Let's Talk Numbers

Objectives

1. Review how we deal with numbers in real life.

Notes

• A quick review of decimal.
  • What are the decimal digits?
  • What do these mean?
  • Count to 15, starting at 0, increasing by 1 each time.
  • What happened after 9 and why?
    • You ran out of symbols!
    • Is 10 a new symbol or a combination of the basic digits?
    • Could you have invented a new symbol for ten items?
    • What would be the problem with a new symbol?
  • Why are the words "ten", "eleven" and "twelve" problematic?
    • Japanese
    • ichi, ni, san, shi, go, roku, shichi, hachi, ku, ju
    • ju-ichi
    • ju-ni
    • ju-san
    • ... niju
    • niju-ichi
• We represent these numbers with the digits 0-9, but do we need to?
  • We probably do this because we have 10 fingers.
  • Can you count to 10 on your fingers?
  • Can you count to 37 on your fingers?
  • Take a look at this page
    • In this system we are encoding the digits differently
      • Let f be a finger
      • Let t be a thumb.
      • Let 0 be a hand with noting down
      • What is the number:
        • 0 f - no tens and 1 ones
        • 0 ff - no tens and two ones
        • 0 fff - no tens and three ones
        • 0 ffff - no tens and four ones
        • 0 t - no tens and five ones
        • 0 tf - no tens, and six ones
        • f 0 - one ten and no ones
        • f f - one ten and one one.
      • Do we need both a 0 and a space?
        • What does ff mean? (0ff or ff)
• Decimal is called base 10, because are counting in powers of 10.
  • What does 387 mean?
    • The positions are the key
    • things to the far right mean just digit or symbol.
      • so 387 has seven of something.
      • But for consistency sake, what is $10^0$?
      • We will then write the seven of something as $7x10^0$
    • What does the 8 mean in 387?
      • You should say 80.
      • since it is in the first position from the right it means $8x10^1$.
  • What does 8694 mean?
  • What does ab mean if a,b ∈ {0, 1, 2, ... 9};
    • $ax10^1 + bx10^0$
  • Generalize this for $a_{n-1}a_{n-2}...a_1a_0$
    • $a_{n-1}x10^{x-1} + a_{n-2}x10^{n-2} ... a_1x10^1 + a_0x10^0$
• Let's look at the hand counting system above.
  • ff tff = ?
  • Is each group a coefficient for a power of 10? (yes)
• Our normal number system is a position based number system.
  • is 387 the same as 783? No, the position indicates the power.
• So our normal system is a base 10 number system.