What is the number from which another number is subtracted is called?

One of the four basic arithmetic operations

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Arithmetic operations v t e

Addition (+) term + term summand + summand addend + addend augend + addend } = {\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\}\,=\,} sum {\displaystyle \scriptstyle {\text{sum}}} Subtraction (−) term − term minuend − subtrahend } = {\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,-\,{\text{term}}\\\scriptstyle {\text{minuend}}\,-\,{\text{subtrahend}}\end{matrix}}\right\}\,=\,} difference {\displaystyle \scriptstyle {\text{difference}}} Multiplication (×) factor × factor multiplier × multiplicand } = {\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{factor}}\,\times \,{\text{factor}}\\\scriptstyle {\text{multiplier}}\,\times \,{\text{multiplicand}}\end{matrix}}\right\}\,=\,} product {\displaystyle \scriptstyle {\text{product}}} Division (÷) dividend divisor   numerator denominator } = {\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\frac {\scriptstyle {\text{dividend}}}{\scriptstyle {\text{divisor}}}}\\\scriptstyle {\text{ }}\\\scriptstyle {\frac {\scriptstyle {\text{numerator}}}{\scriptstyle {\text{denominator}}}}\end{matrix}}\right\}\,=\,} fraction quotient ratio {\displaystyle {\begin{matrix}\scriptstyle {\text{fraction}}\\\scriptstyle {\text{quotient}}\\\scriptstyle {\text{ratio}}\end{matrix}}} Exponentiation (^) base exponent = {\displaystyle \scriptstyle {\text{base}}^{\text{exponent}}\,=\,} power {\displaystyle \scriptstyle {\text{power}}} nth root (√) radicand degree = {\displaystyle \scriptstyle {\sqrt[{\text{degree}}]{\scriptstyle {\text{radicand}}}}\,=\,} root {\displaystyle \scriptstyle {\text{root}}} Logarithm (log) log base ⁡ ( anti-logarithm ) = {\displaystyle \scriptstyle \log _{\text{base}}({\text{anti-logarithm}})\,=\,} logarithm {\displaystyle \scriptstyle {\text{logarithm}}}

Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, −. For example, in the adjacent picture, there are 5 − 2 peaches—meaning 5 peaches with 2 taken away, resulting in a total of 3 peaches. Therefore, the difference of 5 and 2 is 3; that is, 5 − 2 = 3. While primarily associated with natural numbers in arithmetic, subtraction can also represent removing or decreasing physical and abstract quantities using different kinds of objects including negative numbers, fractions, irrational numbers, vectors, decimals, functions, and matrices.[1]

Subtraction follows several important patterns. It is anticommutative, meaning that changing the order changes the sign of the answer. It is also not associative, meaning that when one subtracts more than two numbers, the order in which subtraction is performed matters. Because 0 is the additive identity, subtraction of it does not change a number. Subtraction also obeys predictable rules concerning related operations, such as addition and multiplication. All of these rules can be proven, starting with the subtraction of integers and generalizing up through the real numbers and beyond. General binary operations that follow these patterns are studied in abstract algebra.

Performing subtraction on natural numbers is one of the simplest numerical tasks. Subtraction of very small numbers is accessible to young children. In primary education for instance, students are taught to subtract numbers in the decimal system, starting with single digits and progressively tackling more difficult problems.

In advanced algebra and in computer algebra, an expression involving subtraction like A − B is generally treated as a shorthand notation for the addition A + (−B). Thus, A − B contains two terms, namely A and −B. This allows an easier use of associativity and commutativity.

Notation and terminology

Subtraction is usually written using the minus sign "−" between the terms; that is, in infix notation. The result is expressed with an equals sign. For example,

2 − 1 = 1 {\displaystyle 2-1=1}

4 − 2 = 2 {\displaystyle 4-2=2}

6 − 3 = 3 {\displaystyle 6-3=3}

4 − 6 = − 2 {\displaystyle 4-6=-2}

There are also situations where subtraction is "understood", even though no symbol appears:

• 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 accounting.

Formally, the number being subtracted is known as the subtrahend,[2][3] while the number it is subtracted from is the minuend.[2][3] The result is the difference.[2][3][1][4] That is,

m i n u e n d − s u b t r a h e n d = d i f f e r e n c e {\displaystyle {\rm {minuend}}-{\rm {subtrahend}}={\rm {difference}}}

All of this terminology derives from Latin. "Subtraction" is an English word derived from the Latin verb subtrahere, which in turn is a compound of sub "from under" and trahere "to pull". Thus, to subtract is to draw from below, or to take away.[5] Using the gerundive suffix -nd results in "subtrahend", "thing to be subtracted".[a] Likewise, from minuere "to reduce or diminish", one gets "minuend", which means "thing to be diminished".

Of integers and real numbers

Integers

Imagine a line segment of length b with the left end labeled a and the right end labeled c. Starting from a, it takes b steps to the right to reach c. This movement to the right is modeled mathematically by addition:

From c, it takes b steps to the left to get back to a. This movement to the left is modeled by subtraction:

Now, a line segment labeled with the numbers 1, 2, and 3. From position 3, it takes no steps to the left to stay at 3, so 3 − 0 = 3. It takes 2 steps to the left to get to position 1, so 3 − 2 = 1. 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.

To subtract arbitrary natural numbers, one begins with a line containing every natural number (0, 1, 2, 3, 4, 5, 6, ...). From 3, it takes 3 steps to the left to get to 0, so 3 − 3 = 0. But 3 − 4 is still invalid, since it again leaves the line. The natural numbers are not a useful context for subtraction.

The solution is to consider the integer number line (..., −3, −2, −1, 0, 1, 2, 3, ...). This way, it takes 4 steps to the left from 3 to get to −1:

Natural numbers

Subtraction of natural numbers is not closed: 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:

1. Conclude that 26 cannot be subtracted from 11; subtraction becomes a partial function.
2. Give the answer as an integer representing a negative number, so the result of subtracting 26 from 11 is −15.

Real numbers

The field of real numbers can be defined specifying only two binary operations, addition and multiplication, together with unary operations yielding additive and multiplicative 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, 3 − π = 3 + (−π). Alternatively, instead of requiring these unary operations, the binary operations of subtraction and division can be taken as basic.

Properties

Anti-commutativity

Subtraction is anti-commutative, meaning that if one reverses the terms in a difference left-to-right, the result is the negative of the original result. Symbolically, if a and b are any two numbers, then

Non-associativity

Subtraction is non-associative, which comes up when one tries to define repeated subtraction. In general, the expression

can be defined to mean either (a − b) − c or a − (b − c), but these two possibilities lead to different answers. To resolve this issue, one must establish an order of operations, with different orders yielding different results.

Predecessor

In the context of integers, subtraction of one also plays a special role: for any integer a, the integer (a − 1) is the largest integer less than a, also known as the predecessor of a.

Units of measurement

When subtracting two numbers with units of measurement such as kilograms or pounds, they must have the same unit. In most cases, the difference will have the same unit as the original numbers.

Percentages

Changes in percentages can be reported in at least two forms, percentage change and percentage point change. Percentage change represents the relative change between the two quantities as a percentage, while percentage point change is simply the number obtained by subtracting the two percentages.[6][7][8]

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 20% − 30%/30% = −1/3 = −33+1/3%, while the percentage point change is −10 percentage points.

In computing

The method of complements is a technique used to subtract one number from another using only the addition of positive numbers. This method was commonly used in mechanical calculators, and is still used in modern computers.

To subtract a binary number y (the subtrahend) from another number x (the minuend), the ones' complement of y is added to x and one is added to the sum. The leading digit "1" of the result is then discarded.

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:

becomes the sum:

Dropping the initial "1" gives the answer: 01001110 (equals decimal 78)

The teaching of subtraction in schools

Methods used to teach subtraction to elementary school 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 traditional mathematics, 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 decimal representations of fractional numbers.

In America

Almost all American schools currently teach a method of subtraction using borrowing or regrouping (the decomposition algorithm) and a system of markings called crutches.[9][10] Although a method of borrowing had been known and published in textbooks previously, the use of crutches in American schools spread after William A. Brownell published a study—claiming that crutches were beneficial to students using this method.[11] This system caught on rapidly, displacing the other methods of subtraction in use in America at that time.

In Europe

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.[12][13]

Comparing the two main methods

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:

from the minuend

where each si and mi is a digit, proceeds by writing down m1 − s1, m2 − s2, and so forth, as long as si does not exceed mi. Otherwise, mi 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 mi+1 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 si+1 by one.

Example: 704 − 512.

− 1 C D U 7 0 4 5 1 2 1 9 2 ⟵ c a r r y ⟵ M i n u e n d ⟵ S u b t r a h e n d ⟵ R e s t o r D i f f e r e n c e {\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}}}

The minuend is 704, the subtrahend is 512. The minuend digits are m3 = 7, m2 = 0 and m1 = 4. The subtrahend digits are s3 = 5, s2 = 1 and s1 = 2. 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.

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, and is written down in the result's hundreds place.

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 7, the student is asked to consider what number, when increased by 1, and 5 is added to it, makes 7.

Subtraction by hand

Austrian method

Example:

• 

  1 + ... = 3

• 

  The difference is written under the line.

• 

  9 + ... = 5
  The required sum (5) is too small.

• 

  So, we add 10 to it and put a 1 under the next higher place in the subtrahend.

• 

  9 + ... = 15
  Now we can find the difference as before.

• 

  (4 + 1) + ... = 7

• 

  The difference is written under the line.

• 

  The total difference.

Subtraction from left to right

Example:

• 

  7 − 4 = 3
  This result is only penciled in.

• 

  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.

• 

  15 − 9 = 6

• 

  Because the next digit in the minuend is not smaller than the subtrahend, we keep this number.

• 

  3 − 1 = 2

American method

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 subtracting the next digit and borrowing as needed, until every digit has been subtracted. Example:

• 

  3 − 1 = ...

• 

  We write the difference under the line.

• 

  5 − 9 = ...
  The minuend (5) is too small!

• 

  So, we add 10 to it. The 10 is "borrowed" from the digit on the left, which goes down by 1.

• 

  15 − 9 = ...
  Now the subtraction works, and we write the difference under the line.

• 

  6 − 4 = ...

• 

  We write the difference under the line.

• 

  The total difference.

Trade first

A variant of the American method where all borrowing is done before all subtraction.[14]

Example:

• 

  1 − 3 = not possible.
  We add a 10 to the 1. Because the 10 is "borrowed" from the nearby 5, the 5 is lowered by 1.

• 

  4 − 9 = not possible.
  So we proceed as in step 1.

• 

  Working from right to left:
  11 − 3 = 8

• 

  14 − 9 = 5

• 

  6 − 4 = 2

Partial differences

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 difference.[15]

Example:

• 

  The smaller number is subtracted from the greater:
  700 − 400 = 300
  Because the minuend is greater than the subtrahend, this difference has a plus sign.

• 

  The smaller number is subtracted from the greater:
  90 − 50 = 40
  Because the minuend is smaller than the subtrahend, this difference has a minus sign.

• 

  The smaller number is subtracted from the greater:
  3 − 1 = 2
  Because the minuend is greater than the subtrahend, this difference has a plus sign.

• 

  +300 − 40 + 2 = 262

Nonvertical methods

Counting up

Instead of finding the difference digit by digit, one can count up the numbers between the subtrahend and the minuend.[16]

Example: 1234 − 567 = can be found by the following steps:

• 567 + 3 = 570
• 570 + 30 = 600
• 600 + 400 = 1000
• 1000 + 234 = 1234

Add up the value from each step to get the total difference: 3 + 30 + 400 + 234 = 667.

Breaking up the subtraction

Another method that is useful for mental arithmetic is to split up the subtraction into small steps.[17]

Example: 1234 − 567 = can be solved in the following way:

• 1234 − 500 = 734
• 734 − 60 = 674
• 674 − 7 = 667

Same change

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.[18]

Example:

"1234 − 567 =" can be solved as follows:

• 1234 − 567 = 1237 − 570 = 1267 − 600 = 667

See also

• Absolute difference
• Decrement
• Elementary arithmetic
• Method of complements
• Negative number
• Plus and minus signs

Notes

1. ^ "Subtrahend" is shortened by the inflectional Latin suffix -us, e.g. remaining un-declined as in numerus subtrahendus "the number to be subtracted".

References

1. ^ a b Weisstein, Eric W. "Subtraction". mathworld.wolfram.com. Retrieved 2020-08-26.
2. ^ a b c Schmid, Hermann (1974). Decimal Computation (1 ed.). Binghamton, NY: John Wiley & Sons. ISBN 978-0-471-76180-8.
3. ^ a b c Schmid, Hermann (1983) [1974]. Decimal Computation (1 (reprint) ed.). Malabar, FL: Robert E. Krieger Publishing Company. ISBN 978-0-89874-318-0.
4. ^ "Subtraction". www.mathsisfun.com. Retrieved 2020-08-26.
5. ^ "Subtraction". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.)
6. ^ Paul E. Peterson, Michael Henderson, Martin R. West (2014) Teachers Versus the Public: What Americans Think about Schools and How to Fix Them Brookings Institution Press, p. 163
7. ^ Janet Kolodzy (2006) Convergence Journalism: Writing and Reporting across the News Media Rowman & Littlefield Publishers, p. 180
8. ^ David Gillborn (2008) Racism and Education: Coincidence Or Conspiracy? Routledge p. 46
9. ^ Klapper, Paul (1916). The Teaching of Arithmetic: A Manual for Teachers. pp. 80–. Retrieved 2016-03-11.
10. ^ Susan Ross and Mary Pratt-Cotter. 2000. "Subtraction in the United States: An Historical Perspective," The Mathematics Educator 8(1):4–11. p. 8: "This new version of the decomposition algorithm [i.e., using Brownell's crutch] has so completely dominated the field that it is rare to see any other algorithm used to teach subtraction today [in America]."
11. ^ Ross, Susan C.; Pratt-Cotter, Mary (1999). "Subtraction From a Historical Perspective". School Science and Mathematics. 99 (7): 389–93. doi:10.1111/j.1949-8594.1999.tb17499.x.
12. ^ Klapper 1916, pp. 177–.
13. ^ David Eugene Smith (1913). The Teaching of Arithmetic. Ginn. pp. 77–. Retrieved 2016-03-11.
14. ^ The Many Ways of Arithmetic in UCSMP Everyday Mathematics Archived 2014-02-25 at the Wayback Machine Subtraction: Trade First
15. ^ Partial-Differences Subtraction Archived 2014-06-23 at the Wayback Machine; The Many Ways of Arithmetic in UCSMP Everyday Mathematics Archived 2014-02-25 at the Wayback Machine Subtraction: Partial Differences
16. ^ The Many Ways of Arithmetic in UCSMP Everyday Mathematics Archived 2014-02-25 at the Wayback Machine Subtraction: Counting Up
17. ^ The Many Ways of Arithmetic in UCSMP Everyday Mathematics Archived 2014-02-25 at the Wayback Machine Subtraction: Left to Right Subtraction
18. ^ The Many Ways of Arithmetic in UCSMP Everyday Mathematics Subtraction: Same Change Rule

Bibliography

• Brownell, W.A. (1939). Learning as reorganization: An experimental study in third-grade arithmetic, Duke University Press.
• Subtraction in the United States: An Historical Perspective, Susan Ross, Mary Pratt-Cotter, The Mathematics Educator, Vol. 8, No. 1 (original publication) and Vol. 10, No. 1 (reprint.) PDF

External links

Look up subtraction in Wiktionary, the free dictionary.

Wikimedia Commons has media related to Subtraction.

• "Subtraction", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Printable Worksheets: Subtraction Worksheets, One Digit Subtraction, Two Digit Subtraction, Four Digit Subtraction, and More Subtraction Worksheets
• Subtraction Game at cut-the-knot
• Subtraction on a Japanese abacus selected from Abacus: Mystery of the Bead

Retrieved from "https://en.wikipedia.org/w/index.php?title=Subtraction&oldid=1121909623"

Page 2

This article is about the number. For the year, see AD 1. For other uses, see One (disambiguation) and Number One (disambiguation).

This article contains special characters. Without proper rendering support, you may see question marks, boxes, or other symbols.

Natural number

−1 0 1 2 3 4 5 6 7 8 9 →

• List of numbers
• Integers

← 0 10 20 30 40 50 60 70 80 90 →

1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit of counting or measurement. For example, a line segment of unit length is a line segment of length 1. In conventions of sign where zero is considered neither positive nor negative, 1 is the first and smallest positive integer.[1] It is also sometimes considered the first of the infinite sequence of natural numbers, followed by 2, although by other definitions 1 is the second natural number, following 0.

The fundamental mathematical property of 1 is to be a multiplicative identity, meaning that any number multiplied by 1 equals the same number. Most if not all properties of 1 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 not considered a prime number; this was not universally accepted until the mid-20th century. Additionally, 1 is the smallest possible difference between two distinct natural numbers.

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

Etymology

The word one can be used as a noun, an adjective, and a pronoun.[2]

It comes from the English word an,[2] which comes from the Proto-Germanic root *ainaz.[2] The Proto-Germanic root *ainaz comes from the Proto-Indo-European root *oi-no-.[2]

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

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

As a number

One, sometimes referred to as unity,[3][1] is the first non-zero natural number. It is thus the integer after zero.

Any number multiplied by one remains that number, as one is the identity for multiplication. As a result, 1 is its own factorial, its own square and square root, its own cube and cube root, and so on. One is also the result of the empty product, as any number multiplied by one is itself. It is also the only natural number that is neither composite nor prime with respect to division, but is instead considered a unit (meaning of ring theory).

As a digit

Main article: History of the Hindu–Arabic numeral system

The glyph used today in the Western world to represent the number 1, a vertical line, often with a serif at the top and sometimes a short horizontal line at the bottom, traces its roots back to the Brahmic script of ancient India, where it was a simple 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 confusion with the glyph used for seven in other countries. In styles in which the digit 1 is written with a long upstroke, the digit 7 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 1 has an ascender in most modern typefaces, in typefaces with text figures, the glyph usually is of x-height, as, for example, in

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

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 (also called unity), usually of a group or a ring.

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

In a multiplicative group or monoid, the identity element is sometimes denoted 1, but e (from the German Einheit, "unity") is also traditional. However, 1 is especially common for the multiplicative identity of a ring, i.e., when an addition and 0 are also present. When such a ring has characteristic n not equal to 0, the element called 1 has the property that n1 = 1n = 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 complex number, unit vector, and a unit matrix (more usually called an identity matrix). Note that the term unit matrix is sometimes used to mean something quite different.

By definition, 1 is the probability of an event that is absolutely or almost certain to occur.

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

In number theory, 1 is the value of Legendre's constant, which was introduced in 1808 by Adrien-Marie Legendre in expressing the asymptotic behavior of the prime-counting function. Legendre's constant was originally conjectured to be approximately 1.08366, but was proven to equal exactly 1 in 1899.

Properties

Tallying is often referred to as "base 1", since only one mark – the tally itself – is needed. This is more formally referred to as a unary numeral system. Unlike base 2 or base 10, this is not a positional notation.

Since the base 1 exponential function (1x) always equals 1, 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 as a recurring decimal: as 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 problems, numeric values are typically normalized to fall within the unit interval from 0 to 1, where 1 usually represents the maximum possible value in the range of parameters. Likewise, vectors are often normalized into unit vectors (i.e., vectors of magnitude one), because these often have more desirable properties. Functions, too, are often normalized by the condition that they have integral one, maximum value one, or square integral one, depending on the application.

Because of the multiplicative identity, if f(x) is a multiplicative function, then f(1) must be equal to 1.

It is also the first and second number in the Fibonacci sequence (0 being the zeroth) and is the first number in many other mathematical sequences.

The definition of a field requires that 1 must not be equal to 0. Thus, there are no fields of characteristic 1. Nevertheless, abstract algebra can consider the field with one 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 consequence of Benford's law.

1 is the only known Tamagawa number for a simply connected algebraic group over a number field.

The generating function that has all coefficients 1 is given by

1 1 − x = 1 + x + x 2 + x 3 + … {\displaystyle {\frac {1}{1-x}}=1+x+x^{2}+x^{3}+\ldots }

This power series converges and has finite value if and only if | x | < 1 {\displaystyle |x|<1}

Primality

Main article: Prime number § Primality of one

1 is by convention neither a prime number nor a composite number, but a unit (meaning of ring theory) like −1 and, in the Gaussian integers, i and −i.

The fundamental theorem of arithmetic 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)6 × 123 × 22, among infinitely many similar "factorizations".

1 appears to meet the naïve definition of a prime number, being evenly divisible only by 1 and itself (also 1). As such, some mathematicians considered it a prime number as late as the middle of the 20th century, but mathematical consensus has generally and since then universally been to exclude it for a variety of reasons (such as complicating the fundamental theorem of arithmetic and other theorems related to prime numbers).

1 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 two positive integers, and zero is divisible by all positive integers.

Table of basic calculations

In technology

• The resin identification code used in recycling to identify polyethylene terephthalate.[4]
• The ITU country code for the North American Numbering Plan area, which includes the United States, Canada, and parts of the Caribbean.
• A binary code is a sequence of 1 and 0 that is used in computers for representing any kind of data.
• In many physical devices, 1 represents the value for "on", which means that electricity is flowing.[5][6]
• The numerical value of true 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 atomic number of hydrogen.
• +1 is the electric charge of positrons and protons.
• Group 1 of the periodic table consists of the alkali metals.
• Period 1 of the periodic table consists of just two elements, hydrogen and helium.
• The dwarf planet Ceres has the minor-planet designation 1 Ceres because it was the first asteroid to be discovered.
• The Roman numeral I often stands for the first-discovered satellite of a planet or minor planet (such as Neptune I, a.k.a. Triton). For some earlier discoveries, the Roman numerals originally reflected the increasing distance from the primary 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 (20 BC – AD 50) regarded the number one as God's number, and the basis for all numbers ("De Allegoriis Legum," ii.12 [i.66]).

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

In sports

In many professional sports, the number 1 is assigned to the player who is first or leading in some respect, or otherwise important; the number is printed on his sports uniform or equipment. This is the pitcher in baseball, the goalkeeper in association football (soccer), the starting fullback in most of rugby league, the starting loosehead prop in rugby union and the previous year's world champion in Formula One. 1 may be the lowest possible player number, like 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 crew.
• 1 is the value of an ace in many playing card games, such as cribbage.
• List of highways numbered 1
• List of public transport routes numbered 1
• 1 is often used to denote the Gregorian calendar month of January.
• 1 CE, the first year of the Common Era
• 01, the former dialling code for Greater London (now 020)
• 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 Plan.
• In some countries, a street address of "1" is considered prestigious and developers will attempt to obtain such an address for a building, to the point of lobbying for a street or portion of a street to be renamed, even if this makes the address 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

• 
  Mathematics portal

• −1
• +1 (disambiguation)
• List of mathematical constants
• One (word)
• Root of unity
• List of highways numbered 1

References

Wikimedia Commons has media related to:
1 (number) (category)

1. ^ a b Weisstein, Eric W. "1". mathworld.wolfram.com. Retrieved 2020-08-10.
2. ^ a b c d e f g h "Online Etymology Dictionary". etymonline.com. Douglas Harper.
3. ^ Skoog, Douglas. Principles of Instrumental Analysis. 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 Technology, Evans Brothers, p. 9, ISBN 978-0-237-52725-9
6. ^ Godbole, Achyut S. (1 September 2002), Data Comms & Networks, Tata McGraw-Hill Education, p. 34, ISBN 978-1-259-08223-8
7. ^ Olson, Roger (2017). The Essentials of Christian Thought: Seeing Reality through the Biblical Story. Zondervan Academic. ISBN 9780310521563.
8. ^ British Society for the History of Science (July 1, 1977). "From Abacus to Algorism: Theory and Practice in Medieval Arithmetic". The British Journal for the History of Science. Cambridge University PRess. 10 (2): Abstract. doi:10.1017/S0007087400015375. S2CID 145065082. Retrieved May 16, 2021.

External links

Wikiquote has quotations related to 1 (number).

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

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