Problem 4 - Basic Engineering Mathematics

Question

Convert \(47_{10}\) to a binary number.

Step-by-Step Solution

Answer

The binary representation of 47 is 101111.

Step 1: Understand the Decimal Number System

The number 47 is given in the decimal system (base 10). In the decimal system, each digit represents a power of 10, starting from the rightmost digit representing \(10^0\) and increasing by a power of 1 as you move to the left.

Step 2: Determine the Largest Power of 2

To start converting 47 from decimal to binary, find the largest power of 2 less than or equal to 47. The powers of 2 below 47 are \(1, 2, 4, 8, 16, 32\). The largest power is \(2^5 = 32\).

Step 3: Subtract and Record the Binary Digit

Subtract \(32\) from \(47\), which gives \(15\). Record a binary digit of 1 for \(2^5\) indicating that \(2^5\) is used in the sum to reach 47. Thus, place a 1 in the binary representation for \(2^5\) place.

Step 4: Repeat the Process for Remaining

Next, use the largest power of 2 less than 15, which is \(2^3 = 8\). Subtract \(8\) from \(15\) to get \(7\). Add a 1 for \(2^3\) in the binary representation.

Step 5: Continue with Smaller Powers

Now find the largest power of 2 less than 7, which is \(2^2 = 4\). After subtracting, you have \(3\) remaining. Record a 1 in the \(2^2\) place.

Step 6: Final Subtraction for Smallest Powers

With \(3\) remaining, the largest power is \(2^1 = 2\). Subtract to get \(1\). Record a 1 for the \(2^1\) place. For the final remaining \(1\), which is \(2^0 = 1\), record a 1 in the \(2^0\) place.

Step 7: Assemble the Binary Number

The binary number is constructed by placing a 1 in each place value we have recorded, which results in \(101111\). The binary representation for \(47\) is therefore \(101111_2\).

Key Concepts

Understanding the Decimal System

The decimal system is the numeral system most of us use in our daily lives. It is also known as the base-10 system. The term "base-10" means that each digit in a number has a place value based on the powers of 10. For example, the number 47 can be broken down into its respective place values.

• The digit '4' is in the tens place, representing \(4 \times 10^1 = 40\).
• The digit '7' is in the units place, representing \(7 \times 10^0 = 7\).

This combination gives us 47. Using the powers of 10 helps us to comprehend what each position of a digit stands for as we move from right to left in a number.
It is the foundational system for arithmetic globally.

Exploring the Binary Number System

Unlike the decimal system, the binary number system is based on powers of 2. Hence, it is known as the base-2 system. Rather than using ten different digits (0 through 9) like in the decimal system, binary only uses two digits: 0 and 1. This system is especially pivotal in computer science, as computers operate using binary code.

Each position in a binary number represents a specific power of 2, similar to place values representing powers of 10 in the decimal system. For instance, the binary number 101111 consists of the positions:

• '1' at \(2^5 = 32\)
• '0' at \(2^4 = 16\)
• '1' at \(2^3 = 8\)
• '1' at \(2^2 = 4\)
• '1' at \(2^1 = 2\)
• '1' at \(2^0 = 1\)

When you sum the values represented by the '1's in these positions, you get 47, the decimal equivalent of the binary number 101111.

What are Powers of 2?

Powers of 2 are the building blocks of the binary system. Each place in a binary number corresponds to a power of 2, increasing from right to left. For instance, the sequence starts as follows: \(2^0 = 1\), \(2^1 = 2\), \(2^2 = 4\), \(2^3 = 8\), \(2^4 = 16\), \(2^5 = 32\), and so on.This progression is vital in converting numbers between decimal and binary. When converting from decimal to binary, as in the case of 47, we identify the largest power of 2 less than or equal to the number, subtract it, and continue this process with the remainder until zero is reached. This helps us determine which powers of 2 fall into the composition of the original decimal number.

Grasping Place Value in Binary Conversion

Place value is as crucial in binary numbers as it is in decimal numbers. Each position in a binary sequence holds a specific value dependent on its placement. During conversion from decimal to binary, consistently recording binary digits in the appropriate place is key. For example, when writing the number 47 as 101111 in binary:

• The '1' in the far-left place denotes \(2^5 = 32\)
• The next '0' represents \(2^4 = 16\), which is not part of the total
• Subsequent '1's are in positions \(2^3\), \(2^2\), \(2^1\), and \(2^0\)

Correct placement ensures that when these powers are summed, they equal the original decimal number. Understanding place values is essential for the conversion process.