Introduction to Number Systems

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What is a Number System?

• A number system is a system at which we count or use numbers.

• For instance, the way we count with our fingers is by the decimal number system.

  • This is when each number is represented by combinations of the 10 symbols we use for counting: .
  • This is the most intuitive number system as it is based on what most people have: our ten fingers.

• To further illustrate the idea of a number system, let us look at some other common number systems in computer science.

Common Number Systems in Computer Science

• We commonly use 3 number systems in computer science: binary, octal, and the hexadecimal number system.

Binary

• A computer is made up of thousands or millions of tiny switches that can turn on or turn off.
  • Any combination of states of these switches can represent information, such as numbers, text, or even visual and auditory information.
  • Since they have two states: on and off, we can then represent them as either s or s.
  • Therefore, any information is then processed as long strings of s and s.
• This means that a computer utilizes a binary counting system, which only consists of numbers and .

Binary Counting

• To start counting, we start from zero, then add , to get the next successive number $$
  \array{rlcl}{
  \fn{Decimal:} & 0 &\stackrel{+1}{\longrightarrow} & 1\
  \fn{Binary:} & 0 &\stackrel{+1}{\longrightarrow} & 1
  }
• Since we only use and , adding to in binary results in , but instead, we move to the next place value.

• Continuing this pattern, we can get a table of values for converting numbers between the decimal and the binary number system.