Binary Numbers

Here are the binary representations of numbers from 0 to 10:

These are the binary equivalents for the decimal numbers from 0 to 10. Each binary number represents a power of 2 in the binary system.

Generating binary numbers involves converting decimal numbers (base 10) to their binary equivalent (base 2). Here’s a basic example of the process:

1. Successive division by 2:
   • Start by dividing the decimal number by 2.
   • Record the remainder (the modulus) as the least significant digit of the binary number.
   • Take the quotient of the division and repeat the process until the quotient is 0.
2. Building the binary number:
   • Read the remainders from bottom to top to obtain the binary number.
   • The first remainder you obtained will be the least significant digit, and the last remainder will be the most significant digit.

Let’s see an example with the decimal number 25:

25 ÷ 2 = 12 with remainder 1
12 ÷ 2 = 6 with remainder 0
6 ÷ 2 = 3 with remainder 0
3 ÷ 2 = 1 with remainder 1
1 ÷ 2 = 0 with remainder 1

Now, read the remainders from bottom to top (bottom to top: 11001), so the binary equivalent of 25 in decimal is 11001.

This is a basic method, and there are other approaches and algorithms for binary-decimal conversion and vice versa. There are also online resources and calculators that can assist you with these conversions.

We’ll apply the same process to convert the decimal number 55 to binary:

55 ÷ 2 = 27 with remainder 1
27 ÷ 2 = 13 with remainder 1
13 ÷ 2 = 6 with remainder 1
6 ÷ 2 = 3 with remainder 0
3 ÷ 2 = 1 with remainder 1
1 ÷ 2 = 0 with remainder 1

Now, read the remainders from bottom to top (bottom to top: 110111), so the binary equivalent of 55 in decimal is 110111.

Let’s apply the process to convert the decimal number 77 to binary:

77 ÷ 2 = 38 with remainder 1
38 ÷ 2 = 19 with remainder 0
19 ÷ 2 = 9 with remainder 1
9 ÷ 2 = 4 with remainder 1
4 ÷ 2 = 2 with remainder 0
2 ÷ 2 = 1 with remainder 0
1 ÷ 2 = 0 with remainder 1

Now, read the remainders from bottom to top (bottom to top: 1001101), so the binary equivalent of 77 in decimal is 1001101.