arithmetic

• [104] Unlike rational number arithmetic, real number arithmetic is closed under exponentiation as long as it uses a positive number as its base.

• An affine form is a number together with error terms that describe how the number may deviate from the actual magnitude.

• It represents real numbers similar to the scientific notation through three numbers: a significand, a base, and an exponent.

• [62] Integer arithmetic  Using the number line method, calculating is performed by starting at the origin of the number line then moving five units to right for the

• [97] One way to calculate exponentiation with a fractional exponent is to perform two separate calculations: one exponentiation using the numerator of the exponent followed
by drawing the nth root of the result based on the denominator of the exponent.

• [68] A common technique for multiplication with larger numbers is called long multiplication.

• [109] One basic way to express the degree of certainty about each number’s value and avoid false precision is to round each measurement to a certain number of digits, called
significant digits, which are implied to be accurate.

• [106] In cases where absolute precision is not required, the problem of calculating arithmetic operations on real numbers is usually addressed by truncation or rounding.

• [4] However, the more common view is to include operations on integers, rational numbers, real numbers, and sometimes also complex numbers in its scope.

• Using scientific notation, a number is decomposed into the product of a number between 1 and 10, called the significand, and 10 raised to some integer power, called the exponent.

• [99] The decimal fraction notation is a special way of representing rational numbers whose denominator is a power of 10.

• [79] Number theory  Main article: Number theory Number theory studies the structure and properties of integers as well as the relations and laws between them.

• [118] A common method employed by computers to approximate real number arithmetic is called floating-point arithmetic.

• For example, in the method addition with carries, the two numbers are written one above the other.

• Because of this, there is often no simple and accurate way to express the results of arithmetic operations like or .

• 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 .

• [7] Arithmetic is closely related to number theory and some authors use the terms as synonyms.

• The main kinds of numbers employed in arithmetic are natural numbers, whole numbers, integers, rational numbers, and real numbers.

• They differ from each other based on what type of number they operate on, what numeral system they use to represent them, and whether they operate on mathematical objects
other than numbers.

• Rounding is a similar process in which the last preserved digit is increased by one if the next digit is 5 or greater but remains the same if the next digit is less than 5,
so that the rounded number is the best approximation of a given precision for the original number.

• [89] According to the fundamental theorem of arithmetic, every integer greater than 1 is either a prime number or can be represented as a unique product of prime numbers.

• This means that when dividing one integer by another integer, the result is not always an integer.

• For operations on numbers with more than one digit, different techniques can be employed to calculate the result by using several one-digit operations in a row.

• [100] Modified versions of integer calculation methods like addition with carry and long multiplication can be applied to calculations with decimal fractions.

• [a] The whole numbers are identical to the natural numbers with the only difference being that they include 0.

• Intervals can be used to represent a range of values if one does not know the precise magnitude, for example, because of measurement errors.

• This means that the result of is sometimes different from the result of .

• [92] Rational number arithmetic  Rational number arithmetic is the branch of arithmetic that deals with the manipulation of numbers that can be expressed as a ratio
of two integers.

• For instance, the rational numbers , , and are written as 0.1, 3.71, and 0.0044 in the decimal fraction notation.

• [66] A similar technique is utilized for subtraction: it also starts with the rightmost digit and uses a “borrow” or a negative carry for the column on the left if the result
of the one-digit subtraction is negative.

• If the multiplicand is a natural number then multiplication is the same as repeated addition, as in .

• [91] Fermat’s last theorem is the statement that no positive integer values can be found for , , and , to solve the equation if is greater than .

• Irrational numbers are numbers that cannot be expressed through fractions or repeated decimals, like the root of 2 and π.

• The logarithm of a number to the base is the exponent to which must be raised to produce .

• [45] Counting is a type of repeated addition in which the number 1 is continuously added.

• [65] Other methods used for integer additions are the number line method, the partial sum method, and the compensation method.

• The symbol of the rational numbers is .

• If an arithmetic operation results in a number that requires more bits than are available, the computer rounds the result to the closest representable number.

• Interval arithmetic includes operations like addition and multiplication on intervals, as in and .

• For example, floating-point addition is not associative since the rounding errors introduced can depend on the order of the additions.

• Unlike ordinary decimal notation, where trailing zeros of large numbers are implicitly considered to be non-significant, in scientific notation every digit in the significand
is considered significant, and adding trailing zeros indicates higher precision.

• [40] Two important concepts in relation to arithmetic operations are identity elements and inverse elements.

• [23] The set of rational numbers together with the set of irrational numbers makes up the set of real numbers.

• Arithmetic operations form the basis of many branches of mathematics, such as algebra, calculus, and statistics.

• [84] Further branches of number theory are probabilistic number theory, which employs methods from probability theory,[85] combinatorial number theory, which relies on the
field of combinatorics,[86] computational number theory, which approaches number-theoretic problems with computational methods,[87] and applied number theory, which examines the application of number theory to fields like physics, biology,
and cryptography.

• Dividing one rational number by another can be achieved by multiplying the first number with the reciprocal of the second number.

• [6] When understood in a wider sense, it also includes the study of how the concept of numbers developed, the analysis of properties of and relations between numbers, and
the examination of the axiomatic structure of arithmetic operations.

• [19] Decimal fractions like 0.3 and 25.12 are a special type of rational numbers since their denominator is a power of 10.

• [24] Even wider classes of numbers include complex numbers and quaternions.

• Representing uncertainty using only significant digits is a relatively crude method, with some unintuitive subtleties; explicitly keeping track of an estimate or upper bound
of the approximation error is a more sophisticated approach.

• [73] One way to ensure that the result is an integer is to round the result to a whole number.

• [80] Some of the main branches of modern number theory include elementary number theory, analytic number theory, algebraic number theory, and geometric number theory.

• One way to get an approximate result for the second operation is to employ Newton’s method, which uses a series of steps to gradually refine an initial guess until it reaches
the desired level of accuracy.

• [113] When representing uncertainty by significant digits, uncertainty can be coarsely propagated by rounding the result of adding or subtracting two or more quantities to
the leftmost last significant decimal place among the summands, and by rounding the result of multiplying or dividing two or more quantities to the least number of significant digits among the factors.

• [43] 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.

• This system makes it cumbersome to write large numbers, which is why many non-positional systems include additional symbols to directly represent larger numbers.

• [95] Unlike integer arithmetic, rational number arithmetic is closed under division as long as the divisor is not 0.

• [63] Simple one-digit operations can be performed by following or memorizing a table that presents the results of all possible combinations, like an addition table or a multiplication
table.

• It examines problems like how prime numbers are distributed and the claim that every even number is a sum of two prime numbers.

• There are different kinds of numbers and different numeral systems to represent them.

• [116] The precision of numerical quantities can be expressed uniformly using normalized scientific notation, which is also convenient for concisely representing numbers which
are much larger or smaller than 1.

• The symbol of the real numbers is .

• [18] A number is rational if it can be represented as the ratio of two integers.

• [14][b] Some mathematicians do not draw the distinction between the natural and the whole numbers by including 0 in the set of natural numbers.

• When a number is written using ordinary decimal notation, leading zeros are not significant, and trailing zeros of numbers not written with a decimal point are implicitly
considered to be non-significant.

• The symbol of the natural numbers is .

• Irrational numbers are numbers that cannot be expressed through the ratio of two integers.

• [39] If these operations are performed on variables rather than numbers, they are sometimes referred to as algebraic operations.

• For example, the square root of a number is the same as raising the number to the power of and the cube root of a number is the same as raising the number to the power of
.

• Arithmetic is present in many aspects of daily life, for example, to calculate change while shopping or to manage personal finances.

Works Cited

[‘Other symbols for the natural numbers include , , , and .[13]
2. ^ Other symbols for the whole numbers include , , and .[15]
3. ^ A repeating decimal is a decimal with an infinite number of repeating digits, like 0.111…, which expresses the rational
number .
4. ^ Some authors use a different terminology and refer to the first number as multiplicand and the second number as the multiplier.[51]
5. ^ If the exponent is 0 then the result is 1, as in . The only exception is , which is not defined.[57]
6. ^
Some systems of counting rods include different colors to represent both positive and negative numbers.[133]
7. ^ Some computer scientists see slide rules as the first type of analog computer.[135]
8. ^ The successor of a natural number is the
number that comes after it. For instance, 4 is the successor of 3.
9. ^ There are different versions of the exact formulation and number of axioms. For example, some formulations start with 1 instead of 0 in the first axiom.[149]
10. ^ An influential
argument for Platonism, first formulated by Willard Van Orman Quine and Hilary Putnam, states that numbers exist because they are indispensable to the best scientific theories.[197]
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