University of Bahrain, IT College, CS Department

Course: CSC103

Numbering Systems
1. Numbering Bases
- Base 2: binary base, where the digits are 0, 1
- Base 8: octal base, where the digits are 0, 1, 2, 3, 4, 5, 6, 7
- Base 10: decimal base, where the digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
- Base 16: hexadecimal base, where the digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

➢ An octal digit has a perfect correspondence to a 3-bit binary value number (8 = 23 )

Base 8 Base 2
0 000
1 001
2 010
3 011 Table 1
4 100
5 101
6 110
7 111

➢ A hexadecimal digit has a perfect correspondence to a 4-bit binary value number (16 = 24 )

Base 16 Base 2
0 0000
1 0001
2 0010
3 0011
4 0100
5 0101
6 0110
7 0111 Table 2
8 1000
9 1001
A(10) 1010
B(11) 1011
C(12) 1100
D(13) 1101
E(14) 1110
F(15) 1111

Prepared by: Dr. Youssef Harrath 1

University of Bahrain, IT College, CS Department
Course: CSC103
2. Conversion between bases
➢ Binary(2) → Octal(8)
Separate the binary number in groups of 3 digits each starting from the right side. Complete the most left
group by zeros (in its left side) if it contains less than 3 digits. Convert each group of 3 digits into its
corresponding octal digit (according to the table 1).
Example
(1011100101000)2 = (001 011 100 101 000)2 = (13450)8

➢ Binary(2) → Hexadecimal(16)
Separate the binary number in groups of 4 digits each starting from the right side. Complete the most left
group by zeros (in its left side) if it contains less than 4 digits. Convert each group of 4 digits into its
corresponding hexadecimal digit (according to the table 2).
Example
(10110010011011)2 = (0010 1100 1001 1011)2 = (2𝐶9𝐵)16

➢ Base b(2, 8, or 16) → Decimal(10)

Given a number 𝐴 in a base b (2, 8, or 16), A can be written as: (𝐴)𝑏 = (𝑎𝑛−1 𝑎𝑛−2 … 𝑎0 )𝑏 where
0 ≤ 𝑎𝑖 ≤ 𝑏 − 1 and 0 ≤ 𝑖 ≤ 𝑛 − 1
To convert the number A from a base b (b= 8, 10, or 16) to its equivalent decimal number, we use the
following formula:
𝑛−1

(𝐴)𝑏 = (∑ 𝑎𝑖 𝑏 𝑖 )10
𝑖=0
Example 1: Binary(2) → Decimal(10)

(011101)2 = ((1 × 20 ) + (0 × 21 ) + (1 × 22 ) + (1 × 23 ) + (1 × 24 ) + (0 × 25 ))
10
=( 1 + 0 + 4 + 8 + 16 + 0 )10
= (29)10

Example 2: Octal(8) → Decimal(10)

(1237)8 = ((7 × 80 ) + (3 × 81 ) + (2 × 82 ) + (1 × 83 ))10

=( 7 + 24 + 128 + 512 )10
= (671)10

Example 3: Hexadecimal(16) → Decimal(10)

(1𝐴𝐹2)16 = ((2 × 160 ) + (𝐹 × 161 ) + (𝐴 × 162 ) + (1 × 163 ))10

= ((2 × 160 ) + (15 × 161 ) + (10 × 162 ) + (1 × 163 ))10
=( 2 + 240 + 2560 + 4096 )10
= (6898)10

Prepared by: Dr. Youssef Harrath 2

University of Bahrain, IT College, CS Department
Course: CSC103

➢ Octal(8) or Hexadecimal(16) → Binary(2)

Expansion of each digit into its binary equivalent 3-binary digits for base 8 and 4-binary digits for base 16.

Example 1: Octal(8) → Binary(2)

(743)8 = (111 100 011)2

Example 2: Hexadecimal(16) → Binary(2)

(𝐴𝐹680)16 = (1010 1111 0110 1000 0000)2

➢ Decimal(10) → Binary(2)
- Successive divisions of the quotient by 2
- Stop when the quotient becomes 0
- Copy the reminders starting from the last division

Example
(756)10 = (? )2
756 = 378 × 2 + 𝟎
378 = 189 × 2 + 𝟎
189 = 94 × 2 + 𝟏
94 = 47 × 2 + 𝟎
47 = 23 × 2 + 𝟏
23 = 11 × 2 + 𝟏
11 = 5 × 2 + 𝟏
5= 2×2+𝟏
2= 1×2+𝟎
1= 𝟎×2+𝟏

(756)10 = (1011110100)2

➢ More Conversions
• Octal (8) → Hexadecimal (16)
Convert the octal number to binary and then from binary to hexadecimal.
Example:
(705)8 = (? )16
(705)8 = (111 000 101)2
(705)8 = (1 1100 0101)2
(705)8 = (0001 1100 0101)2
(705)8 = (1𝐶5)16

Prepared by: Dr. Youssef Harrath 3

University of Bahrain, IT College, CS Department
Course: CSC103

• Hexadecimal (16) → Octal (8)

Convert the hexadecimal number to binary and then from binary to octal.
Example:
(𝐷𝐹162)16 = (? )8
(𝐷𝐹162)16 = (1101 1111 0001 0110 0010)2
(𝐷𝐹162)16 = (11 011 111 000 101 100 010)2
(𝐷𝐹162)16 = (011 011 111 000 101 100 010)2
(𝐷𝐹162)16 = (3370542)8

• Decimal (10) → Octal (8)

Convert the decimal number to binary and then from binary to octal.
Example:
(756)10 = (? )8
(756)10 = (1011110100)2
(756)10 = (1 011 110 100)2
(756)10 = (001 011 110 100)2
(756)10 = (1364)8

• Decimal (10) → Hexadecimal

Convert the decimal number to binary and then from binary to hexadecimal.
Example:
(756)10 = (? )16
(756)10 = (1011110100)2
(756)10 = (10 1111 0100)2
(756)10 = (0010 1111 0100)2
(756)10 = (2𝐹4)16

3. Binary addition and multiplication

➢ Addition

We use the following table to perform addition in base 2.

1 + 1 = 0 and reminder 1
1+0=1
0+1=1
0+0=0

Prepared by: Dr. Youssef Harrath 4

University of Bahrain, IT College, CS Department
Course: CSC103

Example 1

1 0 1 1 0
+
0 1 1 1 1
= 10 0 1 0 1

Example 2

0 1 1 1 0 1
+
0 1 0 0 0 1
= 1 0 1 1 1 0

➢ Multiplication

We use the following table to perform multiplication in base 2.

1×1=1
1×0=0
0×1=0
0×0=0

Example

1 0 1
×
0 1 1
1 0 1
1 0 1
0 0 0
= 0 1 1 1 1

4. Exercises
A. Convert the following numbers to the indicated base (show your steps)
a. (0111001)2 = (? )10
b. (𝐹𝐸30)16 = (? )10
c. (1764)8 = (? )2
d. (0111001)2 = (? )8
e. (3908)10 = (? )16
B. Find the binary sum and product of the following two binary numbers:
0111001 and 011

Prepared by: Dr. Youssef Harrath 5