Converting the Decimal Number 45 to Binary: Methods and Explanation

Understanding how to convert a decimal number to its binary form is a fundamental skill in computer science and digital electronics. This guide will walk you through the process of converting the decimal number 45 to binary using multiple methods and explain the underlying logic.

1. Subtraction Method

The subtraction method involves repeatedly subtracting the largest possible power of 2 until you are left with zero. Here's how the conversion of 45 to binary works:

45 - 32 13 2^5

13 - 8 5 2^3

5 - 4 1 2^2

1 - 1 0 2^0

So, the binary representation of 45 is 101101

QED

2. Division by 2 Method

Another way to convert a decimal number to binary is by using division by 2. This method is particularly useful for understanding the positional values of each digit in the binary representation. Here’s a step-by-step process:

45 / 2 22 remainder 1 22 / 2 11 remainder 0 11 / 2 5 remainder 1 5 / 2 2 remainder 1 2 / 2 1 remainder 0 1 / 2 0 remainder 1

Reading the remainders from bottom to top, we get the binary representation: 00101101.

Note: To make it a byte (8 bits), we add zeros to the left. Therefore, the binary representation of 45 is 00101101.

Double-check:

Multiply each 1 by 2^n, where n is the position starting from the right (2^0, 2^1, 2^2, etc.).

0 × 2^7 0 × 2^6 1 × 2^5 0 × 2^4 1 × 2^3 1 × 2^2 0 × 2^1 1 × 2^0 0 0 32 0 8 4 0 1 45

3. Binary Representation of 45

Let's represent each number in the binary form of 45 as a power of two, from right to left:

12 (2^4) - not used

4 (2^2) - used

8 (2^3) - used

16 (2^4) - not used

32 (2^5) - used

64 (2^6) - not needed

256 (2^8) - not needed

Adding the values for the positions that are '1':

1 × 2^5 32

1 × 2^3 8

1 × 2^2 4

1 × 2^0 1

So, 32 8 4 1 45, confirming the binary representation is correct.

4. Comparison of Methods

There are at least two ways to convert from decimal to binary manually:

Method 1 (Subtraction): Split the decimal number into multiples of powers of 2. For 45, the expression is:

4510 1 × 2^5 0 × 2^4 1 × 2^3 1 × 2^2 0 × 2^1 1 × 2^0 1011012

Method 2 (Division by 2): Constantly divide the decimal number by 2 and record the integer remainders. Reading the remainders from bottom to top, we get:

45 div; 2 22 remainder 1 22 div; 2 11 remainder 0 11 div; 2 5 remainder 1 5 div; 2 2 remainder 1 2 div; 2 1 remainder 0 1 div; 2 0 remainder 1

Results in binary: 1011012

Both methods will give the same result, confirming the binary representation of 45 is 101101.