Introduction to Chapter 2.

• This is chapter 2. You should read it.
• You may have figured this from the previous chapter, but
  • The mathematical foundations of computer science are strong.
  • Even at the lowest level, we use mathematics in computing.
  • A number system that employs the idea of on/off will probably come into play
• We need to distinguish numbers systems in many programming applications.
  • The integers are the numbers
    • 1, 2, 3, ...
    • and -1, -2, -3 ...
    • And 0
  • We frequently limit the integers we use because
    • We can not represent infinitely different numbers.
    • And there are infinitely different integers.
    • So we generally only represent a range of numbers.
    • From a minimum to a maximum.
    • Think a scoreboard.
  • The other numbers we frequently represent we call floating point numbers
    • These are a subset of the rational numbers.
    • Think numbers with a decimal point.
    • But again, we can only represent so many digits.
  • We will discuss both of these more in chapter 3, and in programming again and again.
• We need to discuss positional number systems.
  • In our (human) history we have had many different ways to represent numbers.
  • Today we mostly use a base 10 positional number system.
  • This will go back to middle school or earlier.
• What does 437 mean?
  • For example if I have 437 dollars.
  • I really have 400 dollars + 30 dollars + 7 dollars.
  • The number 437 can be expressed as $4\times10^2+3\times10^1+7\times10^0$
  • Starting from the right, we have
    • Single units.
    • Then tens units
    • Then 100s units
    • And so on.
  • The position is important because that tells us what to multiply the digit by to get the number of units it represents.
  • Since we are raising 10 to different powers this is a base 10 number system.
    • Also known as the decimal system.
  • Note in base 10, we use 10 digits, 0, 1, 2, 3, 4, ... , 9 to represent all possible positive numbers.
  • This is a convenient and quick way to represent numbers.
• We will study three bases other than base 10
  • We will discuss in detail base two or binary.
  • We will use base 8 or octal as one shorthand for binary.
  • We will use base 16 or hexadecimal as another shorthand for binary.
• In the end our goals for this chapter are
  • Distinguish between integers and floating point numbers.
  • Describe a positional number system.
  • Convert numbers from other bases to base 10
  • Convert numbers from base 10 to other bases.
  • Describe and use base 2, 8 and 16