binary systems_

Computers use the binary system (0s and 1s) to represent and process all data, since all information must be converted into binary. This is because computers are made of tiny switches that can only be ON (1) or OFF (0), making binary the basic building block of computer systems.

Computers use switches with logic gates to store and process data. The denary system (base 10) uses digits 0–9 and place values like units, tens, and hundreds, while the binary system (base 2) uses only 0 and 1 with place values based on powers of 2 (1, 2, 4, 8, etc.). Binary numbers (e.g. 11101110) can be converted to denary by adding the values of the columns where there is a 1.

Example 1:

11101110 = 128 + 64 + 32 + 8 + 4 + 2, which equals 238.

Example 2:

Example 2 Convert the following binary number, 011110001011, into a denary number. 2048 1024 512 256 128 64 32 16 8 4 2 1 0 1 1 1 1 0 0 0 1 0 1 1 The equivalent denary number is 1024 + 512 + 256 + 128 + 8 + 2 + 1 = 1931

Denary numbers can be converted to binary using two methods: by repeatedly subtracting powers of 2 (such as 128, 64, 32, etc.), or by repeatedly dividing the number by 2 until the result reaches 0.

Example1 (142)

Method 1

To convert 142 from denary to binary using subtraction, subtract the largest powers of 2 step by step (128 + 8 + 4 + 2). This gives the binary number 10001110 by placing 1s in those columns and 0s in the others.

Method 2

To convert 142 to binary using division, repeatedly divide the number by 2, keeping track of the remainders. Then, read the remainders in reverse order (from last to first) to get the binary number 10001110.

Example 2

Method 1

To convert 59 from denary to binary using subtraction, repeatedly subtract the largest powers of 2 (32 + 16 + 8 + 2 + 1) until reaching 0. This gives the 8-bit binary number 00111011.

Method 2

To convert 59 to binary using division, repeatedly divide the number by 2 and record the remainders. Reading the remainders from bottom to top gives the binary number 00111011.