real and complex numbers

Here’s an explanation of real numbers and complex numbers in a notes format, designed for ease of understanding with emojis:

Real Numbers (R) 🌍

• Real numbers are an expansion of rational numbers and fill up the entire number line 📏, including all the “gaps” that rational numbers leave.
• They are denoted by the symbol R.
• What fills the gaps? Irrational Numbers 💫
  • Irrational numbers are those that cannot be written as a simple fraction p/q, where p and q are integers. They are simply numbers that are not rational.
  • A classic example is the square root of 2 (√2). You can physically draw a line segment of length √2 (e.g., the hypotenuse of a square with sides of length 1). However, it cannot be precisely expressed as a ratio of two integers. This fact was known to ancient Greeks like Pythagoras, and its irrationality was reportedly proved by his follower Hippasus around 500 BCE, shocking the Pythagoreans who believed rational numbers formed the basis of all science.
  • In general, the square root of any integer that is not a perfect square (e.g., √3, √5, √6) is an irrational number.
  • Other well-known irrational numbers include pi (π) (the ratio of a circle’s circumference to its diameter) and e (used in natural logarithms). These numbers have infinite non-repeating decimal expansions.
• Density Property 🌊
  • Just like rational numbers, real numbers are dense: you can always find another real number between any two distinct real numbers (for example, by taking their average). This means there are no “gaps” in the real number line.
• Relationship to other Number Sets 🌳
  • Every natural number is an integer, every integer is a rational number, and every rational number is a real number.
  • The set of natural numbers (N) is a subset of integers (Z).
  • The set of integers (Z) is a subset of rational numbers (Q).
  • The set of rational numbers (Q) is a proper subset of real numbers (R). This means that while all rational numbers are real numbers, there are real numbers (the irrationals) that are not rational.
  • This hierarchical relationship can be visualized using Venn diagrams, where N is the innermost circle, followed by Z, then Q, and finally R as the largest encompassing circle.
• “Size” of Infinity ✨
  • Even though rational numbers are dense, the set of real numbers has a larger “size” or cardinality of infinity than the set of natural numbers, integers, or rational numbers. This implies there are vastly more irrational numbers than rational numbers.

Complex Numbers (C) 🌌

• The Need for Expansion 🚧
  • When dealing with operations like square roots of negative numbers, the existing real number system falls short.
  • For instance, if you try to find the square root of -1 (√-1), you cannot find a real number that, when multiplied by itself, yields a negative result. This is because the rule for multiplication of signs states that if two numbers have the same sign (either both positive or both negative), their product is always positive.
  • This limitation is also seen when solving quadratic equations: if the discriminant (b² - 4ac) is less than 0, it means you’d be taking the square root of a negative number, which implies no real solutions.
• Introducing Complex Numbers ✨
  • To allow for the square roots of negative numbers, a new class of numbers called complex numbers was created.
  • Complex numbers extend the real number system.
• Symbol ℂ
  • While the provided sources do not explicitly state a symbol for complex numbers, they follow a pattern of using single letters (N, Z, Q, R) for other number sets. Complex numbers are commonly denoted by C (or ℂ). This information is not directly from the provided sources and you may want to independently verify it.
• Beyond this Course 📚
  • The provided sources indicate that the study of complex numbers is generally beyond the scope of this particular course.