Base conversions #

Last time

• Binary <=> Hex with a lookup table.

Today

• Decimal <=> Binary
• Hex <=> Decimal

Plan A:

• You can pull out the least significant digit by doing an integer division, 4th-grade style, by the new base.
• The remainder is the least significant digit.
• The (integer) quotient contains the remaining digits, continue until done.

Example: Convert 19 to binary

• 19 / 2 = 9, remainder 1
• 9 / 2 = 4, remainder 1
• 4 / 2 = 2, remainder 0
• 2 / 2 = 1, remainder 0
• 1 / 2 = 0, remainder 1
• so 19 in binary is 10011 = 16 + 2 + 1

Plan B:

• You can pull out the most significant digit by dividing by the largest power of the base less than the input.
• The (integer) quotient is the most significant digit
• The remainder contains the remaining digits.
• Continue with the next lower power of base until done, zero quotients are zero digits.

Example: Convert 19 to binary

• 32 is too big, so 16 is the largest power of 2 less than 19.
• 19 / 16 = 1, remainder 3
• 3 / 8 = 0, remainder 3
• 3 / 4 = 0, remainder 3,
• 3 / 2 = 1, remainder 1
• 1 / 1 = 1
• So 19 in binary is 10011
• For binary, this might be easier to think of as “can we subtract out one of the power of 2”, but division handles any base.

Plan C: Just do the implicit arithmetic in the target system.

• Convert hex 3A to decimal
• That’s (3)(16) + (10)(1) = 58

Example: Convert 50 to hexidecimal

Plan A:

• 50 / 16 = 3, remainder 2
• 3 / 16 = 0, remainder 3
• 50 decimal is 32 hex

Plan B:

• 256 is too big, so start with 16
• 50 / 16 = 3, remainder 2
• 2 / 1 = 2, remainder 0
• 50 decimal is 32 hex

Example: Convert binary 110011 to decimal

Plan A:

• 110011 / 1010
• Let’s use repeated subtraction
• Ends up being 5, remainder 1
• 5 < 10, so 10’s place is 5.

Example: Convert hex 3FA9 to decimal

• (3)(4096) + (15)(256) + (10)(16) + 9

Binary Arithmetic #

Addition

Subtraction

Multiplication

Negative Numbers

Plan A: Sign bit

Plan B: Two’s complement

• Flip all bits
• Add one (ignore overflow)
• This flips the sign of a signed integer