1.1.1 Binary

1.1.1 Binary systems

Learning Outcome

      •  recognize the use of binary numbers in computer systems
      • convert positive denary integers into binary and positive binary integers into denary (a maximum of 16 bits will be used)
      •  show understanding of the concept of a byte and how the byte is used to measure memory size
      • use binary in computer registers for a given application (such as in robotics, digital instruments, and counting systems)

The word binary is derived from the Latin word bini (two by two)

This system uses only two symbols (0 1), usually referred as binary digit or binary digits.

The numbers in the binary system are represented to the base 2 and the positional multipliers are the powers of 2.

The leftmost bit in the binary number is called the Most Significant Bit (MSB) and it has the largest positional weight.

The rightmost bit is the Least Significant Bit (LSB) and has the smallest positional weight.

Consider a binary no 101001 the left extreme 1 isMSB and the right extreme 1 is  LSB.

Example The binary sequence (1101)2 has the decimal equivalent:

(1101)2 = 1 u 23 + 1 u 22 + 0 u 21 + 1 u 20 = 8 + 4 + 0 + 1 = (13)10


I Decimal to Binary Conversion

To convert Decimal to Binary “Repeated Division by 2” method can be used. Any Decimal number divided by 2 will leave a remainder of 0 or 1. Repeated division by 2 will leave a sequence of 0s and 1s that become the binary equivalent of the decimal number.

Example 

Convert (65)10 into its equivalent binary number


II Binary to Decimal Conversion

To convert Binary to Decimal we can use the positional notation method.

Step 1: Write down the Binary digits and list the powers of 2 from right to left(Positional Notation)

Step 2: For each positional notation written for the digit, now write the equivalent weight.

Step 3: Multiply each digit with its corresponding weight

Step 4: Add all the values.

For example


III Binary to Octal Conversion

Step 1: Group the given binary number into 3 bits from right to left.

Step 2: You can add preceding 0 to make a group of 3 bits if the leftmost group has less than 3 bits.

Step 3: Convert equivalent octal value using “2’s power positional weight method”


 

IV Practice Problems

Question No 1:

  1. 876 is an Octal number.          Yes or no
  2. 101 is a Binary number.           Yes or no
  3. 304 is an Octal number.          Yes or no
  4. 2001 is a Binary number.        Yes or no

Question No 2:

Convert the following Decimal numbers to its equivalent Binary, Octal, Hexadecimal.

  1. 2810
  2. 265
  3. 125

Question No :

Question No 1:

Convert the given Binary number into its equivalent Decimal, Octal, and Hexadecimal number.

  1. 101110101
  2. 1011010
  3. 101011111