2.2 Binary, Octal, Decimal, and Hexadecimal Conversion

The previous section defined a numeral as a sum of digit-times-weight contributions. Conversion does not change the value; it changes only the symbols used to describe that value.

For example:

All four numerals identify the same quantity.

Choose a conversion route

Different source and destination bases suggest different tools.

Route 1: expand a numeral into decimal

Use the place-value definition directly:

The hexadecimal digit D contributes thirteen times its weight, not the decimal digit sequence 13.

Horner's method performs the same calculation from left to right:

This form is useful in programs because it repeatedly multiplies the running total by the base and adds the next digit.

Route 2: group bits for octal and hexadecimal

Because $8=2^3$, exactly three binary bits describe one octal digit. Because $16=2^4$, exactly four bits describe one hexadecimal digit.

For hexadecimal, group from the radix point outward in sets of four:

Therefore $(101001011100)_2=(A5C)_{16}$.

For octal, regroup the same bits in threes:

Therefore $(101001011100)_2=(5134)_8$.

If the leftmost group is short, pad it with leading zeros. Padding does not change the value.

Lab 1 — translate one bit pattern three ways

Toggle a 12-bit field to meet hexadecimal, octal, and decimal missions. The lab redraws both 3-bit and 4-bit boundaries while preserving the same underlying bits.

After solving a mission, toggle the lowest bit and then the highest bit. Compare the numerical changes.

Route 3: repeated division for decimal integers

To convert a nonnegative decimal integer to base $b$:

1. divide the current value by $b$;

2. record the remainder, which is the next least-significant digit;

3. continue with the quotient;

4. stop when the quotient is zero;

5. read the remainders in reverse production order.

Convert $173$ to hexadecimal:

The first remainder is the low digit, so read upward:

Check by expansion:

The value zero is a boundary case. Repeated division performs no positive-value iteration, but its representation is still 0.

Lab 2 — assemble remainders in the correct direction

Change the decimal input and destination base. The workshop generates all quotient/remainder rows, but you must select the remainder cards in the correct order.

Try a number whose conversion contains repeated digits. The cards have separate identities even when their digit symbols match.

Converting fractional parts

For a fraction, repeated multiplication is the mirror image of repeated division.

To convert decimal fraction $f$ to base $b$:

1. multiply $f$ by $b$;

2. record the integer part as the next digit from left to right;

3. continue with the new fractional part.

Convert $0.625$ to binary:

So $(0.625)_{10}=(0.101)_2$.

Unlike integer remainders, fractional digits are already produced from most significant to least significant; do not reverse them.

Some fractions never terminate in the destination base. A fixed-width system must stop after a chosen number of digits and record that the result is approximate.

Checkpoint

Use a method that exposes a check:

• non-decimal to decimal: expand place values;

• decimal integer to base $b$: repeated division and reverse remainders;

• decimal fraction to base $b$: repeated multiplication and keep digit order;

• binary ↔ octal/hexadecimal: group bits around the radix point.

The next section asks a deeper question: after bits have been converted and stored, should the top bit represent a positive weight or a negative sign?