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Decimal representation

Decimal representation is a positional numeral system that expresses numbers using the base-10 digits 0 through 9, where the value of each digit is determined by its position relative to the decimal point, corresponding to powers of 10.^[1] For integers, the rightmost digit represents the units place (10^0), the next to the left is the tens place (10^1), followed by hundreds (10^2), and so on, allowing compact notation for large numbers such as 345 equaling 3×10^2 + 4×10^1 + 5×10^0.^[1] To the right of the decimal point, digits denote fractional parts as negative powers of 10, starting with tenths (10^{-1}), hundredths (10^{-2}), and continuing indefinitely, enabling the representation of real numbers like 0.24 as 2×10^{-1} + 4×10^{-2}.^[2] In the context of real numbers, every real number possesses a decimal expansion, which may be finite (terminating after a certain number of digits) or infinite; however, some numbers, such as terminating decimals, admit two different representations (e.g., 0.999... = 1.000...), with details covered in the section on uniqueness and ambiguities.^[3] Rational numbers, expressible as fractions a/b where a and b are integers with b ≠ 0, have decimal expansions that either terminate—such as 1/2 = 0.5—or repeat periodically, as in 1/3 = 0.333..., a property arising from the finite remainders in long division processes.^[3] In contrast, irrational numbers like π or √2 feature non-terminating and non-repeating decimal expansions, ensuring their distinction from rationals and highlighting the completeness of the real number system in capturing all points on the number line.^[3] The decimal system's origins trace back to ancient practices of counting with ten fingers, evolving into formalized positional notation by civilizations including the Chinese during the Warring States period (around the 5th century BC), who developed rod-based calculations and the abacus for decimal arithmetic.^[4] It reached its modern form through the Hindu-Arabic numeral system, which incorporated the essential digit zero for placeholders, facilitating efficient computation, and was introduced to Europe by the mathematician Fibonacci in the early 13th century via his work Liber Abaci.^[4] Today, decimal representation dominates mathematical, scientific, and everyday applications due to its alignment with human cognition and simplicity in performing operations like addition and multiplication.^[5]

Basic Structure

Definition and Notation

Decimal representation is a method of expressing real numbers in the base-10 positional numeral system, where a number n is written as n = a + b, with a denoting the integer part and b the fractional part, each expanded as a sum of digits multiplied by powers of 10.^[2]^[6] The integer part uses non-negative powers of 10, while the fractional part employs negative powers, allowing precise representation of both whole numbers and fractions using the same digit set.^[2] In this notation, the digits range from 0 to 9, and a decimal point (.) separates the integer and fractional parts. Place values to the left of the decimal point represent units ($10^0), tens ($10^1), hundreds ($10^2), and so on, while positions to the right denote tenths ($10^{-1}), hundredths ($10^{-2}), thousandths ($10^{-3}), and further subdivisions.^[6] This structure enables the compact encoding of numerical values through positional significance rather than additive symbols.^[6] The decimal system traces its roots to ancient positional numeral systems, particularly the base-10 variant that emerged in India by the 6th century AD, where place-value notation facilitated handling large numbers and calculations.^[7] Its standardization in Europe occurred in the 16th and 17th centuries, notably through the work of Flemish mathematician Simon Stevin, who in 1585 published La Thiende, providing the first comprehensive treatment of decimal fractions and advocating their practical use in measurement and arithmetic.^[8] For example, the number 3.14159 is represented as
$3 \times 10^{0} + 1 \times 10^{-1} + 4 \times 10^{-2} + 1 \times 10^{-3} + 5 \times 10^{-4} + 9 \times 10^{-5}. ^[6] This expansion illustrates how each digit's contribution is determined by its position relative to the decimal point.^[6]

Integer and Fractional Parts

In decimal representation, a real number n is divided into its integer part and fractional part by the decimal point. The integer part, denoted \lfloor n \rfloor, is the greatest integer less than or equal to n, consisting of the digits to the left of the decimal point, which form a non-negative or negative whole number.^[9] For positive n, this is the floor of n; for negative n, it remains the floor, ensuring the part is the largest integer not exceeding n.^[10] The fractional part, denoted \{n\} = n - \lfloor n \rfloor, comprises the digits to the right of the decimal point and always lies in the half-open interval [0, 1). This value represents the remainder after subtracting the integer part from n, making it non-negative even for negative numbers.^[11] For instance, in the decimal 123.456, the integer part is 123 and the fractional part is 0.456, since $123 + 0.456 = 123.456.^[12] Similarly, for -0.7, the integer part is -1 and the fractional part is 0.3, as -1 + 0.3 = -0.7.^[10] The integer part is always finite and unique for any real number in decimal notation, reflecting the place value system's structure for whole numbers. In contrast, the fractional part may be finite (terminating) or infinite (non-terminating), depending on whether the decimal expansion ends or continues indefinitely. For negative numbers, the sign applies to the entire representation, but the fractional part itself remains positive, preserving the [0, 1) range.^[11] Edge cases illustrate these properties clearly. The number zero has an integer part of 0 and a fractional part of 0, often written as 0.000.... Pure integers, such as 5, are represented as 5.0, with an integer part of 5 and a fractional part of 0. For irrational numbers like \pi \approx 3.14159\ldots, the integer part is 3, while the fractional part is the infinite sequence 0.14159....^[9]

Representation Properties

Uniqueness and Ambiguities

In decimal representation, certain real numbers possess two distinct infinite expansions, leading to non-uniqueness. A prominent example is the equivalence between $0.999\ldots and $1.000\ldots, where the former consists of repeating 9s indefinitely, while the latter terminates with infinite 0s.^[13]^[14] This equivalence can be rigorously proven using the geometric series summation formula. The repeating decimal $0.\overline{9} is expressed as the infinite series \sum_{k=1}^{\infty} 9 \times 10^{-k}, which has first term a = 9/10 and common ratio r = 1/10. The sum of this convergent series is given by S = \frac{a}{1 - r} = \frac{9/10}{1 - 1/10} = \frac{9/10}{9/10} = 1. Thus, $0.\overline{9} = 1.^[13] Similar ambiguities arise in finite decimal cases when extended to infinite forms, such as $1.00 = 0.999\ldots in the limit sense, where trailing 0s can be equivalently represented by trailing 9s. These dual representations occur precisely for rational numbers whose decimal expansions terminate, as they can be rewritten with infinite 9s instead of infinite 0s.^[14]^[15] Such non-uniqueness has practical implications in numerical computations, where equality checks between representations like $0.999\ldots and $1 may fail due to finite precision approximations, potentially leading to errors in algorithms. However, for almost all real numbers—specifically, irrational numbers and non-terminating rationals—the decimal expansion is unique, excluding these exceptional cases.^[16]

Standard Notational Conventions

In standard decimal notation, the decimal point (.) serves as the separator between the integer and fractional parts, a convention adopted in English-language mathematical texts to ensure clarity and consistency. This point is placed on the baseline, aligned with the digits, distinguishing it from other punctuation.^[17] For terminating decimals, which end after a finite number of digits, trailing zeros after the last non-zero digit are optional and frequently omitted to simplify representation without altering the value; for instance, 0.5 is equivalent to 0.50 or 0.500./09%3A_Rational_Numbers/9.04%3A_Decimals) Repeating decimals are denoted using a vinculum (a horizontal bar) over the repeating sequence, such as $0.\overline{3} for \frac{1}{3} = 0.333\ldots, though alternative notations include dots above the first and last repeating digits or an explicit label like "0.3 with 3 repeating."^[18]^[19] Leading zeros in the integer part are omitted for numbers greater than or equal to 1, but a leading zero before the decimal point is required for fractions less than 1 to avoid ambiguity, such as writing 0.5 rather than .5 in formal mathematics.^[20] Negative decimals are indicated by placing the minus sign immediately before the integer part, with no additional sign in the fractional part; for example, -3.14 represents the negative of 3.14.^[21] Internationally, while some locales use a comma (,) as the decimal separator, the point is the standard in English mathematical and scientific contexts, as endorsed by resolutions from the General Conference on Weights and Measures for formal documentation.^[22]^[17] For very large or small decimals, integration with scientific notation is conventional, expressing the number as a coefficient between 1 and 10 multiplied by a power of 10, such as $3.14159 \times 10^{0} for pi approximated to five decimal places.^[23]

Classification of Expansions

Terminating Decimals

A terminating decimal is a decimal representation of a number in which the fractional part consists of a finite sequence of digits after the decimal point, followed indefinitely by zeros. This finite sequence distinguishes it from expansions that continue infinitely without termination. Such decimals arise in the representation of certain rational numbers and are fundamental in base-10 notation for expressing exact values without repetition.^[24]^[25] For a rational number expressed as a fraction \frac{p}{q} in lowest terms, where p and q are integers with q > 0, the decimal expansion terminates if and only if the prime factorization of q contains only the primes 2 and/or 5. This condition ensures that the denominator can be expressed as q = 2^a 5^b for non-negative integers a and b, allowing the division to align perfectly with powers of 10, since $10^k = 2^k 5^k. For instance, \frac{1}{2} = 0.5, \frac{1}{4} = 0.25, and \frac{3}{5} = 0.6 all terminate because their denominators factor into 2 and 5 alone; in contrast, \frac{1}{3} yields $0.333\ldots, which does not terminate due to the prime factor 3.^[26]^[27] The process of determining termination can be observed through long division, where the numerator p is divided by the denominator q. The algorithm proceeds by repeatedly dividing and recording remainders; the decimal terminates precisely when a remainder of zero is reached, after which all subsequent digits are zero. This finite process reflects the alignment of the denominator's factors with the base 10.^[28] Terminating decimals possess unique representations in base 10, with the exception of those rational numbers that admit a dual form: one finite expansion ending in zeros and an equivalent infinite expansion of all 9's, such as $1 = 1.000\ldots = 0.999\ldots. To avoid ambiguity, conventions often prefer the terminating form. Furthermore, the set of all terminating decimals forms a countable collection, as it corresponds to the countable union over finite lengths n of the $10^n possible finite digit sequences after the decimal point, mirroring the countability of the rationals they represent.^[29]^[30]

Repeating Decimals

Repeating decimals are infinite decimal expansions featuring a sequence of one or more digits that repeats periodically after an optional initial non-repeating segment. If the repetition starts immediately following the decimal point, the expansion is purely repeating; otherwise, it includes a non-repeating prefix followed by the repeating block.^[31] In the decimal representation of a rational number a/b in lowest terms, the length of the non-repeating prefix equals the maximum of the exponents of 2 and 5 in the prime factorization of b. The repeating block's length, known as the period, is the smallest positive integer k such that $10^k \equiv 1 \pmod{q}, where q is the largest factor of b coprime to 10; this k is the multiplicative order of 10 modulo q.^[32] Representative examples illustrate this structure. The fraction $1/3 yields $0.\overline{3}, a pure repeating decimal with period 1. Similarly, $1/7 = 0.\overline{142857} has period 6. In contrast, $1/6 = 0.1\overline{6} features a non-repeating digit "1" (due to the factor of 2 in the denominator) followed by a repeating "6" of period 1.^[32] A key property is that every rational number possesses a decimal expansion that is either terminating (equivalent to repeating 0s) or repeating, and every such repeating expansion corresponds to a rational number. The period length divides \phi(q), Euler's totient function evaluated at q, as guaranteed by Euler's theorem since \gcd(10, q) = 1.^[32]^[33] The decimal place-value system originated in ancient Indian mathematics, with further developments in the Islamic world before transmission to Europe. The concept was formalized in Europe in the 16th century by Simon Stevin, who in his 1585 treatise La Thiende systematically described infinite repeating decimals as representations of real numbers.^[34]

Non-Terminating Non-Repeating Decimals

Non-terminating non-repeating decimals, also known as infinite aperiodic decimals, are decimal expansions that continue indefinitely without terminating and without any repeating sequence of digits. These expansions are the hallmark of irrational numbers, which cannot be expressed as the ratio of two integers.^[35]^[36] Prominent examples include the decimal expansions of well-known irrational constants. The number \pi, a transcendental irrational, has the expansion \pi \approx 3.1415926535\dots. The algebraic irrational \sqrt{2} expands as \sqrt{2} \approx 1.4142135623\dots. Similarly, the transcendental number e expands as e \approx 2.7182818284\dots. These expansions never repeat or terminate, distinguishing them from rational numbers.^[37]^[38] Key properties of non-terminating non-repeating decimals include their unique representation in the decimal system, free from the dual forms that arise in terminating decimals (such as $0.999\dots = 1.000\dots). There are uncountably many such numbers, reflecting the cardinality of the irrationals. Continued fraction expansions offer an alternative representation that often yields superior rational approximations compared to simple decimal truncations, with every irrational corresponding to a unique infinite continued fraction.^[39]^[40]^[41] To see why these decimals characterize irrationals, note that every rational number has a decimal expansion that either terminates or eventually repeats, as proven by the long division algorithm for fractions. Thus, assuming a non-terminating non-repeating decimal is rational leads to a contradiction. The uncountability follows from Cantor's diagonal argument: the reals are uncountable, the rationals are countable, so their difference—the irrationals—must be uncountable. A variant of the diagonal argument directly constructs irrationals by altering assumed enumerations of decimals to produce non-repeating sequences.^[42]^[40]^[43] Computationally, these expansions are produced via algorithms such as infinite series (e.g., for \pi and e) or iterative methods (e.g., for \sqrt{2}), but they cannot be captured exactly in finite decimal digits, requiring arbitrary-precision arithmetic for extended calculations. As discussed in the uniqueness properties, this uniqueness ensures consistent computational handling without representational ambiguities.^[37]

Conversion and Approximation

Decimal to Fraction Conversion

Converting a decimal number to an exact fraction is possible only for rational numbers, which manifest as either terminating or repeating decimals. This process reverses the division inherent in decimal expansions, yielding a fraction in lowest terms. For terminating decimals, the conversion is straightforward, while repeating decimals require algebraic manipulation to isolate the repeating portion. These methods ensure exact representation without approximation, distinguishing them from irrational decimals, which cannot be expressed as fractions. For terminating decimals, which end after a finite number of digits, the decimal $0.d_1 d_2 \dots d_k is equivalent to the fraction \frac{d_1 d_2 \dots d_k}{10^k}, where d_1 d_2 \dots d_k forms the integer numerator and the denominator is a power of 10 corresponding to the number of decimal places. Simplification follows by dividing numerator and denominator by their greatest common divisor. For example, $0.75 = \frac{75}{100} = \frac{3}{4}.^[44]^[45] Pure repeating decimals, denoted as $0.\overline{ab\dots} with a repeating block of length m, can be converted by setting x = 0.\overline{ab\dots}, multiplying by $10^m to shift the decimal, and subtracting the original equation to eliminate the repeat: $10^m x - x = integer part, solving for x = \frac{\text{integer}}{10^m - 1}. Simplification yields the lowest terms. For instance, let x = 0.\overline{3}; then $10x = 3.\overline{3}, so $9x = 3 and x = \frac{1}{3}.^[27]^[44] Mixed repeating decimals, such as $0.a_1 \dots a_n \overline{b_1 \dots b_m} with n non-repeating digits and an m-digit repeat, involve aligning the shifts for both parts. Set x = the decimal; multiply by $10^n to shift past the non-repeating part and by $10^{n+m} to shift one full cycle further, then subtract appropriately to solve for x. For example, for x = 0.1\overline{6}, $10x = 1.\overline{6} and $100x = 16.\overline{6}, so $90x = 15 and x = \frac{1}{6}.^[45]^[27] A general algorithm employs this algebraic setup or reverses long division to express the decimal as a quotient of integers, handling both cases systematically. Non-uniqueness arises in representations like $0.\overline{9} = 1, resolved by preferring the terminating form when possible. However, irrational numbers, characterized by non-terminating, non-repeating decimals, cannot be converted to exact fractions, as their expansions lack periodicity.^[44]^[35]

Fraction to Decimal Expansion

The decimal expansion of a rational fraction \frac{p}{q}, where p and q are integers with q \neq 0 and the fraction in lowest terms, is obtained primarily through the long division algorithm. This method involves dividing the numerator p by the denominator q, recording the quotient digits after the decimal point, and tracking the remainders at each step. If a remainder becomes zero, the expansion terminates; otherwise, if a remainder repeats, the digits from that point begin a repeating cycle.^[45]^[26] To determine whether the expansion terminates or repeats without performing the full division, factor the denominator q into primes after removing any factors of 2 or 5 (which contribute to the terminating part). A fraction has a terminating decimal if and only if q has no prime factors other than 2 and 5, i.e., q = 2^a 5^b for non-negative integers a and b. In such cases, the number of decimal places required is \max(a, b), as this is the power of 10 needed to express the denominator.^[46]^[47] For example, consider \frac{1}{2}: here q = 2 = 2^1 5^0, so \max(1, 0) = 1 digit, and long division yields $0.5, which terminates. In contrast, for \frac{1}{7}, q = 7 has a prime factor other than 2 or 5, leading to a repeating expansion. Performing long division on \frac{1}{7} produces digits 1, 4, 2, 8, 5, 7 with remainders 1, 3, 2, 6, 4, 5, which cycle back to 1 after six steps, giving $0.\overline{142857}.^[45]^[48] For integer values, where the denominator q = 1, the decimal expansion is trivial: the integer part followed by ".0", as there is no fractional component to expand. In computational settings, modern algorithms enhance efficiency by using modular arithmetic to find the period length of repeating expansions without generating all digits. Specifically, for a fraction \frac{p}{q} with q coprime to 10, the period is the multiplicative order of 10 modulo q, i.e., the smallest positive integer k such that $10^k \equiv 1 \pmod{q}. This approach avoids simulating the full long division and is particularly useful for large denominators.^[49]^[50]

Finite Approximations of Infinite Decimals

Finite approximations of infinite decimals are essential for practical computations involving irrational numbers, which have non-terminating, non-repeating decimal expansions. These approximations transform unending sequences into manageable finite forms through methods like truncation and rounding, enabling their use in numerical analysis, engineering, and scientific modeling.^[51] Truncation, also known as chopping, involves discarding all digits beyond a specified number n in the decimal expansion. For an infinite decimal x = d_0.d_1 d_2 \dots d_n d_{n+1} \dots, the truncated approximation x_t retains only up to d_n, resulting in an absolute error bounded by |x - x_t| < 10^{-n}. For example, truncating \pi \approx 3.1415926535\dots after four decimal places yields $3.1415, with an error less than $10^{-4} = 0.0001. This method is straightforward but systematically underestimates positive numbers greater than the approximation.^[51] Rounding provides a more balanced approximation by adjusting the last retained digit based on the subsequent one. If the next digit is 5 or greater, the last digit is incremented; otherwise, it remains unchanged. The absolute error for rounding to n decimal places satisfies |x - x_r| \leq 0.5 \times 10^{-n}, half the bound for truncation, as it selects the nearest representable value. A variant, banker's rounding (or round half to even), resolves ties at exactly 5 by rounding to the nearest even digit in the last position, reducing bias in repeated operations. For instance, 2.5 rounds to 2 and 3.5 to 4 under banker's rules. Applying standard rounding to e \approx 2.7182818284\dots to five decimal places gives $2.71828, with error at most $0.5 \times 10^{-5} = 0.000005.^[51]^[52] These techniques find widespread application in numerical computations, where infinite decimals must be stored in finite-precision formats like floating-point arithmetic, and in scientific notation to express large or small values concisely. For superior approximations minimizing error relative to denominator size, continued fractions generate the best rational approximations to irrationals; their convergents p_k / q_k satisfy |x - p_k / q_k| < 1/(q_k q_{k+1}) and outperform other fractions with similar denominators. For \sqrt{2} \approx 1.4142135623\dots, the continued fraction [1; 2, 2, 2, ...] yields the convergent 99/70 ≈ 1.4142857, a decimal approximation with error less than 10^{-4} (specifically, < 1/(70 × 169) ≈ 8.5 × 10^{-5}) using just seven places. Such methods handle the non-repeating nature of irrationals through iterative convergent calculations, ensuring high accuracy for applications like root-finding algorithms.^[51]^[53]^[53]

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How To Express A Repeating Decimal Number As A Fraction
Sep 22, 1997 · If you multiply top and bottom by two more 5's, i.e. by 25, then the denominator will become 2 x 2 x 2 x 5 x 5 x 5 = 10 x 10 x 10 = 1000 so 37/ ...Missing: places | Show results with:places
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[PDF] Math Circle Beginners Group October 11, 2015
Oct 11, 2015 · Fractions whose denominators have factors other than and have decimal expansions that go on forever (non-terminating decimals). 10. Can you give ...
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[PDF] Contents 2 Modular Arithmetic in Z - Evan Dummit
repeats with period 6. • It turns out that there is a simple way to determine the period of any repeating decimal expansion. We start with decimal ...
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[PDF] orders of units in modular arithmetic - Keith Conrad
The period length is the minimal possible size of a repeating part. Example 4.6. The order of 10 mod 27 is 3: 101 ≡ 10 mod 27, 102 = 100 ≡ 19 mod 27, 103 = 190 ...
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[PDF] Round-off/Truncation Errors & Numerical Cancellation Outline Notes
Nov 6, 2003 · Round-off errors occur when numbers are rounded to a finite number of digits. Truncation errors arise when an infinite process is replaced by a ...
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rounding - PlanetMath
Mar 22, 2013 · Rounding is a general technique for approximating a real number by a decimal fraction. ... rounding (or rounding for short), and banker's rounding ...<|separator|>
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[PDF] Continued Fractions - Cornell Mathematics
Aug 22, 2016 · Here are some continued fraction expansions of some special numbers which have been known for a long time: π =3+. 1. 7 +. 1. 15 +. 1. 1 +. 1.