Integers and Rational Numbers

Positive and Negative Numbers

When representing temperature, we use as a reference point. For example, above zero is written as , and below zero is written as . When expressing quantities with opposite properties, such as “above and below zero,” “increase and decrease,” or “income and expense,” it is convenient to assign for one direction and for the other.
Here, is called the positive sign, and is called the negative sign. For instance, is read as “positive 5,” and is read as “negative 3.”
For numbers:
  • Numbers greater than are denoted with a positive sign ( ), and
  • Numbers less than are denoted with a negative sign ( ).
Examples:
Numbers with a positive sign (e.g., , ) are called positive numbers, while numbers with a negative sign e.g., , are called negative numbers. Note that is neither positive nor negative.

Integers and Rational Numbers

  • Natural numbers with a positive sign ( ) are called positive integers.
  • Natural numbers with a negative sign ( ) are called negative integers.
Integers include positive integers, , and negative integers. Positive integers are often written without the sign, making them identical to natural numbers.
  • Fractions with positive signs e.g., are called positive rational numbers.
  • Fractions with negative signs e.g., are called negative rational numbers.
Rational numbers include positive rational numbers, , and negative rational numbers. Positive rational numbers, like positive integers, can be written without the sign.
Since integers can be expressed as fractions (e.g., ), all integers are rational numbers.

Classification of rational numbers:

Number Line

Integers can be represented on a straight line.
On this line, designate a reference point as , called the origin. Assign positive integers ( ) to points at equal intervals to the right of and negative integers ( ) to points at equal intervals to the left of . This line is called the number line.
All rational numbers can be represented as points on the number line. For example, the points for , , and are shown on the number line as follows: