Rational numbers

Here’s an explanation of rational numbers, presented in a notes format with emojis for easy understanding:

Rational Numbers (Q) 💭

• What are they?

  • Rational numbers are numbers that can be written as a ratio of two integers.
  • They are expressed in the form p/q ➗, where ‘p’ and ‘q’ are both integers.
  • The top number, ‘p’, is called the numerator ⬆️, and the bottom number, ‘q’, is called the denominator ⬇️.
  • The set of rational numbers is denoted by the symbol Q.
    • 💡 The ‘Q’ stands for quotient, linking to the idea of a ratio.

• Extending from other number sets

  • Rational numbers extend natural numbers and integers.
  • Every integer is also a rational number because it can be written with a denominator of 1 (e.g., 7 can be written as 7/1).

• Representation is not unique 🤔

  • Unlike integers, the same rational number can be written in many different ways.
  • For example, 3/5 is the same as 6/10, and 30/50.
  • You can get equivalent fractions by multiplying both the numerator and the denominator by the same quantity.
  • This property is extremely useful for arithmetic operations like addition and subtraction, and for comparing two fractions, as it allows you to convert them to equivalent fractions with the same denominator.

• Reduced Form (Canonical Form) 🧩

  • To have a “best” or unique way to represent a rational number, it’s often written in its reduced form.
  • In reduced form, the numerator and denominator have no common factors other than 1.
  • This means their Greatest Common Divisor (GCD) is 1.
  • Prime factorization can be used to find the GCD and reduce a fraction. For instance, 18/60 reduces to 3/10 because the GCD of 18 and 60 is 6.

• Density Property 🌌

  • Unlike integers and natural numbers, which are “discrete” (meaning there’s a clear “next” or “previous” number with nothing in between, e.g., between 2 and 3, there are no other integers), rational numbers are dense.
  • This means that between any two distinct rational numbers, you can always find another rational number.
  • A simple way to find a rational number between two others is to take their average (sum divided by 2).
  • Because of this density, you cannot talk about a “next” or “previous” rational number.

• Cardinality (Size of Infinity) 📏

  • Despite being dense and seeming “larger” than integers, the set of rational numbers actually has the same “size” or cardinality as the set of integers (and natural numbers).
  • This counter-intuitive fact is demonstrated by finding a bijection (a one-to-one and onto mapping) between them, showing that rational numbers can be “counted” or enumerated, even though they are infinite.