Decimal To Binary And Binary To Decimal Conversion Explained

by ADMIN 61 views

Understanding Decimal and Binary Systems

In the realm of mathematics and computer science, different number systems serve as the foundation for representing numerical values. Among these, the decimal and binary systems hold paramount importance. The decimal system, which we use in our daily lives, is a base-10 system, employing ten distinct digits (0 through 9) to represent numbers. In contrast, the binary system, the language of computers, is a base-2 system, utilizing only two digits: 0 and 1. Understanding the conversion between these systems is crucial for anyone venturing into the world of programming, digital electronics, or computer architecture.

The Conversion Process

Converting a decimal number to its binary equivalent involves a systematic process of repeated division by 2. Here's a step-by-step breakdown:

  1. Divide the decimal number by 2. Note the quotient and the remainder. The remainder will always be either 0 or 1, representing the binary digits.
  2. Divide the quotient obtained in the previous step by 2. Again, note the quotient and the remainder.
  3. Repeat this process of dividing the quotient by 2 until the quotient becomes 0.
  4. Write down the remainders in reverse order. The sequence of remainders, read from bottom to top, forms the binary equivalent of the decimal number.

Let's illustrate this process with an example. Suppose we want to convert the decimal number 42 to binary:

  1. 42 ÷ 2 = 21, remainder 0
  2. 21 ÷ 2 = 10, remainder 1
  3. 10 ÷ 2 = 5, remainder 0
  4. 5 ÷ 2 = 2, remainder 1
  5. 2 ÷ 2 = 1, remainder 0
  6. 1 ÷ 2 = 0, remainder 1

Reading the remainders in reverse order, we get 101010, which is the binary equivalent of 42.

Example Conversions

Let's apply this conversion process to the given decimal numbers:

1. 1234

  • 1234 ÷ 2 = 617, remainder 0
  • 617 ÷ 2 = 308, remainder 1
  • 308 ÷ 2 = 154, remainder 0
  • 154 ÷ 2 = 77, remainder 0
  • 77 ÷ 2 = 38, remainder 1
  • 38 ÷ 2 = 19, remainder 0
  • 19 ÷ 2 = 9, remainder 1
  • 9 ÷ 2 = 4, remainder 1
  • 4 ÷ 2 = 2, remainder 0
  • 2 ÷ 2 = 1, remainder 0
  • 1 ÷ 2 = 0, remainder 1

Binary equivalent: 10011010010

2. 5603

  • 5603 ÷ 2 = 2801, remainder 1
  • 2801 ÷ 2 = 1400, remainder 1
  • 1400 ÷ 2 = 700, remainder 0
  • 700 ÷ 2 = 350, remainder 0
  • 350 ÷ 2 = 175, remainder 0
  • 175 ÷ 2 = 87, remainder 1
  • 87 ÷ 2 = 43, remainder 1
  • 43 ÷ 2 = 21, remainder 1
  • 21 ÷ 2 = 10, remainder 1
  • 10 ÷ 2 = 5, remainder 0
  • 5 ÷ 2 = 2, remainder 1
  • 2 ÷ 2 = 1, remainder 0
  • 1 ÷ 2 = 0, remainder 1

Binary equivalent: 1010111100011

3. 1854

  • 1854 ÷ 2 = 927, remainder 0
  • 927 ÷ 2 = 463, remainder 1
  • 463 ÷ 2 = 231, remainder 1
  • 231 ÷ 2 = 115, remainder 1
  • 115 ÷ 2 = 57, remainder 1
  • 57 ÷ 2 = 28, remainder 1
  • 28 ÷ 2 = 14, remainder 0
  • 14 ÷ 2 = 7, remainder 0
  • 7 ÷ 2 = 3, remainder 1
  • 3 ÷ 2 = 1, remainder 1
  • 1 ÷ 2 = 0, remainder 1

Binary equivalent: 11100111110

4. 3975

  • 3975 ÷ 2 = 1987, remainder 1
  • 1987 ÷ 2 = 993, remainder 1
  • 993 ÷ 2 = 496, remainder 1
  • 496 ÷ 2 = 248, remainder 0
  • 248 ÷ 2 = 124, remainder 0
  • 124 ÷ 2 = 62, remainder 0
  • 62 ÷ 2 = 31, remainder 0
  • 31 ÷ 2 = 15, remainder 1
  • 15 ÷ 2 = 7, remainder 1
  • 7 ÷ 2 = 3, remainder 1
  • 3 ÷ 2 = 1, remainder 1
  • 1 ÷ 2 = 0, remainder 1

Binary equivalent: 111110000111

5. 1239

  • 1239 ÷ 2 = 619, remainder 1
  • 619 ÷ 2 = 309, remainder 1
  • 309 ÷ 2 = 154, remainder 1
  • 154 ÷ 2 = 77, remainder 0
  • 77 ÷ 2 = 38, remainder 1
  • 38 ÷ 2 = 19, remainder 0
  • 19 ÷ 2 = 9, remainder 1
  • 9 ÷ 2 = 4, remainder 1
  • 4 ÷ 2 = 2, remainder 0
  • 2 ÷ 2 = 1, remainder 0
  • 1 ÷ 2 = 0, remainder 1

Binary equivalent: 10011010111

Understanding Place Value in Binary

Just as each digit in a decimal number represents a power of 10, each digit in a binary number represents a power of 2. The rightmost digit represents 2⁰ (1), the next digit to the left represents 2¹ (2), then 2² (4), and so on. This place value system is the key to converting binary numbers to their decimal equivalents.

The Conversion Process

Converting a binary number to decimal involves multiplying each digit by its corresponding power of 2 and summing the results. Here's a step-by-step process:

  1. Identify the place value of each digit. Start from the rightmost digit, which has a place value of 2⁰, and move leftward, increasing the power of 2 for each subsequent digit.
  2. Multiply each digit by its corresponding place value.
  3. Sum the products obtained in the previous step. The sum is the decimal equivalent of the binary number.

For instance, let's convert the binary number 101101 to decimal:

  • 1 × 2⁵ = 32
  • 0 × 2⁴ = 0
  • 1 × 2³ = 8
  • 1 × 2² = 4
  • 0 × 2¹ = 0
  • 1 × 2⁰ = 1

Summing these products, we get 32 + 0 + 8 + 4 + 0 + 1 = 45, which is the decimal equivalent of 101101.

Example Conversions

Let's convert the given binary numbers to decimal:

1. 1001110

  • 0 × 2⁰ = 0
  • 1 × 2¹ = 2
  • 1 × 2² = 4
  • 1 × 2³ = 8
  • 0 × 2⁴ = 0
  • 0 × 2⁵ = 0
  • 1 × 2⁶ = 64

Decimal equivalent: 64 + 0 + 0 + 8 + 4 + 2 + 0 = 78

2. 01110

  • 0 × 2⁰ = 0
  • 1 × 2¹ = 2
  • 1 × 2² = 4
  • 1 × 2³ = 8
  • 0 × 2⁴ = 0

Decimal equivalent: 0 + 2 + 4 + 8 + 0 = 14

3. 1000111

  • 1 × 2⁰ = 1
  • 1 × 2¹ = 2
  • 1 × 2² = 4
  • 0 × 2³ = 0
  • 0 × 2⁴ = 0
  • 0 × 2⁵ = 0
  • 1 × 2⁶ = 64

Decimal equivalent: 64 + 0 + 0 + 0 + 4 + 2 + 1 = 71

4. 1001001

  • 1 × 2⁰ = 1
  • 0 × 2¹ = 0
  • 0 × 2² = 0
  • 1 × 2³ = 8
  • 0 × 2⁴ = 0
  • 0 × 2⁵ = 0
  • 1 × 2⁶ = 64

Decimal equivalent: 64 + 0 + 0 + 8 + 0 + 0 + 1 = 73

5. 111011011

  • 1 × 2⁰ = 1
  • 1 × 2¹ = 2
  • 0 × 2² = 0
  • 1 × 2³ = 8
  • 1 × 2⁴ = 16
  • 0 × 2⁵ = 0
  • 1 × 2⁶ = 64
  • 1 × 2⁷ = 128
  • 1 × 2⁸ = 256

Decimal equivalent: 256 + 128 + 64 + 0 + 16 + 8 + 0 + 2 + 1 = 475

Converting between decimal and binary number systems is a fundamental skill in various fields, especially in computer-related disciplines. The process involves repeated division by 2 for decimal-to-binary conversion and multiplying each digit by its corresponding power of 2 for binary-to-decimal conversion. Mastering these techniques enables a deeper understanding of how computers represent and manipulate numerical data.