The Number System

  • The Number System

The Number System

The number system is divided into two main categories: Real Numbers and the Imaginary Number.

Roman Numeral Round Analog Clock
The number system diagram

Positive Numbers $(N)$

The counting numbers 1, 2, 3, ... are called the positive numbers. Sometimes they are also called Natural or Counting Numbers

$$ N = \{1,2,3,...\} $$

Integers $(I)$

The integers consists of natural numbers, zero, and the negatives of natural numbers.

$$ I = \{...,-3,-2,-1,0,1,2,3,...\} $$

Rational Numbers $(Q)$

A number is a rational number if it is a ratio of whole numbers which if expanded in decimal form is terminating or non-terminating and non-repeating.

$$\begin{align} & \text{ Q = \{x|x can be expressed as $\frac{a}{b}$, where a and b are integers and b $\neq$ 0 \} } \\ & \text{ Q = \{x|x is a repeating, non-terminating decimal expansion \} } \\ & \text{ Q = \{x|x is a terminating decimal \} } \\ \end{align} $$

Examples:

  1. $ Q= \{ 1, 2, 1/2, -5 \} $
  2. $ 2 \frac{1}{2} = 2.5 = 2.5000 $
  3. $ \frac{1}{3} = 0.33333.. = 0.\bar{3} $
  4. $ \frac{199}{333} = 0.597597597.. = 0.\overline{597} $

Irrational Numbers $(Q')$

Irrational numbers are numbers that can't be expressed as a fraction or quotient of two integers. They are also non-terminating and non-repeating in decimal form.

$$\begin{align} & \text{ $Q'$ = \{x|x can not be expressed as $\frac{a}{b}$ \} } \\ & \text{ $Q'$ = \{x|x is a non-repeating, non-terminating decimal expansion \} } \end{align} $$

Examples: $ \pi, \sqrt{2}, e $

Imaginary Number $(i)$

The imaginary number is a number of the form b$i$ where b is a real number and $i = \sqrt{-1}$.

  • $-i = \frac{1}{i}$
  • $i^{-1} = -i$
  • $i^2 = -1$
  • $i^3 = -i$
  • $i^4 = 1$

Complex Numbers $(C)$

A complex number is a number composed of Real and Imaginary numbers in the form: $a+bi$, where a and b are real numbers. a is called the real part and b$i$ is called the imaginary part.

If $a=0$, then the number is purely imaginary. If $b=0$, then it is a purely real number.

Examples:

  1. $ 3+i $
  2. $ \frac{4+\sqrt{2}i}{3} $
  3. $ 5-\frac{3i}{4} $