Decimal Numbers

For example, the number 13.25 is a decimal number.

$$ 13.25 $$

The digits before the decimal point (13) form the integer part of the number, while the digits after the decimal point (25) make up the decimal part, also known as the mantissa.

Decimal numbers can be classified according to the structure of their decimal expansion as follows:

• Terminating decimal numbers
  These have a finite number of nonzero digits after the decimal point, for example \(0.25\). They are always rational numbers, since they can be expressed as fractions whose denominator is a power of \(10\). For instance: $$ 0.25 = \frac{25}{100} = \frac{1}{4} $$

  Note. Integers are a special case of decimal numbers in which the decimal part is zero. For example, the integer \(5\) can be written as \(5.0\). Similarly, the integer \(-12\) can be written as \(-12.0\), and so on. Therefore, integers belong to the class of terminating decimal numbers, since they have a finite number of decimal digits, specifically zero.

• Non-terminating decimal numbers
  These have an infinite number of digits after the decimal point. They can be further subdivided into:
  • Repeating decimals: the decimal digits repeat according to a regular and predictable pattern, for example \(0.333...\). By convention, the repeating block of digits after the decimal point is written once, with a bar placed above it. For example, \(0.333... = 0.\overline{3}\). Such numbers are always rational, because any repeating decimal expansion can be expressed exactly as a fraction. For example: $$ 0.\overline{3} = \frac{1}{3}, \quad 7.25\overline{3} = \frac{725}{99} $$
  • Non-repeating decimals: the decimal digits do not follow any repeating pattern. In this case, the number is irrational, as in \( \pi = 3.14159...\). Non-terminating, non-repeating decimals are not rational numbers but instead define the class of irrational numbers. Examples include \( \pi = 3.14159...\) and the square root of \(2\), \( \sqrt{2} = 1.41421...\).

From this classification, it follows that every rational number admits a decimal representation, either terminating or repeating.

However, the converse is not true. Not all decimal numbers are rational, because non-repeating decimals are irrational.

Repeating decimals

Repeating decimals are decimal numbers whose digits after the decimal point repeat according to a fixed pattern. The block of digits that repeats is called the repeating period.

By convention, the repeating period is written only once, with a bar placed above the digits that repeat. For example:

$$ 0.252525... = 0.\overline{25} $$

Repeating decimals are traditionally divided into two main categories:

• Pure repeating decimals: the repeating period coincides with the entire decimal part, with no non-repeating digits preceding it. For example, \(0.666...\) is a pure repeating decimal with period \(6\): $$ 0.666... = 0.\overline{6} $$
• Mixed repeating decimals: the decimal part contains, before the repeating period, a block of non-repeating digits called the non-repeating part or antiperiod. For example, in the number \(7.25333...\), the antiperiod is \(25\) and the period is \(3\):  $$ 7.25333... = 7.25\overline{3} $$

This distinction between pure and mixed repeating decimals makes it possible to describe the structure of a decimal expansion with greater precision.

Relationship between fractions and decimal numbers

As stated earlier, all rational numbers can be written as decimal numbers, but the converse does not hold. Not all decimal numbers are rational.

Rational numbers are numbers that can be expressed as a ratio of two integers, \( \frac{a}{b} \), and every such fraction admits a decimal expansion.

A fraction written in lowest terms can give rise to four distinct types of decimal numbers:

• Integer: If the denominator is equal to \(1\), the result is an integer.

  Example: \( \frac{7}{1} = 7.0 \).

• Terminating decimal: When the denominator is composed exclusively of powers of \(2\) and or \(5\), the decimal expansion is finite.

  Example: \( \frac{7}{20} = \frac{7}{2^2 \cdot 5} = 0.35 \).

• Pure repeating decimal: If the denominator contains only prime factors other than \(2\) and \(5\), the resulting decimal expansion is purely repeating.

  Example: \( \frac{6}{21} = \frac{6}{3 \cdot 7} = 0.\overline{285714} \).

• Mixed repeating decimal: When the denominator includes \(2\) and or \(5\) together with other prime factors, the decimal expansion is mixed repeating.

  Example: \( \frac{7}{12} = \frac{7}{2^2 \cdot 3} = 0.58\overline{3} \). Here, the antiperiod is \(58\), which consists of two digits due to the factor \(2^2\), while the period is \(3\), which has one digit corresponding to the prime factor \(3\). The factors \(2\) and \(5\) in the denominator determine the number of digits in the antiperiod, whereas the remaining prime factors determine the length of the period. In this case, there are two factors of \(2\) in the denominator, so the antiperiod has two digits, \(58\). There is only one prime factor other than \(2\) and \(5\), so the period consists of a single digit, \(3\), which repeats indefinitely.

How to convert a repeating decimal into a fraction

To determine the fraction that generates a repeating decimal, the following formula can be applied: $$ \text{Generating fraction} = \frac{N - A}{D} $$ where:

• \( N \): the complete number written without the decimal point.
• \( A \): the digits that appear before the repeating period, also written without the decimal point.
• \( D \): the denominator, formed by as many \(9\)s as there are digits in the repeating period, followed by as many \(0\)s as there are digits in the non-repeating part.

Example

The pure repeating decimal \(0.\overline{3}\) is generated by the following fraction:

$$ \frac{N - A}{D} = \frac{3 - 0}{9} = \frac{3}{9} = \frac{1}{3} $$

The decimal number \(0.1\overline{23}\) is generated by the fraction:

$$ \frac{N - A}{D} = \frac{123 - 1}{99} = \frac{122}{990} = \frac{61}{495} $$

The decimal number \( 13.2\overline{31} \) has the following generating fraction:

$$ \frac{N - A}{D} = \frac{13231 - 132}{990} = \frac{13099}{990} = $$

In this case, the complete number without the decimal point is $ N = 13231 $, the digits preceding the repeating period are $ A = 132 $, and the denominator consists of as many \(9\)s as there are digits in the period (\(2\)), followed by as many \(0\)s as there are digits in the non-repeating part (\(1\)).

Therefore, the generating fraction of the number \( 13.231\overline{31} \) is \( \frac{13099}{990} \).

This procedure offers a clear and systematic way to recover the generating fraction of any repeating decimal.

The length of the decimal expansion

To determine the length of a decimal expansion, the fraction \( \frac{a}{b} \) is first written in lowest terms by simplifying both the numerator and the denominator as much as possible.

The next step is to examine the prime factorization of the denominator \( b \).

• If the denominator contains only the prime factors \( 2 \) and or \( 5 \)
  The decimal expansion is terminating. The number of digits in the decimal part is equal to the larger exponent of \( 2 \) or \( 5 \) appearing in the prime factorization of the denominator.

  Example. Consider the fraction $$ \frac{7}{40} = 0.175 $$ Here, the denominator factors as \( b = 2^3 \cdot 5 = 40 \). The largest exponent among the prime factors is \(3\), so the decimal expansion contains three digits (175).

• If the denominator contains prime factors other than \( 2 \) and \( 5 \)
  The decimal expansion is a pure repeating decimal. The length of the repeating period is determined by finding the smallest positive integer \( k \) such that $$ 10^k \equiv 1 \pmod{b} $$

  Example. Consider the fraction $$ \frac{6}{7} = 0.\overline{857142} $$ In this case, the denominator in lowest terms is \( b = 7 \). To determine the number of digits in the repeating period, one must find the integer $ k $ satisfying $$ 10^k \equiv 1 \pmod{7} $$ In practical terms, this amounts to identifying the smallest power of \(10\) that leaves a remainder of \(1\) when divided by \(7\).

  • For k = 1, $ 10^1 = 10 $, which leaves a remainder of 3 when divided by 7.
  • For k = 2, $ 10^2 = 100 $, which leaves a remainder of 2.
  • For k = 3, $ 10^3 = 1000 $, which leaves a remainder of 6.
  • For k = 4, $ 10^4 = 10000 $, which leaves a remainder of 4.
  • For k = 5, $ 10^5 = 100000 $, which leaves a remainder of 5.
  • For k = 6, $ 10^6 = 1000000 $, which leaves a remainder of 1. This is the desired value. Therefore, the repeating period consists of k = 6 digits. Indeed, the decimal expansion $ \frac{6}{7} = 0.\overline{857142} $ has a repeating block of six digits (857142).
     
• If the denominator contains both \( 2 \) and or \( 5 \) together with other prime factors
  The decimal expansion is a mixed repeating decimal. In this case, the finite part, known as the non-repeating part or antiperiod, is determined by the largest exponent of the prime factors \( 2 \) and or \( 5 \). The length of the repeating period is then computed as in the pure repeating case, considering only the prime factors other than \( 2 \) and \( 5 \).
  

  Example. Consider the fraction $$ \frac{7}{132} = 0.05\overline{30} $$ The denominator reduced to lowest terms is \( b = 44 \), which factors as \( b = 2^2 \cdot 11 \). To determine the number of digits in the non-repeating part, one identifies the largest exponent of the prime factors \(2\) or \(5\). In this case, the largest exponent is \(2\), so the non-repeating part has two digits (05). To determine the length of the repeating period, only the prime factors other than \(2\) and \(5\) are considered, which here reduce to \(11\). One then finds the smallest integer $ k $ such that $$ 10^k \equiv 1 \pmod{11} $$ This corresponds to the smallest power of \(10\) that leaves a remainder of \(1\) when divided by \(11\).

  • For k = 1, $ 10^1 = 10 $, which leaves a remainder of 10 when divided by 11.
  • For k = 2, $ 10^2 = 100 $, which leaves a remainder of 1. This is the desired value. Therefore, the repeating period consists of k = 2 digits. Indeed, the decimal expansion $ \frac{7}{132} = 0.05\overline{30} $ has a repeating block of two digits (30).

  More generally, when several prime factors are involved, the congruence $ 10^k \equiv 1 \pmod{m} $ can be analyzed separately for each prime factor of \( m \). One computes $$ 10^{k_1} \equiv 1 \pmod{p_1}, \quad 10^{k_2} \equiv 1 \pmod{p_2} $$ and then takes the least common multiple of the corresponding exponents: $$ k = \text{lcm}(k_1, k_2) $$ The final result is the same.

Significant digits in decimal numbers

The significant digits of a decimal number are the digits that convey meaningful and reliable information about its numerical value, excluding leading zeros that serve merely as placeholders.

Zeros that appear to the left of the decimal point, before any other significant digit, are not considered significant.

For example, writing 013.25 or 13.25 represents exactly the same number. In both cases, the significant digits are 13.25.

The leading zero is not significant, since it does not contribute any information about the magnitude of the number.

Note. Zeros that appear between significant digits or at the end of a number can be significant, because they indicate the level of precision or the result of rounding. For example, \(0.270\) has three significant digits, indicating a value rounded between \(0.265\) and \(0.274\).

When a number is written in scientific notation, the significant part consists of the digits that appear before the power of ten.

For example, \(3.17 \times 10^{-1}\) has a significant part equal to \(3.17\).

The number \(3.170 \times 10^{-1}\) has a significant part equal to \(3.170\), which includes the information conveyed by the trailing zero, as it specifies the precision of the value.

This representation makes it possible to express both the numerical value and its degree of accuracy or uncertainty with clarity and rigor.

And so on.