Number Systems: Base Ten

• We use the decimal number system
  • This number system uses base 10
  • Another word for base is radix
  • And the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
• This is a positional number system
  • The position of the digit in a number has meaning
  • IE the number 42 and the number 24 are different.
• In this system
  • The leftmost digit is called the Most Significant Digit (MSD)
  • The rightmost digit is called the Least Significant Digit (LSD)
  • In the number 7356
    • 7 is the most significant digit
    • 6 is the lest significant digit
• Each digit in a number has an associated weight.
  • The LSD has a weight of 1, actually $10^0$
    • The 0 is because we say the LSD is at position 0
    • The 10 is because we are in base 10.
    • (Remember $a^0 = 1, a \neq 0$)
  • For numbers to the left of the LSB, the position increases by one each time we move to the left.
  • In the number 7356
    • 6 is at position 0 and has weight $10^0 = 1$
    • 5 is at position 1 and has weight $10^1 = 10$
    • 3 is at position 2 and has weight $10^2 = 100$
    • 7 is at position 3 and has weight $10^3 = 1000$
  • So the number 7356 is really $7\times10^3 + 3\times10^2 + 5\times10^1 + 6\times10^0$
  • What are the digits past the third position?
• This works for numbers to the right of the decimal point as well.
  • Consider 6.742
  • 6 is at position 0 with weight $10^0$
  • 7 is at position -1 with weight $10^{-1}$
  • 4 is at position -2 with weight $10^{-2}$
  • 2 is at position -3 with weight $10^{-3}$
  • And the number is $6\times10^0 + 7\times10^{-1} + 4\times10^{-2} + 2\times10^{-3}$
• Why do you think we use base 10?
• Exercise: For a given number
  1. Give the MSD and LSD
  2. provide the polynomial in terms of base and powers that the number represents.