Decimal to Binary Conversion Examples

Decimal-to-binary conversion is a core concept in computer science and digital electronics. Understanding how to convert base-10 numbers to base-2 helps students grasp how computers store and process data efficiently. This process converts numbers from base 10 (human-readable system) to base 2 (machine-readable system). This concept is essential for programming, digital logic design, and computer architecture.

List of methods and examples of decimal to binary conversion are given below.

1. Repeated Division by 2 Method

This is the most common and fundamental method used to convert decimal numbers into binary form. It involves dividing the number repeatedly by 2 and recording remainders.

Algorithm for Repeated Division by 2 Method

This algorithm clearly defines each step to ensure accuracy and avoid common mistakes during conversion.

• Step 1: Start with the given decimal number N
• Step 2: Divide $N$ by 2
• Step 3: Record the remainder (it will always be 0 or 1)
• Step 4: Update the quotient obtained from the division
• Step 5: Repeat Steps 2 to 4 until N=0
• Step 6: Write all recorded remainders in reverse order (from last to first)
• Step 7: The resulting sequence is the binary equivalent

The following diagram explains the entire algorithm of decimal to binary conversion

Important: The digits 0 and 1 represent the same numeric values in all number systems (including decimal, binary, octal, hexadecimal, etc.).

Decimal to Binary Conversion Examples

Here are the top 15 examples of Decimal to Binary conversions

Example 1: Convert Decimal (6)₁₀ to Binary

Solution:

The following diagram shows the conversion of the decimal number (6)₁₀ into its equivalent binary

Description:

6 is divided by 2, and the quotient is 3 with a remainder of 0.
3 is divided by 2, and the quotient is 1 with a remainder of 1.
1 is divided by 2, and the quotient is 0 with a remainder of 1.

Now, we write all remainders from bottom to top to get the final binary number. The binary representation of the decimal number (6)₁₀ is (110)₂.

Example 2: Convert Decimal (16)₁₀ to Binary

Solution:

The following diagram shows the conversion of the decimal number (16)₁₀ into its equivalent binary

Description:

16 is divided by 2, and the quotient is 8 with a remainder of 0.
8 is divided by 2, and the quotient is 4 with a remainder of 0.
4 is divided by 2, and the quotient is 2 with a remainder of 0.
2 is divided by 2, and the quotient is 1 with a remainder of 0.
1 is divided by 2, and the quotient is 0 with a remainder of 1.

Now, we write all remainders from bottom to top. The binary representation of the decimal number (16)₁₀ is (10000)₂.

Example 3: Convert Decimal (25)₁₀ to Binary

Solution:

The following diagram shows the conversion of the decimal number (25)₁₀ into its equivalent binary

Description:

25 is divided by 2, and the quotient is 12 with a remainder of 1.
12 is divided by 2, and the quotient is 6 with a remainder of 0.
6 is divided by 2, and the quotient is 3 with a remainder of 0.
3 is divided by 2, and the quotient is 1 with a remainder of 1.
1 is divided by 2, and the quotient is 0 with a remainder of 1.

Now, we write all remainders from bottom to top. The binary representation of the decimal number (25)₁₀ is (11001)₂.

Example 4: Convert Decimal (42)₁₀ to Binary

Solution:

The following diagram shows the conversion of the decimal number (42)₁₀ into its equivalent binary

Description:

42 is divided by 2, and the quotient is 21 with a remainder of 0.
21 is divided by 2, and the quotient is 10 with a remainder of 1.
10 is divided by 2, and the quotient is 5 with a remainder of 0.
5 is divided by 2, and the quotient is 2 with a remainder of 1.
2 is divided by 2, and the quotient is 1 with a remainder of 0.
1 is divided by 2, and the quotient is 0 with a remainder of 1.

Now, we write all remainders from bottom to top. The binary representation of the decimal number (42)₁₀ is (101010)₂.

Example 5: Convert Decimal (58)₁₀ to Binary

Solution:

The following diagram shows the conversion of the decimal number (58)₁₀ into its equivalent binary

Description:

58 is divided by 2, and the quotient is 29 with a remainder of 0.
29 is divided by 2, and the quotient is 14 with a remainder of 1.
14 is divided by 2, and the quotient is 7 with a remainder of 0.
7 is divided by 2, and the quotient is 3 with a remainder of 1.
3 is divided by 2, and the quotient is 1 with a remainder of 1.
1 is divided by 2, and the quotient is 0 with a remainder of 1.

Now, we write all remainders from bottom to top. The binary representation of the decimal number (58)₁₀ is (111010)₂.

Example 6: Convert Decimal (100)₁₀ to Binary

Solution:

The following diagram shows the conversion of the decimal number (100)₁₀ into its equivalent binary

Description:

100 is divided by 2, and the quotient is 50 with a remainder of 0.
50 is divided by 2, and the quotient is 25 with a remainder of 0.
25 is divided by 2, and the quotient is 12 with a remainder of 1.
12 is divided by 2, and the quotient is 6 with a remainder of 0.
6 is divided by 2, and the quotient is 3 with a remainder of 0.
3 is divided by 2, and the quotient is 1 with a remainder of 1.
1 is divided by 2, and the quotient is 0 with a remainder of 1.

Now, we write all remainders from bottom to top. The binary representation of the decimal number (100)₁₀ is (1100100)₂.

Example 7: Convert Decimal (156)₁₀ to Binary

Solution:

The following diagram shows the conversion of the decimal number (156)₁₀ into its equivalent binary

Description:

156 is divided by 2, and the quotient is 78 with a remainder of 0.
78 is divided by 2, and the quotient is 39 with a remainder of 0.
39 is divided by 2, and the quotient is 19 with a remainder of 1.
19 is divided by 2, and the quotient is 9 with a remainder of 1.
9 is divided by 2, and the quotient is 4 with a remainder of 1.
4 is divided by 2, and the quotient is 2 with a remainder of 0.
2 is divided by 2, and the quotient is 1 with a remainder of 0.
1 is divided by 2, and the quotient is 0 with a remainder of 1.

Now, we write all remainders from bottom to top. The binary representation of the decimal number (156)₁₀ is (10011100)₂.

Example 8: Convert Decimal (200)₁₀ to Binary

Solution:

The following diagram shows the conversion of the decimal number (200)₁₀ into its equivalent binary

Description:

200 is divided by 2, and the quotient is 100 with a remainder of 0.
100 is divided by 2, and the quotient is 50 with a remainder of 0.
50 is divided by 2, and the quotient is 25 with a remainder of 0.
25 is divided by 2, and the quotient is 12 with a remainder of 1.
12 is divided by 2, and the quotient is 6 with a remainder of 0.
6 is divided by 2, and the quotient is 3 with a remainder of 0.
3 is divided by 2, and the quotient is 1 with a remainder of 1.
1 is divided by 2, and the quotient is 0 with a remainder of 1.

Now, we write all remainders from bottom to top. The binary representation of the decimal number (200)₁₀ is (11001000)₂.

Example 9: Convert Decimal (255)₁₀ to Binary

Solution:

The following diagram shows the conversion of the decimal number (255)₁₀ into its equivalent binary

Description:

255 is divided by 2, and the quotient is 127 with a remainder of 1.
127 is divided by 2, and the quotient is 63 with a remainder of 1.
63 is divided by 2, and the quotient is 31 with a remainder of 1.
31 is divided by 2, and the quotient is 15 with a remainder of 1.
15 is divided by 2, and the quotient is 7 with a remainder of 1.
7 is divided by 2, and the quotient is 3 with a remainder of 1.
3 is divided by 2, and the quotient is 1 with a remainder of 1.
1 is divided by 2, and the quotient is 0 with a remainder of 1.

Now, we write all remainders from bottom to top. The binary representation of the decimal number (255)₁₀ is (11111111)₂.

Example 10: Convert Decimal (1024)₁₀ to Binary

Solution:

The following diagram shows the conversion of the decimal number (1024)₁₀ into its equivalent binary

Description:

1024 is divided by 2, and the quotient is 512 with a remainder of 0.
512 is divided by 2, and the quotient is 256 with a remainder of 0.
256 is divided by 2, and the quotient is 128 with a remainder of 0.
128 is divided by 2, and the quotient is 64 with a remainder of 0.
64 is divided by 2, and the quotient is 32 with a remainder of 0.
32 is divided by 2, and the quotient is 16 with a remainder of 0.
16 is divided by 2, and the quotient is 8 with a remainder of 0.
8 is divided by 2, and the quotient is 4 with a remainder of 0.
4 is divided by 2, and the quotient is 2 with a remainder of 0.
2 is divided by 2, and the quotient is 1 with a remainder of 0.
1 is divided by 2, and the quotient is 0 with a remainder of 1.

Now, we write all remainders from bottom to top. The binary representation of the decimal number (1150)₁₀ is (10000000000)₂.

Example 11: Convert Decimal (1150)₁₀ to Binary

Solution:

The following diagram shows the conversion of the decimal number (1150)₁₀ into its equivalent binary

Description:

1150 is divided by 2, and the quotient is 575 with a remainder of 0.
575 is divided by 2, and the quotient is 287 with a remainder of 1.
287 is divided by 2, and the quotient is 143 with a remainder of 1.
143 is divided by 2, and the quotient is 71 with a remainder of 1.
71 is divided by 2, and the quotient is 35 with a remainder of 1.
35 is divided by 2, and the quotient is 17 with a remainder of 1.
17 is divided by 2, and the quotient is 8 with a remainder of 1.
8 is divided by 2, and the quotient is 4 with a remainder of 0.
4 is divided by 2, and the quotient is 2 with a remainder of 0.
2 is divided by 2, and the quotient is 1 with a remainder of 0.
1 is divided by 2, and the quotient is 0 with a remainder of 1.

Now, we write all remainders from bottom to top. The binary representation of the decimal number (1150)₁₀ is (10001111110)₂.

Example 12: Convert Decimal (11234)₁₀ to Binary

Solution:

The following diagram shows the conversion of the decimal number (11234)₁₀ into its equivalent binary

Description:

11234 is divided by 2, and the quotient is 5617 with a remainder of 0.
5617 is divided by 2, and the quotient is 2808 with a remainder of 1.
2808 is divided by 2, and the quotient is 1404 with a remainder of 0.
1404 is divided by 2, and the quotient is 702 with a remainder of 0.
702 is divided by 2, and the quotient is 351 with a remainder of 0.
351 is divided by 2, and the quotient is 175 with a remainder of 1.
175 is divided by 2, and the quotient is 87 with a remainder of 1.
87 is divided by 2, and the quotient is 43 with a remainder of 1.
43 is divided by 2, and the quotient is 21 with a remainder of 1.
21 is divided by 2, and the quotient is 10 with a remainder of 1.
10 is divided by 2, and the quotient is 5 with a remainder of 0.
5 is divided by 2, and the quotient is 2 with a remainder of 1.
2 is divided by 2, and the quotient is 1 with a remainder of 0.
1 is divided by 2, and the quotient is 0 with a remainder of 1.

Now, we write all remainders from bottom to top.
The binary representation of the decimal number (11234)₁₀ is (10101111100010)₂.

Example 13: Convert Decimal (55890)₁₀ to Binary

Solution:

The following diagram shows the conversion of the decimal number (55890)₁₀ into its equivalent binary

Description:

55890 is divided by 2, and the quotient is 27945 with a remainder of 0.
27945 is divided by 2, and the quotient is 13972 with a remainder of 1.
13972 is divided by 2, and the quotient is 6986 with a remainder of 0.
6986 is divided by 2, and the quotient is 3493 with a remainder of 0.
3493 is divided by 2, and the quotient is 1746 with a remainder of 1.
1746 is divided by 2, and the quotient is 873 with a remainder of 0.
873 is divided by 2, and the quotient is 436 with a remainder of 1.
436 is divided by 2, and the quotient is 218 with a remainder of 0.
218 is divided by 2, and the quotient is 109 with a remainder of 0.
109 is divided by 2, and the quotient is 54 with a remainder of 1.
54 is divided by 2, and the quotient is 27 with a remainder of 0.
27 is divided by 2, and the quotient is 13 with a remainder of 1.
13 is divided by 2, and the quotient is 6 with a remainder of 1.
6 is divided by 2, and the quotient is 3 with a remainder of 0.
3 is divided by 2, and the quotient is 1 with a remainder of 1.
1 is divided by 2, and the quotient is 0 with a remainder of 1.

Now, we write all remainders from bottom to top. The binary representation of the decimal number (55890)₁₀ is (1101101001010010)₂.

Example 14: Convert Decimal (881123)₁₀ to Binary

Solution:

The following diagram shows the conversion of the decimal number (881123)₁₀ into its equivalent binary

Description:

881123 is divided by 2, and the quotient is 440561 with a remainder of 1.
440561 is divided by 2, and the quotient is 220280 with a remainder of 1.
220280 is divided by 2, and the quotient is 110140 with a remainder of 0.
110140 is divided by 2, and the quotient is 55070 with a remainder of 0.
55070 is divided by 2, and the quotient is 27535 with a remainder of 0.
27535 is divided by 2, and the quotient is 13767 with a remainder of 1.
13767 is divided by 2, and the quotient is 6883 with a remainder of 1.
6883 is divided by 2, and the quotient is 3441 with a remainder of 1.
3441 is divided by 2, and the quotient is 1720 with a remainder of 1.
1720 is divided by 2, and the quotient is 860 with a remainder of 0.
860 is divided by 2, and the quotient is 430 with a remainder of 0.
430 is divided by 2, and the quotient is 215 with a remainder of 0.
215 is divided by 2, and the quotient is 107 with a remainder of 1.
107 is divided by 2, and the quotient is 53 with a remainder of 1.
53 is divided by 2, and the quotient is 26 with a remainder of 1.
26 is divided by 2, and the quotient is 13 with a remainder of 0.
13 is divided by 2, and the quotient is 6 with a remainder of 1.
6 is divided by 2, and the quotient is 3 with a remainder of 0.
3 is divided by 2, and the quotient is 1 with a remainder of 1.
1 is divided by 2, and the quotient is 0 with a remainder of 1.

Now, we write all remainders from bottom to top. The binary representation of the decimal number (881123)₁₀ is (11010111000111100011)₂.

Decimal Fraction to Binary Conversion

Decimal fractions (numbers with decimal points) require a different method involving multiplication. This is crucial for representing real numbers in computing.
A list of fractional conversion steps and examples is given below.

Algorithm for Multiplication by 2 Method (Fractional Part)

This method converts decimal fractions into binary by repeatedly multiplying by 2 and extracting integer parts.

• Step 1: Take the decimal fractional number (for example, 0.x)
• Step 2: Multiply the fractional number by 2
• Step 3: Note the integer part of the result (either 0 or 1)
• Step 4: Keep only the fractional part of the result
• Step 5: Repeat Steps 2 to 4 with the new fractional part
• Step 6: Continue the process until the fractional part becomes 0, or the required precision (number of bits mostly 3,4 bits only) is achieved.
• Step 7: Write all the recorded integer parts in the same order to get the binary fraction

This method is widely used for floating-point number conversion.

Example 01: Convert (0.625)₁₀ to Binary

Solution:

The following diagram shows the conversion of the decimal number (0.625)₁₀ into its equivalent binary

Description:

0.625 is multiplied by 2 and the result is 1.25, so the integer part is 1
0.25 is multiplied by 2 and the result is 0.5, so the integer part is 0
0.5 is multiplied by 2 and the result is 1.0, so the integer part is 1

Binary fraction = 0.101₂

Example 02: Convert (0.375)₁₀ to Binary

Solution:

The following diagram shows the conversion of the decimal number (0.375)₁₀ into its equivalent binary

Description:

0.375 is multiplied by 2 and the result is 0.75, so the integer part is 0
0.75 is multiplied by 2 and the result is 1.5, so the integer part is 1
0.5 is multiplied by 2 and the result is 1.0, so the integer part is 1

Binary fraction = 0.011₂

Example 03: Convert (0.8125)₁₀ to Binary

Solution:

The following diagram shows the conversion of the decimal number (0.8125)₁₀ into its equivalent binary

Description:

0.8125 is multiplied by 2 and the result is 1.625, so the integer part is 1
0.625 is multiplied by 2 and the result is 1.25, so the integer part is 1
0.25 is multiplied by 2 and the result is 0.5, so the integer part is 0
0.5 is multiplied by 2 and the result is 1.0, so the integer part is 1

Binary fraction = 0.1101₂

Example 04: Convert (0.1)₁₀ to Binary (Approximate)

Solution:

The following diagram shows the conversion of the decimal number (0.1)₁₀ into its equivalent binary

Description:

0.1 is multiplied by 2 and the result is 0.2, so the integer part is 0
0.2 is multiplied by 2 and the result is 0.4, so the integer part is 0
0.4 is multiplied by 2 and the result is 0.8, so the integer part is 0
0.8 is multiplied by 2 and the result is 1.6, so the integer part is 1
0.6 is multiplied by 2 and the result is 1.2, so the integer part is 1
0.2 is multiplied by 2 and the result is 0.4, so the integer part is 0

(Binary starts repeating)

Binary fraction ≈ (0.000110)₂

Example 05: Convert (0.45)₁₀ to Binary (Approximate up to 4 bits)

Solution:

The following diagram shows the conversion of the decimal number (0.45)₁₀ into its equivalent binary

Description:

0.45 is multiplied by 2 and the result is 0.9, so the integer part is 0
0.9 is multiplied by 2 and the result is 1.8, so the integer part is 1
0.8 is multiplied by 2 and the result is 1.6, so the integer part is 1
0.6 is multiplied by 2 and the result is 1.2, so the integer part is 1

Binary fraction ≈ 0.0111₂

Decimal (Integer + Fraction) to Binary Conversion

When a decimal number contains both an integer part and a fractional part, each part must be converted separately using different methods. After conversion, the results are combined to obtain the final binary value.

• Step 01: Convert the integer part using the division method
• Step 02: Convert the fractional part using the multiplication method
• Step 03: Combine both results of step 1 and step 2

Example 01: Convert (16.625)₁₀ to Binary

Solution:

The following diagram shows the conversion of the decimal number (16.625)₁₀ into its equivalent binary

Integer Part Description:

16 is divided by 2, and the quotient is 8 with a remainder of 0.
8 is divided by 2, and the quotient is 4 with a remainder of 0.
4 is divided by 2, and the quotient is 2 with a remainder of 0.
2 is divided by 2, and the quotient is 1 with a remainder of 0.
1 is divided by 2, and the quotient is 0 with a remainder of 1.

Now, we write all remainders from bottom to top. The binary representation of the decimal number (16)₁₀ is (10000)₂.

Fraction Part Description:

0.625 is multiplied by 2 and the result is 1.25, so the integer part is 1
0.25 is multiplied by 2 and the result is 0.5, so the integer part is 0
0.5 is multiplied by 2 and the result is 1.0, so the integer part is 1

Binary fraction = 0.101₂

Final Answer:

(16.625)₁₀ = (10000.101)₂

Example 02: Convert (25.375)₁₀ to Binary

Solution:

The following diagram shows the conversion of the decimal number (25.375)₁₀ into its equivalent binary

Integer Part Description:

25 is divided by 2 and the quotient is 12 with a remainder of 1
12 is divided by 2 and the quotient is 6 with a remainder of 0
6 is divided by 2 and the quotient is 3 with a remainder of 0
3 is divided by 2 and the quotient is 1 with a remainder of 1
1 is divided by 2 and the quotient is 0 with a remainder of 1

Binary integer part = 11001₂

Fraction Part Description:

0.375 is multiplied by 2 and the result is 0.75, so the integer part is 0
0.75 is multiplied by 2 and the result is 1.5, so the integer part is 1
0.5 is multiplied by 2 and the result is 1.0, so the integer part is 1

Binary fraction = 0.011₂

Final Answer:

(25.375)₁₀ = (11001.011)₂

Example 03: Convert (56.8125)₁₀ to Binary

Solution:

The following diagram shows the conversion of the decimal number (56.8125)₁₀ into its equivalent binary

Integer Part Description:

56 is divided by 2 and the quotient is 28 with a remainder of 0
28 is divided by 2 and the quotient is 14 with a remainder of 0
14 is divided by 2 and the quotient is 7 with a remainder of 0
7 is divided by 2 and the quotient is 3 with a remainder of 1
3 is divided by 2 and the quotient is 1 with a remainder of 1
1 is divided by 2 and the quotient is 0 with a remainder of 1

Binary integer part = 111000₂

Fraction Part Description:

0.8125 is multiplied by 2 and the result is 1.625, so the integer part is 1
0.625 is multiplied by 2 and the result is 1.25, so the integer part is 1
0.25 is multiplied by 2 and the result is 0.5, so the integer part is 0
0.5 is multiplied by 2 and the result is 1.0, so the integer part is 1

Binary fraction = 0.1101₂

Final Answer:

(56.8125)₁₀ = (111000.1101)₂

Example 04: Convert (100.1)₁₀ to Binary (Approximate)

Solution:

The following diagram shows the conversion of the decimal number (100.1)₁₀ into its equivalent binary

Integer Part Description:

100 is divided by 2 and the quotient is 50 with a remainder of 0
50 is divided by 2 and the quotient is 25 with a remainder of 0
25 is divided by 2 and the quotient is 12 with a remainder of 1
12 is divided by 2 and the quotient is 6 with a remainder of 0
6 is divided by 2 and the quotient is 3 with a remainder of 0
3 is divided by 2 and the quotient is 1 with a remainder of 1
1 is divided by 2 and the quotient is 0 with a remainder of 1

Binary integer part = 1100100₂

Fraction Part Description:

0.1 is multiplied by 2 and the result is 0.2, so the integer part is 0
0.2 is multiplied by 2 and the result is 0.4, so the integer part is 0
0.4 is multiplied by 2 and the result is 0.8, so the integer part is 0
0.8 is multiplied by 2 and the result is 1.6, so the integer part is 1
0.6 is multiplied by 2 and the result is 1.2, so the integer part is 1
0.2 is multiplied by 2 and the result is 0.4, so the integer part is 0

(Binary starts repeating)

Binary fraction ≈ 0.000110₂

Final Answer:

(100.1)₁₀ ≈ (1100100.000110)₂

Example 05: Convert (255.45)₁₀ to Binary (Approximate up to 4 bits)

Solution:

The following diagram shows the conversion of the decimal number (255.45)₁₀ into its equivalent binary

Integer Part Description:

255 is divided by 2 and the quotient is 127 with a remainder of 1
127 is divided by 2 and the quotient is 63 with a remainder of 1
63 is divided by 2 and the quotient is 31 with a remainder of 1
31 is divided by 2 and the quotient is 15 with a remainder of 1
15 is divided by 2 and the quotient is 7 with a remainder of 1
7 is divided by 2 and the quotient is 3 with a remainder of 1
3 is divided by 2 and the quotient is 1 with a remainder of 1
1 is divided by 2 and the quotient is 0 with a remainder of 1

Binary integer part = 11111111₂

Fraction Part Description:

0.45 is multiplied by 2 and the result is 0.9, so the integer part is 0
0.9 is multiplied by 2 and the result is 1.8, so the integer part is 1
0.8 is multiplied by 2 and the result is 1.6, so the integer part is 1
0.6 is multiplied by 2 and the result is 1.2, so the integer part is 1

Binary fraction ≈ 0.0111₂

Final Answer:

(255.45)₁₀ ≈ (11111111.0111)₂

This method is essential for representing real-world numerical data in binary systems.

Decimal to Binary Conversion Examples for Quick Practice

Practicing different decimal values improves speed and accuracy in conversions. A list of decimal to binary examples is given below.

Decimal	Binary	Decimal	Binary	Decimal	Binary	Decimal	Binary
0	0	25	11001	50	110010	75	1001011
1	1	26	11010	51	110011	76	1001100
2	10	27	11011	52	110100	77	1001101
3	11	28	11100	53	110101	78	1001110
4	100	29	11101	54	110110	79	1001111
5	101	30	11110	55	110111	80	1010000
6	110	31	11111	56	111000	81	1010001
7	111	32	100000	57	111001	82	1010010
8	1000	33	100001	58	111010	83	1010011
9	1001	34	100010	59	111011	84	1010100
10	1010	35	100011	60	111100	85	1010101
11	1011	36	100100	61	111101	86	1010110
12	1100	37	100101	62	111110	87	1010111
13	1101	38	100110	63	111111	88	1011000
14	1110	39	100111	64	1000000	89	1011001
15	1111	40	101000	65	1000001	90	1011010
16	10000	41	101001	66	1000010	91	1011011
17	10001	42	101010	67	1000011	92	1011100
18	10010	43	101011	68	1000100	93	1011101
19	10011	44	101100	69	1000101	94	1011110
20	10100	45	101101	70	1000110	95	1011111
21	10101	46	101110	71	1000111	96	1100000
22	10110	47	101111	72	1001000	97	1100001
23	10111	48	110000	73	1001001	98	1100010
24	11000	49	110001	74	1001010	99	1100011

2. Positional Weight (Sum of Powers of 2) Method

The Positional Weight Method (also known as the Sum of Powers of 2) is a technique used to convert decimal numbers into binary (Base-2) by breaking the decimal value down into a sum of unique powers of 2. 

Algorithm for Positional Weight (Sum of Powers of 2) Method

This structured algorithm helps students accurately convert decimal numbers using powers of 2.

Step 1. Identify the Powers of 2

Step 2. Compare and Subtract

Step 2.1: 

Compare the decimal number to the largest power of 2 in your list.

Step 2.2:

If the power of 2 is less than or equal to the number:

• Place a 1 in that position.
• Subtract the power of 2 from the decimal number.

Otherwise, if the power of 2 is greater than the number:

• Place a 0 in that position.
• Move to the next smaller power of 2.

Repeat until you reach 2⁰

Step 3: Assemble the bits to get the result of the conversion from decimal to binary

Option Step 4: (Verification Result): If you add the positional values (powers of 2) where the binary bit is 1, you will always get the original decimal number.

Example 01: Convert (107)₁₀ to Binary

Step 1. Identify the Powers of 2

The given decimal number is 107, the largest power of 2 we can take is 2⁶ = 64 (since 2⁷ = 128 exceeds 107). So we list the powers of 2 in descending order, starting from 2⁶ to 2⁰

26	25	24	23	22	21	20
64	32	16	8	4	2	1

Step 2. Compare and Subtract

By using step 2 in the algorithm

• For 2^6 = 64
  •  64 ≤ 107 → True,  Use it. Write 1. Remainder: 107 − 64 = 43
• 2^5 = 32:
  • 32 ≤ 43 → True, Use it. Write 1. Remainder: 43 − 32 = 11
• 2^4 =16:
  • 16 ≤ 11 → False, Skip it. Write 0. Remainder remains the same as already: 11
• 2^3 = 8:
  • 8 ≤ 11 → Use it. Write 1. Remainder: 11 − 8 = 3
• 2^2 = 4:
  • 4 ≤ 3 → False, Skip it. Write 0. Remainder remains the same as already: 3
• 2^1 = 2:
  • 2 ≤ 3 → Use it. Write 1. Remainder: 3 − 2 = 1
• 2^0 = 1:
  • 1 ≤ 1 → Use it. Write 1. Remainder: 0

 Step 3: Assemble the Bits

The binary representation is formed by the bits recorded in each step: 1101011

Final Answer:

(107)₁₀ = (1101011)₂

Verification:

Binary	1	1	0	1	0	1	1
Power	64	32	16	8	4	2	1

64+32+8+2+1 =107

Common Mistakes in Decimal to Binary Conversion

Avoiding common mistakes in decimal to binary conversion is essential for achieving accurate results in exams and real-world applications. Even small errors can completely change the binary output.
List of common errors in conversion is given below.

1. Writing Remainders in Wrong Order

This is one of the most frequent mistakes where students write remainders from top to bottom instead of reversing them.

• Explanation:
  • In the division method, remainders must be written from bottom to top
  • Writing them in the same order gives an incorrect binary number
• Example:
  • Correct: 25 → 11001
  • Incorrect: 25 → 10011
• Tip:
  • Always double-check the order of remainders before finalizing the answer

2. Skipping Steps in Division

Students sometimes skip intermediate division steps to save time, which often leads to calculation errors.

• Explanation:
  • Each division step contributes a binary digit
  • Missing even one step results in an incomplete binary representation
• Example:
  • Skipping 6 ÷ 2 step in conversion of 25 can produce wrong output
• Tip:
  • Write all steps clearly, especially in exams, to avoid mistakes

3. Incorrect Fraction Handling

Handling decimal fractions incorrectly is a common issue, especially when mixing methods.

• Explanation:
  • Integer part uses division by 2
  • Fractional part uses multiplication by 2
  • Confusing these methods leads to incorrect results
• Example:
  • 0.75 should be converted using multiplication, not division
• Tip:
  • Treat integer and fractional parts separately, then combine results

4. Misunderstanding Powers of 2

A weak understanding of powers of 2 can result in incorrect binary values.

• Explanation:
  • Binary is based entirely on powers of 2 (1, 2, 4, 8, 16, …)
  • Choosing wrong powers leads to incorrect bit placement
• Example:
  • 19 = 16 + 2 + 1 → 10011 (correct)
  • Wrong selection of powers gives incorrect binary
• Tip:
  • Memorize basic powers of 2 up to at least 2¹⁰ for quick calculations

5. Ignoring Leading Zeros and Bit Length

Sometimes students ignore required bit lengths in problems, especially in digital systems.

• Explanation:
  • Certain applications require fixed-bit representation (e.g., 8-bit, 16-bit)
  • Missing leading zeros changes the format
• Example:
  • 5 = 101 (but in 8-bit → 00000101)
• Tip:
  • Always check if a fixed number of bits is required

Applications of Decimal to Binary Conversion

Understanding decimal to binary conversion is essential for various real-world computing and digital system applications. It forms the foundation of how machines process and store data.
List of key applications is given below.

1. Digital Electronics

Binary numbers are the backbone of digital circuits and electronic systems.

• Explanation:
  • Circuits use two states: ON (1) and OFF (0)
  • Logic gates (AND, OR, NOT) operate using binary inputs
• Use Cases:
  • Microcontrollers
  • Embedded systems
  • Circuit design

2. Computer Programming

All high-level programming languages ultimately rely on binary instructions at the machine level.

• Explanation:
  • Programs are converted into machine code (binary) before execution
  • Binary operations are used in algorithms and data manipulation
• Use Cases:
  • Bitwise operations
  • Memory management
  • Low-level programming

3. Data Storage

Every type of data in a computer system is stored in binary format.

• Explanation:
  • Text, images, audio, and video are all converted into binary
  • Storage devices use bits (0s and 1s) to represent information
• Use Cases:
  • Hard drives and SSDs
  • RAM and cache memory
  • File encoding systems

4. Networking and Communication

Binary encoding is essential for transmitting data across networks and communication systems.

• Explanation:
  • Data is sent as binary signals over cables or wireless systems
  • Encoding schemes convert data into binary streams
• Use Cases:
  • Internet data transfer
  • Wireless communication
  • Protocols like TCP/IP

5. Cybersecurity and Encryption

Binary plays a crucial role in securing data through encryption techniques.

• Explanation:
  • Encryption algorithms convert data into binary patterns
  • Security systems rely on binary-level operations
• Use Cases:
  • Data encryption
  • Password hashing
  • Secure communications

Conclusion

Decimal to binary conversion is a fundamental skill in computer science that builds the foundation for understanding how computers work internally. By mastering different methods such as division, subtraction, and fractional conversion, students can confidently solve complex problems and apply these concepts in real-world scenarios.