Math 467 Short Note on Base Conversions

Short Note on Base Conversions Is there a convenient method for converting decimal to binary? Here is one technique: For the integer part: Example- Convert 49 to binary. Divide by powers of two 49/2 24 r 1 24/2 12 r 0 12/2 6 r 0 6/2 3r0 3/2 1r1 1/2 0r1 Now read the remainders from bottom to top: (49)10 = (110001)2 For the fractional part: Multiply by 2, take the remaining fractional part. For example, convert 0.7 into base 2: 0.7 0.4 0.8 0.6 0.2 0.4 * * * * * * 2 2 2 2 2 2 1.4 0.8 1.6 1.2 0.4 0.8 At this point, we see that this will repeat- Our number, base 2 is (read from top to bottom): 0.1011001100110011... To go from any base to base 10, it’s probably easiest to do the straight conversion. For example,


(110001)2 = 1 × 20 + 1 × 24 + 1 × 25 = 1 + 16 + 32 = 49 For repeating fractional parts, we could convert using a geometric series. Recall that: ∞ X 1 1+ rn = 1 − r n=1 Example: Convert the base 4 representation: 0.03030303... into base 10 (0.03030303...)4 = 3 ∞  2n X 1 n=1 4 =3  n  ∞  X 1 1 1 − 1 = =3 1 5 16 1 − 16 n=1 1