Conversion of Octal to any other radix system

CONVERSION OF OCTAL TO ANY OTHER RADIX SYSTEM

Octal to Binary Conversion

Each digit of the octal number is individually converted to its binary equivalent to obtain conversion of the number.

Problem: 22

Convert the octal number (723)₈ to binary number.

(723)₈ = (111 001 011)₂

Problem: 23

Convert the octal number (426.43)₈ to binary number.

(426.43)₈ = (100010110.100011)₂

Octal to Hexadecimal Conversion

The conversion of octal to hexadecimal number is given below

Step (i) convert octal number to its binary

Step (ii) convert that binary number to its hexadecimal equivalent.

Convert octal number (623), to Hexadecimal number system.

Problem: 24

Step (i)

(623)₈ = 110010011

Step (ii)

(110010011)₂ = (193)₁₆

Problem: 25

Convert octal number (564)₈ to Hexadecimal number system.

Step (i)

Step (ii)

(564)₈⇒ (1.74)₁₆

Octal to Decimal Conversion

The octal number system uses first eight digits of decimal number system (0-7).

The conversion is obtained by multiplay the power of 8 with each digit.

Problem: 26

Convert the octal number (623)₈ to decimal number system.

(623)₈ ⇒ 6 × 8² + 2 × 8¹ + 3 × 8⁰

= 384 + 16 + 3

(623)₈ = (403)₁₀

Problem: 27

Convert the octal number (46.24)₈ to decimal number system.

(46.24)₈ ⇒ 4 × 8¹ + 6 × 8⁰ + 2 × 8^-1 + 4 × 8⁻²

= 32 + 6 + 0.25 + 0.0625.

(46.24)₈ =  (38.3125)₁₀

Problem: 28

Convert the octal number (127.6)₈  to decimal number system.

 (127.6)₈ = 1 × 8² + 2 × 8¹ + 7 × 8⁰ + 6 × 8^-1

= 64+ 16 + 7 + 0.75

(127.6)₈ ⇒ (87.75)₁₀

Example Problems

Problem: 29

Convert the following decimal number to the indicated bases.

(a) 7562.45 to octal

(b) 1938.25 to hexadecimal

(c) 175.175 to binary

Solution:

(a) (7562.45)10 octal

Integer part

7562 ⇒ 16612

Fractional part

0.45 × 8 = 3.6 = 3

0.6 × 8 = 4.8 = 4

0.8 × 8 = 6.4 = 6

0.4 × 8 = 3.2 = 3

0.2 × 8 = 1.6 = 1

(7562.45)₁₀ = (16612.34631)₈

(b) (1938.25)₁₀ to hexadecimal

Integer part:

(1938)₁₀ ⇒ (792)₁₆

Fractional part:

0.25 × 16 = 4.0 = 4

(0.25)₁₀ = (0.4)₁₆

(1938.25)₁₀ = (792.4)₁₆

 (c) (175.175)₁₀ to binary

Integer part

 (175)₁₀ ⇒ (10101111)₂

Fraction part:

0.175 × 2 = 0.35 = 0

0.35 × 2 = 0.70 = 0

0.7 × 2 = 1.4 = 1

0.4 × 2 = 0.8 = 0

0.8 × 2 = 1.6 = 1

 (0.175)₁₀ = (0.00101)₂ ↓

(175.175)₁₀ = (10101111.00101)₂

Problem: 30

Convert the following numbers from the given base to the other three bases indicated.

(a) Decimal 225 to binary, octal and hexadecimal.

(b) Binary 11010111 to decimal, octal and hexadecimal.

Solution:

(a) Decimal (225)₁₀ to binary

 (225)₁₀ ⇒ (11100001)₂

Decimal (225)₁₀ to octal:

(225)₁₀ ⇒ (341)₈

Decimal (225)₁₀ to hexadecimal

(225)₁₀ ⇒ (E1)₁₆

(b) (11010111)₂ to decimal

1 × 2⁷ + 1 × 2⁶ + 0 × 2⁵ + 1 × 2⁴ + 0 × 2³ + 1 × 2² + 1 × 2¹ + 1 × 2⁰

= 128 + 64 + 0 + 16 + 0 + 4 + 2 + 1

(11010111)₂ ⇒ (215)₁₀.

(11010111)₂ to octal

(11010111)₂ → (327)₈

(11010111)₂ to hexadecimal

 (11010111)₂  → (D7)₁₆

Problem: 31

Convert the Hexadecimal number (A26)₁₆ to decimal octal and binary.

Solution:

Hexadecimal (A26)₁₆ to decimal

= A × 16² + 2 × 16¹ + 6 × 16⁰

= 10 × 16² + 2 × 16¹ + 6 × 16⁰

= 2560 + 32 + 6

= 2598

(A26)₁₆ → (2598)₁₀

Hexadecimal (A26)₁₆ to octal

Step (i)

(A26)₁₆ → (101000100110)₂

Step (ii)

(A26)₁₆ → (5046)₈

Hexadecimal (A26)₁₆ to binary

(A26)₁₆ → (101000100110)₂

Problem: 32

Convert the octal number (421)₈  to decimal, binary and hexadecimal. Octal (421)₈ to decimal

(421)₈

= 4 × 8² + 2 × 8¹ + 1 × 8⁰

= 256 + 16 + 1

(421)₈ → (273)₁₀

Octal (421)₈ to binary

(421)₈ → (100010001)₂

Octal (421)₈ to Hexadecimal

Step (i)

Step (ii)

(421)₈ → (111)₁₆

Problem: 33

Convert (10111.11), binary to decimal, octal and hexadecimal.

Solution:

(10111.11)₂ to decimal

1 × 2⁴ + 0 × 2³ + 1 × 2² + 1 × 2¹ + 1 × 2⁰ + 1 × 2^-1 + 1 × 2^-2

16 + 0 + 4 + 2 + 1 + 0.5 + 0.25

(23.75)₁₀

(10111.11)₂ → (23.75)₁₀

(10111.11)₂ to octal

(10111.11)₂  →  (27.6)₈

(10111.11)₂ to hexadecimal

 (10111.11)₂ → (17. C)₁₆

Problem: 34

Convert hexadecimal number (C22.A)₁₆ to decimal, binary and octal.

Solution:

(C22.A)₁₆ to decimal

= C × 16² +2 × 16¹ + 2 × 16⁰ + A × 16^-1

= 12 × 16² + 2 × 16¹ + 2 × 16⁰ + 10 × 16^-1

= 3072 + 32 + 2 + 0.625

(C22.A)₁₆ → (3106.625)₁₀

(C22.A)₁₆ to binary

(C22.A)₁₆ → (110000100010.1010)₂

(C22.A)₁₆ to Octal

(C22.A)₁₆ → (6042.50)₈