Question

Convert (7345321)8 to its binary expansion and (10 1011 1011)2 to its octal expansion.

Short answer

(1 1101 1100 1010 1101 0001${)}_2$

(1273${)}_8$

Step-by-step solution

Step 1:

Each octal digit can be uniquely represented as a block of 3 binary digits.

Binary Decimal Octal

000 0 0

001 1 1

010 2 2

011 3 3

100 4 4

101 5 5

110 6 6

111 7 7

Step 2:

SOLUTION:

(7345321${)}_8$${)}_8$

Let us first determine the block of 3 binary digits that corresponds with each of the octal digits.

(7${)}_8$ = 111

(3${)}_8$= 011

(4${)}_8$= 100

(5${)}_8$= 101

(3${)}_8$= 011

(2${)}_8$= 010

(1${)}_8$= 001

The binary expansion of (7345321)8 is then the obtained by replacing each octal digit by the corresponding block of 3 binary digits.

(7345321${)}_8$= (111 011 100 101 011 010 001${)}_8$

= (1 1101 1100 1010 1101 0001${)}_2$

Step 3:

(10 1011 1011${)}_2$

Divide the binary expansion in blocks of 3(and add 0’s to the first block until it contains 3 digits)

(001 010 111 011${)}_2$

Let us next determine the octal digit that corresponds with each block of 3 binay digits

(001data-custom-editor="chemistry" ${)}_2$= 1

(010${)}_2$= 2

(111${)}_2$= 7

(011${)}_2$ = 3

The octal expansion of (11 1011 1011)2 is then the obtained by replacing each block of 3 binary digits by the corresponding octal digit.

(001 010 111 011${)}_2$= (1 6 7 3${)}_8$

= (1273${)}_8$