Introduction to Binary

• This is Jorgensen chapter 3 as well as Carter chapter 1
• In base 10, we use the digits 0-9
• And the position of the digit lets us know the weight of the digit.
  • $582 = 5\times 10^2 + 8\times10^1 + 2\times10^0$
• Ok, I got this but really?
• Let's discuss simple addition
  • You have memorized that 3 + 1 = 4
  • 0 + 1 = 1 , 1 + 1 = 2, 2 + 1 = 3, ... ,8 + 1 = 9
  • Note each time we add one to the symbol, we know which symbol to go to next.
  • But when we get to 9 + 1 we are out of digits.
  • So we add a new rule, 9 + 1 = 10
  • Ten 10 is constructed from the digits 1 and 0
• Again in base 10, we have 10 digits (or symbols) 0-9, and a base or radix of 10
  • We do not have a special digit for 10, we construct it out of our other digits and position.
  • We probably use base 10 because we have 10 fingers.
• why is 25 + 39 = 64?
  • 25 is $2\times10^1 + 5 $
  • 39 is $3\times10^1 + 9$
  • When we add 5 + 9 we get $1\times10^1$ + 4 or a ones digit of 4
  • When we add $2\times10^1 + 3\times10^1 + 1\times10^1$
  • we get $6\times10^1$
  • So the answer is 64
• In general $d_5d_4d_3d_2d_1d_0 = d_5\times10^5+d_4\times10^4+d_3\times1^3+d_2\times10^2+d_1\times10^1+d_0\times10^0$
  • The digit determines the position multiplier
  • The position of the digit, from the right starting at 0, determines the weight of the value.
• There is no reason to use base 10.
  • The Mayans used base 20 and the Sumerians used base 60
  • In computing we use base 2 or binary.
• Binary
  • The digits are {0,1}
  • The base or radix is 2.
  • $101101_2 = 1\times2^5+0\times2^4+1\times2^3+1\times2^2+0\times2^1+1\times2^0$
• At this point, it is probably reasonable to memorize the powers of 2 up to $2^{10}$
• And the binary patters for 0-15
• Converting From Binary to Decimal
  • To convert $101101_2$ to base 10
  • Write the expansion and work it out.
    • $101101_2 = 1\times2^5+0\times2^4+1\times2^3+1\times2^2+0\times2^1+1\times2^0$
    • $= 1\times32 + 0\times16+1\times8+1\times4+0\times2+1\times1$
    • $= 32+8+4+1 = 45$
  • It is reasonable to check with an on-line calculator
• To convert decimal to binary
  • Constantly divide by 2, noting the remainder.
  • Convert 243 to binary

      2|243
      2|121 R 1
       2|60 R 1
       2|30 R 0
       2|15 R 0
        2|7 R 1
        2|3 R 1
          1 R 1
            

  • Read the remainders backwards
  • $243 = 11110011_2$