A rational number is any number that can be expressed as a fraction p over q, where p is an integer called the numerator, and q is a non-zero integer called the denominator. Examples include one half, three fourths, negative five sevenths, eight over one which equals eight, and zero over three which equals zero.
Rational numbers can be classified into three types. First, positive rational numbers are greater than zero, such as two thirds, seven fourths, and nine over one. Second, negative rational numbers are less than zero, like negative one half, negative three fifths, and negative eight thirds. Third, zero itself is a rational number that is neither positive nor negative, and can be written as zero over any non-zero integer.
Every rational number has a specific position on the number line. For example, negative three halves is at negative one point five, negative one half is at negative zero point five, three fourths is at zero point seven five, and so on. An important property is that between any two rational numbers, there are infinitely many other rational numbers. This makes rational numbers dense on the number line, and they can always be ordered from smallest to largest.
Rational numbers follow specific rules for arithmetic operations. For addition and subtraction, we find a common denominator and then add or subtract the numerators. For example, one third plus two fifths equals five fifteenths plus six fifteenths, which equals eleven fifteenths. For multiplication, we multiply numerators together and denominators together. Two thirds times four fifths equals eight fifteenths. For division, we multiply by the reciprocal of the divisor. Three fourths divided by two sevenths equals three fourths times seven halves, which equals twenty-one eighths.
To summarize what we have learned about rational numbers: They are any numbers that can be expressed as fractions where the numerator is an integer and the denominator is non-zero. Rational numbers include positive numbers, negative numbers, and zero. They are dense on the number line, meaning between any two rational numbers there are infinitely many others. They follow specific rules for arithmetic operations, and understanding rational numbers provides an essential foundation for advanced mathematics.