Understanding Fractions and Decimal Expressions

Fractions and Their Classifications

A fraction is a ratio of two integers.

Classification

• Proper: The numerator is smaller than the denominator, representing a number less than 1.
• Improper: The numerator is larger than the denominator, representing a number greater than 1.
• Apparent: The multiple of the denominator and numerator represents an integer.
• Mixed Numbers: These have a whole number part and a fractional part. They arise from improper fractions. To convert an improper fraction to a mixed number, divide the numerator by the denominator. The quotient is the whole number, the remainder is the new numerator, and the denominator stays the same.

Decimal Expression

A decimal expression is the result of dividing the numerator by the denominator of a fraction.

Decimals: Accurate and Regular Expressions

These emerge to resolve the division that represents a fraction.

• Exact: The remainder of the division is ZERO.
• Recurring: The remainder is never zero.

Classification of Recurring Decimals

• Mixed: Possess a non-recurring decimal period before the recurring part (indicated by an arc above the numbers).
• Pure: The period begins immediately after the decimal point.

Generating a Fraction from a Decimal Number

Generating a fraction from a decimal number is the fractional expression of that decimal number.

• Exact Decimals: The numerator is the integer part followed by the decimal digits without the comma. The denominator is a 1 followed by as many zeros as there are decimal places.
• Recurring Decimals (Mixed): The numerator is the subtraction of the whole number formed by the integer and decimal parts from the integer part. The denominator consists of as many nines as there are recurring digits, followed by as many zeros as there are non-recurring decimal digits. Simplify the fraction afterward.
• Recurring Decimals (Pure): The numerator is the integer part followed by the non-recurring decimal and the period minus the integer part followed by the non-recurring decimal. The denominator consists of as many nines as there are digits in the period.

Scientific Notation

Scientific notation is a concise way of representing numbers using powers of base ten. Numbers are written as a product: a × 10ⁿ, where a is a number greater than or equal to 1 and less than 10, and n is any integer. This notation is used to easily express very large or very small numbers.