User:Phlsph7/Arithmetic operations

Arithmetic operations

Arithmetic operations are ways of combining, transforming, or manipulating numbers. They are functions that have numbers both as input and output. The most important operations in arithmetic are addition, subtraction, multiplication, division, exponentiation, and logarithm. If these operations are performed on variables rather than numbers, they are sometimes referred to as algebraic operations.

Two important concepts in relation to arithmetic operations are identity elements and inverse elements. The identity element or neutral element of an operation does not cause any change if it is applied to another element. For example, the identity element of addition is 0 since any sum of a number and 0 results in the same number. The inverse element is the element that results in the identity element when combined with another element. For instance, the additive inverse of the number 6 is -6 since their sum is 0.

There are not only inverse elements but also inverse operations. In an informal sense, one operation is the inverse of another operation if it undoes the first operation. For example, subtraction is the inverse of addition since a number returns to its original value if a second number is first added and subsequently subtracted, as in $13+4-4=13$. Defined more formally, the operation "$\star$" is an inverse of the operation "$\circ$" if it fulfills the following condition: $t\star s=r$ if and only if $r\circ s=t$.

Commutativity and associativity are laws governing the order in which some arithmetic operations can be carried out. An operation is commutative if the order of the arguments can be changed without affecting the results. This is the case for addition, for example, $7+9$ is the same as $9+7$. Associativity is a rule that affects the order in which a series of operations can be carried out. An operation is associative if in a series of two operations, it does not matter which operation is carried out first. This is the case for multiplication, for example, since $(5\times 4)\times 2$ is the same as $5\times (4\times 2)$.

Addition and subtraction

Main articles: Addition and Subtraction

Addition and subtraction

Addition is an arithmetic operation in which two numbers, called the addends, are combined into a single number, called the sum. The symbol of addition is $+$. Examples are $2+2=4$ and $6.3+1.26=7.56$. The term summation is used if several additions are performed in a row. Counting is a type of repeated addition in which the number 1 is continuously added.

Subtraction is the inverse of addition. In it, one number, known as the subtrahend, is taken away from another, known as the minuend. The result of this operation is called the difference. The symbol of subtraction is $-$. Examples are $14-8=6$ and $45-1.7=43.3$. Subtraction is often treated as a special case of addition: instead of subtracting a positive number, it is also possible to add a negative number. For instance $14-8=14+(-8)$. This helps to simplify mathematical computations by reducing the number of basic arithmetic operations needed to perform calculations.

The additive identity element is 0 and the additive inverse of a number is the negative of that number. For example, $13+0=13$ and $13+(-13)=0$. Addition is both commutative and associative.

Multiplication and division

Main articles: Multiplication and Division (mathematics)

Multiplication and division

Multiplication is an arithmetic operation in which two numbers, called the multiplier and the multiplicant, are combined into a single number called the product. The symbols of multiplication are $\times$, $\cdot$, and *. Examples are $2\times 3=6$ and $0.3\cdot 5=1.5$. If the multiplicant is a natural number then multiplication is the same as repeated addition, as in $2\times 3=2+2+2$.

Division is the inverse of multiplication. In it, one number, known as the dividend, is split into several equal parts by another number, known as the divisor. The result of this operation is called the quotient. The symbols of division are $\div$ and $/$. Examples are $48\div 8=6$ and $29.4/1.4=21$. Division is often treated as a special case of multiplication: instead of dividing by a number, it is also possible to multiply by its reciprocal. The reciprocal of a number is 1 divided by that number. For instance, $48\div 8=48\times {\tfrac {1}{8}}$.

The multiplicative identity element is 1 and the multiplicative inverse of a number is the reciprocal of that number. For example, $13\times 1=13$ and $13\times {\tfrac {1}{13}}=1$. Multiplication is both commutative and associative.

Exponentiation and logarithm

Main articles: Exponentiation and Logarithm

Exponentiation and logarithm

Exponentiation is an arithmetic operation in which a number, known as the base, is raised to the power of another number, known as the exponent. The result of this operation is called the power. Exponentiation is sometimes expressed using the symbol ^ but the more common way is to write the exponent in superscript right after the base. Examples are $2^{4}=16$ and $3$^$3=27$. If the exponent is a natural number then exponentiation is the same as repeated multiplication, as in $2^{4}=2\times 2\times 2\times 2$.

Roots are a special type of exponentiation using a fractional exponent. For example, the square root of a number is the same as raising the number to the power of ${\tfrac {1}{2}}$ and the cube root of a number is the same as raising the number to the power of ${\tfrac {1}{3}}$. Examples are ${\sqrt {4}}=4^{\tfrac {1}{2}}=2$ and ${\sqrt[{3}]{27}}=27^{\tfrac {1}{3}}=3$.

Logarithm is the inverse of exponentiation. In it, the base and the argument of the logarithm are used to determine the power to which the base must be raised to produce the argument. The result of this operation is known as the anti-logarithm. For example, to determine the logarithm of 16 in relation to the base 2, it is necessary to find what exponent should be used on the base 2 for the result to be 16. The anti-logarithm of this operation is 4 since 2 raised by 4 results in 16.

Exponentiation and logarithm do not have general identity elements and inverse elements like addition and multiplication. The neutral element of exponentiation in relation to the exponent is 1, as in $14^{1}=14$. However, exponentiation does not have a general identity element since 1 is not the neutral element for the base. Exponentiation and logarithm are neither commutative nor associative.

• Smyth, William (1864). Elementary Algebra: For Schools and Academies. Bailey and Noyes.
• Husserl, Edmund; Willard, Dallas (6 December 2012). "Translator's Introduction". Philosophy of Arithmetic: Psychological and Logical Investigations with Supplementary Texts from 1887–1901. Springer Science & Business Media. ISBN 978-94-010-0060-4.
• O'Leary, Michael L. (8 September 2015). A First Course in Mathematical Logic and Set Theory. John Wiley & Sons. ISBN 978-0-470-90588-3.
• Khan, Khalid; Graham, Tony Lee (12 June 2018). Engineering Mathematics with Applications to Fire Engineering. CRC Press. ISBN 978-1-351-59761-6.
• Rising, Gerald R.; Matthews, James R.; Schoaff, Eileen; Matthew, Judith (2021). About Mathematics. Linus Learning. ISBN 978-1-60797-892-3.
• Kay, Anthony (14 September 2021). Number Systems: A Path into Rigorous Mathematics. CRC Press. ISBN 978-0-429-60776-9.
• Tarasov, Vasily (6 June 2008). Quantum Mechanics of Non-Hamiltonian and Dissipative Systems. Elsevier. ISBN 978-0-08-055971-1.
• Mazzola, Guerino; Milmeister, Gérard; Weissmann, Jody (2004). Comprehensive Mathematics For Computer Scientists 1: Sets And Numbers, Graphs And Algebra, Logic And Machines, Linear Geometry. Springer Science & Business Media. ISBN 978-3-540-20835-8.
• Krenn, Stephan; Lorünser, Thomas (28 March 2023). An Introduction to Secret Sharing: A Systematic Overview and Guide for Protocol Selection. Springer Nature. ISBN 978-3-031-28161-7.
• Sally, Judith D.; Sally (Jr.), Paul J. (2012). Integers, Fractions, and Arithmetic: A Guide for Teachers. American Mathematical Soc. ISBN 978-0-8218-8798-1.
• Burgin, Mark (22 April 2022). Trilogy Of Numbers And Arithmetic - Book 1: History Of Numbers And Arithmetic: An Information Perspective. World Scientific. ISBN 978-981-12-3685-3.
• Wheater, Carolyn (2 June 2015). Algebra I. Dorling Kindersley Limited. ISBN 978-0-241-88779-0.
• Wright, Robert J.; Ellemor-Collins, David; Tabor, Pamela D. (4 November 2011). Developing Number Knowledge: Assessment,Teaching and Intervention with 7-11 year olds. SAGE. ISBN 978-1-4462-8927-3.
• Achatz, Thomas; Anderson, John G. (2005). Technical Shop Mathematics. Industrial Press Inc. ISBN 978-0-8311-3086-2.
• Klose, Orval M. (16 May 2014). The Number Systems and Operations of Arithmetic: An Explanation of the Fundamental Principles of Mathematics Which Underlie the Understanding and Use of Arithmetic, Designed for In-Service Training of Elementary School Teachers Candidates Service Training of Elementary School Teacher Candidates. Elsevier. ISBN 978-1-4831-3709-4.
• Rodda, Harvey J. E.; Little, Max A. (2 November 2015). Understanding Mathematical and Statistical Techniques in Hydrology: An Examples-based Approach. John Wiley & Sons. ISBN 978-1-119-07659-9.