Understanding Rational Numbers

• Rational numbers are numbers that can be expressed in the form p/q, where p and q are integers and q ≠ 0.

Definition and Types of Rational Numbers

• A rational number is any number that can be written as a fraction p/q, where p and q are integers and q is not zero.
• Integers include all whole numbers and their negatives (e.g., -3, -2, -1, 0, 1, 2, 3).
• Natural numbers are positive integers (e.g., 1, 2, 3...).
• Whole numbers include all natural numbers and zero (e.g., 0, 1, 2, 3...).
• Fractions are numbers representing a part of a whole and are rational.
• Terminating decimals are rational numbers that have a finite number of digits after the decimal point (e.g., 0.25).
• Repeating decimals are rational numbers with a pattern of digits that repeat indefinitely (e.g., 0.333...).
• Integers are rational numbers and can be written with a denominator of 1 (e.g., 5 = 5/1).

Operations: Addition and Subtraction

• To add or subtract rational numbers with a common denominator, add or subtract the numerators and keep the denominator the same: a/c + b/c = (a+b)/c and a/c - b/c = (a-b)/c.
• To add or subtract rational numbers with different denominators, find a common denominator: a/b + c/d = (ad + bc)/bd.
• The additive inverse of a rational number a/b is -a/b, such that a/b + (-a/b) = 0.

Operations: Multiplication and Division

• To multiply rational numbers, multiply the numerators and the denominators: (a/b) * (c/d) = (ac)/(bd).
• To divide rational numbers, multiply by the reciprocal of the divisor: (a/b) ÷ (c/d) = (a/b) * (d/c) = (ad)/(bc), where c ≠ 0.
• The multiplicative inverse (reciprocal) of a rational number a/b is b/a, such that (a/b) * (b/a) = 1.

Representation on the Number Line

• A number line is a straight line on which numbers are marked at intervals.
• To represent a rational number on the number line, divide the segment between two integers into the number of equal parts indicated by the denominator.
• Locate the point corresponding to the numerator. For example, to represent 1/4, divide the segment between 0 and 1 into four equal parts and mark the first part.
• Negative rational numbers are represented to the left of zero. For example, -1/4 is located to the left of 0, one-fourth of the way to -1.

Comparison and Ordering

• To compare rational numbers, convert them to fractions with a common denominator. Then, compare the numerators.
• If a/b and c/d are two rational numbers, and bd > 0, then:
  • If ad > bc, then a/b > c/d.
  • If ad < bc, then a/b < c/d.
  • If ad = bc, then a/b = c/d.
• Rational numbers can be ordered from least to greatest.
• Between any two distinct rational numbers, there are infinitely many rational numbers. One way to find a rational number between two given rational numbers a and b is to calculate their average: (a+b)/2.