Introduction to Number Systems

Reading

Sections 1.1, 1.2

Practice Problems

1.1

5, 8, 9, 10, 11

1.2

1, 2, 3, 9

Notes

First a short survey of number systems.

• We all know the natural numbers: 1, 2, 3, 4, ... Start from 1, and add it into itself over and over. These will be the main focus of this course.
• By including a zero and additive inverses, we get negative integers: -1, -2, 0
• By taking quotients of those we form rational numbers: \(2/3\), \(-42/11\)
• In Calculus we learn about the real numbers: \(e\), \(\pi\), \(\sqrt{2}\)
• One of the fundamental results is that there are many real numbers that are not rational (irrational numbers). In fact almost every real number is not rational.
• Complex numbers have the form: \(a + bi\) where \(i = \sqrt{-1}\) represents a "square root of -1". They are an extension of the reals.
  • We can define addition and multiplication of complex numbers by extending the properties for reals.
  • There is a zero, \(0 + 0i\).
  • Cool fact: Every polynomial equation has solutions in the complex numbers. For example \(x^2 + 1 = 0\) has solutions \(i\) and \(-i\).
• There is a special subset of the complex numbers, called Gaussian integers. These consist of all complex numbers \(a+bi\) where \(a\) and \(b\) are both integers.

• The standard number systems:

  \(\mathbb{N}\)

  The natural numbers: \(\mathbb{N} = \{ 1, 2, 3, \ldots \}\)

  \(\mathbb{Z}\)

  The integers: \(\mathbb{Z} = \{ \ldots, -2, -1, 0, 1, 2, 3, \ldots \}\)

  \(\mathbb{Q}\)

  The rational numbers: \(\mathbb{Q} = \left\{ \frac{a}{b} \mid a, b\in \mathbb{Z}, b\neq 0 \right\}\)

  \(\mathbb{R}\)

  The real numbers. Precisely defining them is more difficult.

  \(\mathbb{C}\)

  The complex numbers: \(\mathbb{C} = \{ a + bi \mid a, b\in\mathbb{R} \}\)