cs2010 Notes: 09-19 Base Conversions

We’ve been working with binary numbers for a bit. Let’s look explicitly at how to convert between different numerical bases.

Review: Digit places

Base 10: 1’s, 10’s, 100’s, etc.

Base 2: 1’s, 2’s, 4’s, 8’s, 16’s, etc.

Base 4: 1’s, 4’s, 16’s, 64’s, etc.

Base 16: 1’s, 16’s, 256’s, etc.

Converting from base 10 to base 10

base 10 (3572), in base 10 is

2*1 + 7*10 + 5*100 + 3*1000 = 3572

Converting from base 2 to base 10

base 2 (100101), in base 10 is

1*1 + 0*2 + 1*4 + 0*8 + 0*16 + 1*32 = 37

Converting from base 16, hexidecimal, to base 10

Hex (4A37) to base 10:

7*1 + 3*16 + (A=10)*256 + 4*(16^4) = 18999

Converting from base 10 to base 2

base 10 (23) in base 2 is

(just do the arithmetic in the target base, this time base 2)

3*1 + 2*10

=

11*1 + 10*1010

=

11 + 10100 = 10111

Converting from base 10 to base 2, plan b

x = 23
while x > 0:
  if odd?(x):
    write down last digit = 1
  else:
    write down last digit = 0
  x = x / 2 (round down)

• 23 is odd => 1
• 11 is odd => 1
• 5 is odd => 1
• 2 is even => 0
• 1 is odd => 1
• 0 is zero, done

result: 10111

Converting from base 10 to base 16

while x > 0
  divide by 16
    - take remainder as last digit
    - take quotient as new x

decimal input = 18999

digits from remainder column, reversed: 4A37