Converting Decimal to Binary

Converting decimal numbers into binary is an essential skill in computer science, particularly if you're delving into programming, networking, or digital electronics. Whether you're a seasoned programmer or a beginner in the world of computing, understanding how to make this conversion will deepen your grasp of how computers operate. In this article, we’ll cover the key methods for converting decimal to binary with step-by-step examples.

What is the Decimal System?

Before we dive into conversion, let’s briefly recall the decimal system (Base 10), which is what we use every day. The decimal system consists of ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Each digit's position represents a power of ten. For example, in the number 345, the digit 3 represents 300 (3 × 10^2), the 4 represents 40 (4 × 10^1), and the 5 stands for 5 (5 × 10^0).

What is the Binary System?

The binary system (Base 2) comprises only two digits: 0 and 1. Each position in a binary number represents a power of two. Understanding how to convert between these two systems allows you to see how computers interpret and process data at the most fundamental level.

Method 1: Division by 2

One of the most straightforward methods to convert a decimal number to binary is by using division by 2. Here’s a step-by-step guide for converting a decimal number to binary using this method:

Step-by-Step Example

Let’s take the decimal number 19 as our example.

  1. Divide the number by 2.

    • 19 ÷ 2 = 9, remainder 1

  2. Record the remainder.

    • Remainder: 1

  3. Repeat the process with the quotient.

    • 9 ÷ 2 = 4, remainder 1
    • Remainder: 1

  4. Continue dividing until the quotient is 0.

    • 4 ÷ 2 = 2, remainder 0
      • Remainder: 0
    • 2 ÷ 2 = 1, remainder 0
      • Remainder: 0
    • 1 ÷ 2 = 0, remainder 1
      • Remainder: 1

  5. Compile the remainders in reverse order.

    • Reading from the last remainder to the first gives you: 10011

Therefore, the decimal number 19 in binary is 10011.

Practice Problem

Convert the decimal number 23 to binary using the same method.

  1. 23 ÷ 2 = 11, remainder 1
  2. 11 ÷ 2 = 5, remainder 1
  3. 5 ÷ 2 = 2, remainder 1
  4. 2 ÷ 2 = 1, remainder 0
  5. 1 ÷ 2 = 0, remainder 1

Reading the remainders gives us 10111. So, 23 in binary is 10111.

Method 2: Subtraction Method

The subtraction method is another technique to convert decimal numbers to binary. It's a bit more intuitive and involves subtracting powers of 2.

Step-by-Step Example

Let’s convert decimal 13 to binary using the subtraction method.

  1. List the powers of 2:

    • 2^0 = 1
    • 2^1 = 2
    • 2^2 = 4
    • 2^3 = 8
    • 2^4 = 16 (this is greater than 13, so we won't use it)

  2. Find the largest power of 2 less than or equal to the number (13):

    • The largest is 8 (2^3). Subtract 8 from 13.
    • 13 - 8 = 5; we mark a ‘1’ in the 2^3 position.

  3. Repeat for the remaining number (5):

    • The largest power of 2 less than or equal to 5 is 4 (2^2).
    • 5 - 4 = 1; we mark a ‘1’ in the 2^2 position.

  4. Continue with the remainder (1):

    • The largest power of 2 less than or equal to 1 is 1 (2^0).
    • 1 - 1 = 0; we mark a ‘1’ in the 2^0 position.
    • The remaining 2^1 (2) was not used, so we mark a ‘0’.

  5. Assemble the binary number:

    • Written out: 2^3 (1), 2^2 (1), 2^1 (0), 2^0 (1) = 1101

Thus, the decimal number 13 in binary is 1101.

Practice Problem

Convert the decimal number 10 to binary using the subtraction method.

  1. Powers of 2: 1, 2, 4, 8 (10 is greater than 8)
    • 10 - 8 = 2; mark a ‘1’ at 2^3 position.
  2. Largest for 2 now is 2 (2^1).
    • 2 - 2 = 0; mark a ‘1’ at 2^1 position.
  3. 2^2 and 2^0 are not used, so mark as ‘0’.

Reading the marks gives us 1010. Thus, 10 in binary is 1010.

Method 3: Using Binary Place Values

In this method, we leverage the binary place values, just like in the decimal system, but here, we look for combinations of 1s that add up to the decimal number.

Example

Let’s convert 30 to binary using this place value method.

  1. Place values of binary:

    • 16 (2^4), 8 (2^3), 4 (2^2), 2 (2^1), 1 (2^0)

  2. Find combinations:

    • 30 can be expressed as:
      • 16 (1) + 8 (1) + 4 (1) + 2 (0) + 1 (0)
    • So we write: 1 1 1 1 0 (from left to right).

Thus, 30 in binary is 11110.

Converting Larger Decimal Numbers

For larger decimal numbers, repeating any of the methods we’ve discussed can be useful. For instance, if you're dealing with decimal 255:

  1. Using division by 2:

    • 255 ÷ 2 gives:
      • Remainders will show 11111111.

  2. Using subtraction:

    • 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1.

Thus, you confirm 255 is indeed 11111111.

Summary of Methods

  • Division by 2: Repeatedly divide by 2 and take remainders.
  • Subtraction Method: Subtract the largest powers of 2 from the decimal value.
  • Place Value Method: Identify which powers of 2 sum to the decimal number.

Conclusion

Converting decimal to binary is a valuable skill for anyone involved in computer science or programming. With methods such as division by 2, subtraction, and place values, you can confidently make the conversions needed for various applications. Practicing these methods through examples will bolster your understanding and mastery of this fundamental concept. Keep practicing, and you'll find yourself converting with ease!