Number Systems

Denary to binary

The standard method is the place-value table:

1. Write a row of place-value headings (1, 2, 4, 8, 16, 32, 64, 128) from right to left.
2. Working from the left of the table, ask: does the largest column heading fit into the denary number? If yes, write a 1 under it and subtract that value from the denary number. If no, write a 0.
3. Move to the next column heading and repeat with the remainder.
4. Continue until you reach the rightmost column.
5. The string of 1s and 0s along the bottom is the binary value.

Example — Convert the denary number 45 to binary.

• 32 fits into 45 (write 1, remainder 13)
• 16 does not fit into 13 (write 0, remainder still 13)
• 8 fits into 13 (write 1, remainder 5)
• 4 fits into 5 (write 1, remainder 1)
• 2 does not fit into 1 (write 0)
• 1 fits into 1 (write 1, remainder 0)

So denary 45 = binary 101101.

If the exam asks for an 8-bit answer, pad on the left with zeros: 00101101.

Binary to denary

The reverse is simpler:

1. Write the binary number under a row of place-value headings.
2. For every column that has a 1, write down the column heading.
3. Add them all up.

Example — Convert the binary number 01100011 to denary.

Add the columns where there is a 1: 64 + 32 + 2 + 1 = 99.

Quick sanity check: if a binary number ends in 1, the denary equivalent must be odd. If it ends in 0, the denary equivalent must be even.