Converting Between Denary & Binary (OCR GCSE Computer Science)

What is denary?

• Denary is a number system that is made up of 10 digits (0-9)
• Denary is referred to as a Base-10 number system
• Each digit has a weight factor of 10 raised to a power, the rightmost digit is 1s (10⁰), the next digit to the left 10s (10¹) and so on
• Humans use the denary system for counting, measuring and performing maths calculations
• Using combinations of the 10 digits we can represent any number

• In this example, (3 x 1000) + (2 x 100) + (6 x 10) + (8 x 1) = 3268
• To represent bigger number we add more digits

What is binary?

• Binary is a number system that is made up of two digits (1 and 0) 
• Binary is referred to as a Base-2 number system
• Each digit has a weight factor of 2 raised to a power, the rightmost digit is 1s (2⁰), the next digit to the left 2s (2¹) and so on
• Using combinations of the 2 digits we can represent any number

• In this example, (1 x 8) + (1 x 4) = 12
• To represent bigger numbers we add more binary digits (bits)

128	64	32	16	8	4	2	1
27	26	25	24	23	22	21	20

Why do computers use binary?

• The CPU is made up of billions of tiny transistors, transistors can only be in a state of on or off
• Computers use binary numbers to represent data (1 = on, 0 = off)

Denary to binary conversion

• It is important to know the process of converting from denary to binary to understand how computers interpret and process data

Example 1

• To convert the denary number 45 to binary, start by writing out the binary headings from right to left

128	64	32	16	8	4	2	1

• Start at the leftmost empty column heading (128). Is the denary number > column heading? (45 > 128) No, put a 0 in the 128 column. Repeat until you put a 1 under a heading. In this example it would be 32
  

128	64	32	16	8	4	2	1
0	0	1

• Next subtract column heading from denary value, 45-32 = 13
• Repeat previous two steps until you have a value under each column heading

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

• 32 + 8 + 4 + 1 = 45
• Denary 45 is 00101101 in Binary

Example 2

• To convert the denary number 213 to binary, start by writing out the binary headings from right to left

128	64	32	16	8	4	2	1

• Start at the leftmost empty column heading (128). Is denary number > column heading? (213 > 128) Yes, put a 1 under the heading.
  

128	64	32	16	8	4	2	1
1

• Next subtract column heading from denary value, 213-128 = 85
• Repeat process until you have a value under each column heading

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

• 128 + 64 + 16 + 4 + 1 = 213
• Denary 213 is 11010101 in Binary

Example 1

• To convert the binary number 1011 to denary, start by writing out the binary headings from right to left

8	4	2	1

• Write in the binary digits under the headings from left to right

8	4	2	1
1	0	1	1

• Add together any value with a 1 under it
• (1 x 8) + (1 x 2) + (1 x 1) = 11
• Binary 1011 is 11 in Denary 

Example 2

• To convert the binary number 01100011 to denary, start by writing out the binary headings from right to left

128	64	32	16	8	4	2	1

• Write in the binary digits under the headings from left to right

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

• Add together any value with a 1 under it
• (1 x 64) + (1 x 32) + (1 x 2) + (1 x 1) = 99
• Binary 01100011 is 99 in Denary