Decimal numbers
Decimal numbers are used to represent quantities that aren’t necessarily whole numbers. Each decimal number consists of an integer part and a decimal part, both of which can also be zero.
This means a decimal number can take different forms depending on its components.
For instance, consider the number \( 123.45 \):
- The integer part is \( 123 \), which represents the digits to the left of the decimal point. This part accounts for the "whole" number of units, tens, hundreds, and so on.
- The decimal (or fractional) part is \( 45 \), which represents the digits to the right of the decimal point. It indicates a fraction of the whole number, representing values smaller than 1.
The decimal point (or, in some countries, a comma) separates these two parts.
The decimal part is particularly useful for expressing fractional quantities such as $ \frac{a}{b} $ where $ b \neq 0 $.
For example, \( 0.5 \) represents half of 1 (or \( \frac{1}{2} \)), while the decimal number \( 0.25 \) represents one-fourth of 1 (or \( \frac{1}{4} \)).
Decimal numbers are generally categorized into two main types:
- Finite decimal numbers
These are numbers with a decimal part that ends after a specific number of digits. They result from straightforward divisions where the outcome doesn’t generate an infinitely repeating remainder. Finite decimals include whole numbers, which have no decimal part (e.g., 1, 2, 3, ...), or fractions $ \frac{a}{b} $ in their simplest form, where the denominator is made up solely of the prime factors 2 and/or 5.For example, $ \frac{3}{4} = 0.75 $ is a finite decimal because the division $ 3:4=0.75 $ results in a quotient with no remainder. Note that the denominator of the fraction (4) is a multiple of 2. The number of decimal places corresponds to the highest exponent of the prime factors 2 and/or 5 in the denominator. In this case, the highest exponent is 2 because the denominator is (4 = 22), so the decimal part has two digits.
- Recurring decimal numbers
These are numbers with a decimal part that repeats endlessly in a regular pattern. The repeating sequence of digits is known as the period. Recurring decimals fall into two categories:- Pure recurring decimals: These have a period that begins immediately after the decimal point. They are generated by fractions in their simplest form where the denominator is made up of prime factors other than 2 and 5.
For instance, the fraction $ \frac{17}{9} $ produces a pure recurring decimal because the denominator consists of factors other than 2 and/or 5: $$ \frac{17}{9} = \frac{17}{3^2} = 1.8888888... = 1.\overline{8} $$ The period, which is the repeating sequence of digits, has a length equal to the number of digits in the largest prime factor of the denominator. In this case, the largest prime factor of 9 is 3, which has one digit. Therefore, the period consists of a single digit: 8.
- Mixed recurring decimals: These have a period that is preceded by a finite sequence of digits called the pre-period. They are generated by fractions in their simplest form where the denominator includes both the prime factors 2 and/or 5 and other primes.
For example, the fraction $ \frac{17}{6} $ produces a mixed recurring decimal because the denominator contains both factors of 2 and/or 5 and other prime numbers: $$ \frac{17}{6} = \frac{17}{2 \cdot 3} = 2.833333... = 2.8\overline{3} $$ Here, the pre-period is 8, and the period is 3. The number of digits in the pre-period is determined by the highest exponent of the prime factors 2 and/or 5 in the denominator. In this case, the highest exponent is 1 (21), so the pre-period consists of one digit: 8.
- Pure recurring decimals: These have a period that begins immediately after the decimal point. They are generated by fractions in their simplest form where the denominator is made up of prime factors other than 2 and 5.
Decimal numbers are invaluable for representing both precise whole numbers and rational numbers (fractions).
