Converting Between Denary & Binary

Key Concepts

• Denary (Decimal): The number system we use every day, with digits from 0 to 9.
• Binary: A number system used by computers, consisting of only two digits: 0 and 1.

Binary Ranges

• Binary numbers up to 8 bits can range from 00000000 (which is 0 in denary) to 11111111 (which is 255 in denary).

Significant Bits

• Most Significant Bit (MSB): The bit furthest to the left. It holds the largest value.
• Least Significant Bit (LSB): The bit furthest to the right. It holds the smallest value.

Converting Denary to Binary (up to 8 bits)

Step-by-Step Explanation

To convert a denary number (0-255) to binary, you will use a table. Here's how to do it.

1. Draw a table like this: | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 | |---|---|---|---|---|---|---|---| | | | | | | | | |

Each number at the top is the value of that column. We're going to figure out if the denary number can fit into these values.

2. Start from the left (128) and ask: Can the denary number fit into this value? If yes, write a 1 and subtract that value from the denary number. If no, write a 0.
3. Move to the next column.

Example: Convert 56 to Binary

• Is 56 larger than or equal to 128? No, so put 0 under 128.
• Is 56 larger than or equal to 64? No, so put 0 under 64.
• Is 56 larger than or equal to 32? Yes, so put 1 under 32. Now subtract 32 from 56. The remainder is 24.
• Is 24 larger than or equal to 16? Yes, so put 1 under 16. Subtract 16 from 24. The remainder is 8.
• Is 8 larger than or equal to 8? Yes, so put 1 under 8. Subtract 8 from 8. The remainder is 0.
• Is 0 larger than or equal to 4, 2, or 1? No, so put 0 under 4, 2, and 1.

Here's how the table looks:

128	64	32	16	8	4	2	1
0	0	1	1	1	0	0	0

So, 56 in binary is 00111000.

Explanation

• Each 1 in the binary number means the denary number is large enough to fit in that column. Each 0 means it's too small.
• The answer 00111000 tells the computer how to store the number 56 in a way it understands.

Converting Binary to Denary

Step-by-Step Explanation

To convert from binary to denary, you do the opposite. Use a table, place each binary digit into its column, and then add up the values of the columns with a 1.

Example: Convert 01010101 to Denary

1. Write out the binary digits in the table: | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 | |---|---|---|---|---|---|---|---| | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |

2. Look at the columns with a 1. Add those values:

• 64 (from the 1 under 64)
• 16 (from the 1 under 16)
• 4 (from the 1 under 4)
• 1 (from the 1 under 1) So, 64 + 16 + 4 + 1 = 85.

Answer: 01010101 in denary is 85.

Explanation

• Each time you see a 1 in a column, that means that value is included in the number. Adding up those values gives you the denary number.

Dealing with Shorter Binary Numbers

• Binary numbers can have less than 8 bits. For example, 11010 is only 5 bits long.
• To make it easier to work with, you can add leading zeros to make it 8 bits: 00011010. This helps keep things clear when comparing or converting.

Quick Reference Conversion Table

Denary	Binary	Hexadecimal
0	00000000	00
56	00111000	38
85	01010101	55
255	11111111	FF