathematica, it is defined as the set of all singletons (sets with 1 element), and in the Von Neumann cardinal consignment of natural numbers, it is defined as the gear up {0}.

In a multiplicative group or monoid, the identity element is sometimes denoted ane, but e (from the German Einheit, "unity") is also traditional. Yet, ane is especially common for the multiplicative identity of a band, i.e., when an addition and 0 are also present. When such a ring has feature n non equal to 0, the element chosen one has the holding that ni = anen = 0 (where this 0 is the additive identity of the ring). Important examples are finite fields.

By definition, 1 is the magnitude, absolute value, or norm of a unit circuitous number, unit of measurement vector, and a unit matrix (more than ordinarily called an identity matrix). Note that the term unit of measurement matrix is sometimes used to mean something quite different.

By definition, ane is the probability of an event that is absolutely or well-nigh certain to occur.

In category theory, 1 is sometimes used to denote the terminal object of a category.

In number theory, one is the value of Legendre'south constant, which was introduced in 1808 by Adrien-Marie Legendre in expressing the asymptotic behavior of the prime-counting part. Legendre'due south constant was originally conjectured to be approximately 1.08366, but was proven to equal exactly ane in 1899.

Properties

Tallying is often referred to as "base of operations 1", since only ane mark – the tally itself – is needed. This is more than formally referred to as a unary numeral organisation. Unlike base 2 or base ten, this is not a positional notation.

Since the base of operations i exponential function (1 ¹⁰ ) always equals ane, its inverse does not exist (which would be called the logarithm base 1 if it did exist).

There are two ways to write the real number 1 equally a recurring decimal: every bit 1.000..., and as 0.999.... 1 is the first figurate number of every kind, such as triangular number, pentagonal number and centered hexagonal number, to name just a few.

In many mathematical and engineering issues, numeric values are typically normalized to fall within the unit interval from 0 to 1, where ane ordinarily represents the maximum possible value in the range of parameters. Likewise, vectors are oft normalized into unit of measurement vectors (i.e., vectors of magnitude one), because these often accept more desirable properties. Functions, too, are frequently normalized past the status that they have integral one, maximum value one, or square integral one, depending on the application.

Considering of the multiplicative identity, if f(ten) is a multiplicative function, then f(ane) must be equal to one.

It is also the commencement and 2d number in the Fibonacci sequence (0 existence the zeroth) and is the first number in many other mathematical sequences.

The definition of a field requires that one must not exist equal to 0. Thus, in that location are no fields of characteristic ane. Nevertheless, abstract algebra can consider the field with one chemical element, which is not a singleton and is not a set at all.

1 is the most common leading digit in many sets of data, a outcome of Benford's law.

ane is the only known Tamagawa number for a merely continued algebraic grouping over a number field.

The generating part that has all coefficients 1 is given past

${\displaystyle {\frac {i}{1-x}}=1+ten+x^{ii}+10^{3}+\ldots }$

This power serial converges and has finite value if and merely if ${\displaystyle |10|<1}$ .

Primality

ane is by convention neither a prime number nor a composite number, but a unit (meaning of ring theory) like −ane and, in the Gaussian integers, i and −i.

The cardinal theorem of arithmetics guarantees unique factorization over the integers only up to units. For example, 4 = 2² , but if units are included, is also equal to, say, (−1)^vi × one²³ × twoⁱⁱ, among infinitely many similar "factorizations".

i appears to encounter the naïve definition of a prime number, being evenly divisible but by 1 and itself (also 1). As such, some mathematicians considered information technology a prime every bit belatedly every bit the middle of the 20th century, but mathematical consensus has generally and since then universally been to exclude it for a diversity of reasons (such every bit complicating the cardinal theorem of arithmetic and other theorems related to prime numbers).

i is the only positive integer divisible by exactly one positive integer, whereas prime numbers are divisible by exactly two positive integers, composite numbers are divisible by more than than two positive integers, and zero is divisible past all positive integers.

Tabular array of bones calculations

Multiplication	1	2	3	4	5	vi	seven	8	9	ten		xi	12	thirteen	14	fifteen	16	17	18	xix	xx		21	22	23	24	25		50	100	yard
1 × ten	1	2	3	4	5	vi	vii	8	9	x		xi	12	13	14	15	sixteen	17	xviii	nineteen	20		21	22	23	24	25		50	100	1000

Sectionalization	i	2	3	four	5	6	7	viii	nine	x		11	12	13	xiv	15
i ÷ x	1	0.5	0.3	0.25	0.two	0.16	0.142857	0.125	0.1	0.1		0.09	0.08three	0.076923	0.0714285	0.06
x ÷ 1	1	two	3	4	5	6	vii	8	9	10		11	12	thirteen	fourteen	15

Exponentiation	1	2	3	iv	v	vi	7	8	ix	ten		11	12	13	14	xv	16	17	18	19	20
1 ten	i	1	ane	i	i	1	i	1	1	i		1	one	1	1	one	1	1	1	ane	1
x 1	1	2	iii	4	five	6	vii	eight	9	10		11	12	xiii	14	15	16	17	18	19	20

In engineering

• The resin identification code used in recycling to place polyethylene terephthalate.^[iv]
• The ITU country lawmaking for the North American Numbering Plan surface area, which includes the Usa, Canada, and parts of the Caribbean.
• A binary code is a sequence of one and 0 that is used in computers for representing any kind of data.
• In many concrete devices, 1 represents the value for "on", which means that electricity is flowing.^{[5] [6]}
• The numerical value of truthful in many programming languages.
• 1 is the ASCII code of "Start of Header".

In science

• Dimensionless quantities are also known as quantities of dimension one.
• 1 is the diminutive number of hydrogen.
• +i is the electric charge of positrons and protons.
• Grouping 1 of the periodic table consists of the brine metals.
• Period 1 of the periodic table consists of just two elements, hydrogen and helium.
• The dwarf planet Ceres has the small-planet designation 1 Ceres because it was the first asteroid to be discovered.
• The Roman numeral I often stands for the outset-discovered satellite of a planet or minor planet (such as Neptune I, a.k.a. Triton). For some before discoveries, the Roman numerals originally reflected the increasing distance from the principal instead.

In philosophy

In the philosophy of Plotinus (and that of other neoplatonists), The One is the ultimate reality and source of all existence.^[7] Philo of Alexandria (xx BC – AD 50) regarded the number one as God'southward number, and the basis for all numbers ("De Allegoriis Legum," 2.12 [i.66]).

The Neopythagorean philosopher Nicomachus of Gerasa affirmed that 1 is not a number, but the source of number. He also believed the number two is the apotheosis of the origin of otherness. His number theory was recovered past Boethius in his Latin translation of Nicomachus's treatise Introduction to Arithmetic.^[8]

In sports

In many professional sports, the number one is assigned to the thespian who is kickoff or leading in some respect, or otherwise of import; the number is printed on his sports uniform or equipment. This is the pitcher in baseball, the goalkeeper in clan football game (soccer), the starting fullback in most of rugby league, the starting loosehead prop in rugby wedlock and the previous twelvemonth'southward globe champion in Formula One. i may be the lowest possible player number, similar in the American–Canadian National Hockey League (NHL) since the 1990s^{[ when? ]} or in American football.

In other fields

• Number One is Royal Navy informal usage for the chief executive officer of a ship, the captain's deputy responsible for discipline and all normal operation of a ship and its coiffure.
• 1 is the value of an ace in many playing card games, such as cribbage.
• List of highways numbered 1
• Listing of public transport routes numbered 1
• ane is often used to denote the Gregorian calendar month of January.
• 1 CE, the first year of the Common Era
• 01, the quondam dialing code for Greater London
• For Pythagorean numerology (a pseudoscience), the number 1 is the number that means beginning, new beginnings, new cycles, it is a unique and absolute number.
• PRS One, a German paraglider design
• +1 is the code for international telephone calls to countries in the North American Numbering Program.
• In some countries, a street address of "1" is considered prestigious and developers will attempt to obtain such an address for a edifice, to the point of lobbying for a street or portion of a street to be renamed, fifty-fifty if this makes the accost less useful for wayfinding. The construction of a new street to serve the development may also provide the possibility of a "1" address. An example of such an address is the Apple Campus, located at 1 Infinite Loop, Cupertino, California.

See also

• −1
• +ane (disambiguation)
• Listing of mathematical constants
• One (give-and-take)
• Root of unity
• List of highways numbered i

References

1. ^ ^{a b} Weisstein, Eric W. "1". mathworld.wolfram.com . Retrieved 2020-08-ten .
2. ^ ^{a b c d due east f thousand h} "Online Etymology Dictionary". etymonline.com. Douglas Harper.
3. ^ Skoog, Douglas. Principles of Instrumental Assay. Brooks/Cole, 2007, p. 758.
4. ^ "Plastic Packaging Resins" (PDF). American Chemistry Council. Archived from the original (PDF) on 2011-07-21.
5. ^ Woodford, Chris (2006), Digital Engineering science, Evans Brothers, p. nine, ISBN978-0-237-52725-9
6. ^ Godbole, Achyut S. (ane September 2002), Data Comms & Networks, Tata McGraw-Hill Instruction, p. 34, ISBN978-1-259-08223-8
7. ^ Olson, Roger (2017). The Essentials of Christian Thought: Seeing Reality through the Biblical Story. Zondervan Academic. ISBN9780310521563.
8. ^ British Society for the History of Science (July 1, 1977). "From Abacus to Algorism: Theory and Do in Medieval Arithmetic". The British Journal for the History of Science. Cambridge University Press. 10 (two): Abstract. doi:10.1017/S0007087400015375. S2CID 145065082. Retrieved May 16, 2021.

External links

• The Number one
• The Positive Integer 1
• Prime curiosities: 1

sloanextrany.blogspot.com

Source: https://en.wikipedia.org/wiki/1