Binary and decimal are two of the most commonly used number systems in computing and daily life. While humans typically work with the decimal system, which is base-10, computers process information in the binary system, which is base-2. Understanding how to convert binary numbers to decimal is an essential skill in fields like computer science, electronics, and digital logic design. In this post, we will explore the conversion of binary to decimal from the basics to more advanced concepts.
What is Binary?
Binary is a number system that uses only two digits: 0 and 1. These digits are known as bits (binary digits), and they form the foundation of all digital computing. Each digit in a binary number represents a power of 2, starting from the rightmost digit.
For example:
- The binary number 1101 represents the sum of the following powers of 2: 1×23+1×22+0×21+1×201 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^01×23+1×22+0×21+1×20 This can be simplified as: 1×8+1×4+0×2+1×1=8+4+0+1=131 \times 8 + 1 \times 4 + 0 \times 2 + 1 \times 1 = 8 + 4 + 0 + 1 = 131×8+1×4+0×2+1×1=8+4+0+1=13 Therefore, the binary number 1101 is equal to the decimal number 13.
What is Decimal?
The decimal system is the number system that most people use daily. It is based on 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Each digit in a decimal number represents a power of 10, starting from the rightmost digit.
For example, the decimal number 153 can be expressed as:1×102+5×101+3×1001 \times 10^2 + 5 \times 10^1 + 3 \times 10^01×102+5×101+3×100
This simplifies to:1×100+5×10+3×1=100+50+3=1531 \times 100 + 5 \times 10 + 3 \times 1 = 100 + 50 + 3 = 1531×100+5×10+3×1=100+50+3=153
The decimal system is the default number system we use in everyday life, but computers rely on binary to perform calculations and store data.
Basic Method for Converting Binary to Decimal
Now, let’s focus on the process of converting binary numbers to decimal. The most straightforward way to convert a binary number to decimal is by expanding it as a sum of powers of 2.
Steps for Conversion:
- Start from the rightmost bit (the least significant bit).
- Assign powers of 2 to each bit, starting from 202^020 for the rightmost bit, and increasing the power by 1 as you move left.
- Multiply each bit by the corresponding power of 2. A bit of 1 contributes the value of the power of 2, while a bit of 0 contributes 0.
- Add the results to get the decimal equivalent.
Example: Convert 1011 (binary) to decimal:
- Start from the right: 1×23+0×22+1×21+1×201 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^01×23+0×22+1×21+1×20
- Simplify: 1×8+0×4+1×2+1×1=8+0+2+1=111 \times 8 + 0 \times 4 + 1 \times 2 + 1 \times 1 = 8 + 0 + 2 + 1 = 111×8+0×4+1×2+1×1=8+0+2+1=11
Thus, the binary number 1011 equals the decimal number 11.
Using Powers of 2 in Binary to Decimal Conversion
As shown in the examples above, each position in a binary number represents a power of 2. The rightmost digit represents 202^020, the next digit represents 212^121, then 222^222, and so on. To convert binary to decimal, it is important to understand how powers of 2 work:
- The rightmost bit is 20=12^0 = 120=1,
- The next bit is 21=22^1 = 221=2,
- The next is 22=42^2 = 422=4,
- The next is 23=82^3 = 823=8,
- The next is 24=162^4 = 1624=16,
- And so on.
The binary system is very efficient for computers because it can be directly mapped to the two states of electrical circuits (on and off), which correspond to 1 and 0, respectively.
Advanced Concepts: Fractional Binary Numbers
In addition to converting whole numbers from binary to decimal, there are also fractional binary numbers that need to be converted. These numbers have a fractional part to the right of the binary point (similar to the decimal point in the decimal system). The process of converting these numbers is similar, but we work with negative powers of 2 for the fractional part.
Example: Convert 101.11 (binary) to decimal:
- Whole part: Start by converting the integer part 101:1×22+0×21+1×20=4+0+1=51 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 4 + 0 + 1 = 51×22+0×21+1×20=4+0+1=5
- Fractional part: Now convert the fractional part .11:1×2−1+1×2−2=12+14=0.5+0.25=0.751 \times 2^{-1} + 1 \times 2^{-2} = \frac{1}{2} + \frac{1}{4} = 0.5 + 0.25 = 0.751×2−1+1×2−2=21+41=0.5+0.25=0.75
- Combine: Add the whole part and the fractional part:5+0.75=5.755 + 0.75 = 5.755+0.75=5.75
Thus, the binary number 101.11 equals the decimal number 5.75.
Converting Large Binary Numbers to Decimal
For larger binary numbers, the method of conversion remains the same, but it may take longer to manually compute. To make the process more efficient, you can use tools like calculators or write a program that automates the conversion. For example, a simple program in Python could convert binary numbers to decimal using built-in functions.
Python Example:
binary_number = “1101101”
decimal_number = int(binary_number, 2)
print(decimal_number)
This will output:
109
Applications of Binary to Decimal Conversion
Understanding binary to decimal conversion is crucial in many areas of computing:
- Computer Architecture: Binary is used to represent data in memory and in processors.
- Networking: IP addresses are often written in binary for routing purposes, and converting them to decimal helps with readability and understanding.
- Programming: Binary values are used in low-level programming, and knowing how to convert between binary and decimal allows you to work efficiently with data.