Binary Conversion

In today’s lesson we will link together the binary and algorithm topics by looking at how to get the computer to:

• convert binary to decimal

• convert decimal to binary

• do other conversion tasks – e.g. decimal to Roman Numbers

Binary

The binary system is the same, but is based on two:

1 0 1 1

28 14

x2x2 x2

As we move left, the column headings increase by a factor of two

In each column we can have two different digits (0 or 1)

This number is:

8 + 2 + 1 = 11

It’s still eleven, it’s just written down differently

Binary Examples

Do all of the odd numbers have anything in common?

0 1 0 028 14

0 0 0 132128 1664

1 1 1 10 0 0 0

0 1 0 10 1 1 0

1 1 1 00 0 0 1

= 20

= 101

= 15

= 30

Converting to Decimal

• To convert a binary number to decimal, all you need to do is:– add the binary column headings

– calculate the column values by multiplying the digits by their headings

– add up all of the column values

• This only requires simple arithmetic, so would be easy to do using either a programming language or a spreadsheet.

Converting to Binary

• Converting decimal to binary is a bit more tricky

• There are two different methods that you can use:– a more “common sense” method,

comparing the value with the column headings

– a more mathematical approach using bitwise AND and the column headings

Comparison Method

• If we are converting a number to binary, we don’t need to use any column where the heading is bigger than the number.

• e.g. for 20, the first column we need to use is 16

• If we have 1 x 16, then we have 4 left, so we don’t need an 8, but we can have a 4…

• i.e. each time we use a column heading, we subtract the heading value from the number

Bitwise AND• For example, 20 AND 4:

0 1 0 028 14

0 0 0 132128 1664

0 1 0 00 0 0 0 = 4

0 1 0 00 0 0 0

= 20

= 4

Bitwise AND Method

• If we do a bitwise AND with the number and the column heading, that will tell us whether we need a 1 in that column.

• e.g. 20 AND 4 = 4 – if the answer is anything other than zero, then we need to put a one in that column.

• In Python we use & for bitwise AND, e.g. if 20 & 4 > 0:

• In Just Basic we use AND, e.g. 20 AND 4

Other Conversions

• We can also use the comparison method to convert decimal numbers to Roman numerals:

1. Order the Roman letters into order of size

2. If the decimal number is larger than the value of the letter, we need to use that letter

3. Subtract the value of the letter from the decimal number

4. Repeat from 2. until you reach 0

• NB. It’s slightly more complicated than that, because, e.g. 9 is IX, but not much.