Binary to Decimal Conversion

Binary
A binary number is a number expressed in the base 2 numeral system. Binary number's digits have 2 symbols: zero (0) and one (1). Each digit of a binary number counts as a power of 2.

Binary number example:
1101₂ = 1×2³ + 1×2² + 0×2¹ + 1×2⁰ = 13₁₀

Decimal
A decimal number is a number expressed in the base 10 numeral system. Decimal number's digits have 10 symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Each digit of a decimal number counts as a power of 10.

Decimal number example:
653₁₀ = 6×10² + 5×10¹ + 3×10⁰

How to convert binary to decimal

For a binary number with n digits:

d_n-1 ... d₃ d₂ d₁ d₀

The decimal number is equal to the sum of binary digits (d_n) times their power of 2 (2ⁿ):

decimal = d₀×2⁰ + d₁×2¹ + d₂×2² + ...

Example

Understanding Binary-to-Decimal Conversion

Binary-to-decimal conversion involves converting numbers from base 2 (binary) to base 10 (decimal). Binary numbers are composed of 0s and 1s, while decimal numbers are composed of digits from 0 to 9.

The general approach to converting binary numbers to decimal includes:

Identifying the position (or place value) of each binary digit starting from the right (positions start at 0).
Multiplying each binary digit by \( 2^{\text{position}} \), where "position" is the digit's distance from the rightmost end.
Summing up all the products to obtain the decimal equivalent.

Steps for Binary-to-Decimal Conversion

Step 1: Write down the binary number and label the position of each digit, starting from 0 on the far right.

Step 2: Multiply each binary digit by \( 2^{\text{position}} \), where the position starts at 0 and increases from right to left.

Step 3: Add the results of all the multiplications to find the decimal equivalent.

Example: Converting Binary to Decimal

Convert \( 101101 \) to decimal:

Write the positions for each digit: \( 1 \, 0 \, 1 \, 1 \, 0 \, 1 \) → positions: \( 5 \, 4 \, 3 \, 2 \, 1 \, 0 \).
Calculate:
- \( 1 \times 2^5 = 32 \)
- \( 0 \times 2^4 = 0 \)
- \( 1 \times 2^3 = 8 \)
- \( 1 \times 2^2 = 4 \)
- \( 0 \times 2^1 = 0 \)
- \( 1 \times 2^0 = 1 \)
Sum: \( 32 + 0 + 8 + 4 + 0 + 1 = 45 \).
Final decimal result: \( 45 \).

Conversion Table for Quick Reference

Here is a quick reference for small binary numbers and their decimal equivalents:

0000 → 0
0001 → 1
0010 → 2
0011 → 3
0100 → 4
0101 → 5
0110 → 6
0111 → 7
1000 → 8
1001 → 9

Applications of Binary-to-Decimal Conversion

Binary-to-decimal conversion is commonly used in:

Understanding binary data in programming and software development.
Interpreting machine-level instructions and debugging code.
Digital electronics for human-readable representations of binary data.

Practice Problem

Convert \( 11001110 \) to decimal:

Solution: Label the positions: \( 1 \, 1 \, 0 \, 0 \, 1 \, 1 \, 1 \, 0 \) → positions: \( 7 \, 6 \, 5 \, 4 \, 3 \, 2 \, 1 \, 0 \).
Calculate:
- \( 1 \times 2^7 = 128 \)
- \( 1 \times 2^6 = 64 \)
- \( 0 \times 2^5 = 0 \)
- \( 0 \times 2^4 = 0 \)
- \( 1 \times 2^3 = 8 \)
- \( 1 \times 2^2 = 4 \)
- \( 1 \times 2^1 = 2 \)
- \( 0 \times 2^0 = 0 \)
Sum: \( 128 + 64 + 0 + 0 + 8 + 4 + 2 + 0 = 206 \).
Final decimal result: \( 206 \).

Binary-to-Decimal Conversion Examples Table
Problem Type	Description	Steps to Solve	Example
Basic Conversion	Converting a binary number to its decimal equivalent.	Write down the binary number and assign powers of 2 to each digit, starting from the rightmost digit (2⁰). Multiply each binary digit by its corresponding power of 2. Add the results to get the decimal value.	For \( 1101 \): Assign powers: \( 1×2³, 1×2², 0×2¹, 1×2⁰ \). Calculate: \( 8 + 4 + 0 + 1 = 13 \). Result: \( 13 \) in decimal.
Handling Leading Zeros	Understanding that leading zeros do not affect the decimal value.	Ignore any leading zeros in the binary number. Follow the basic conversion process.	For \( 00101 \): Ignore leading zeros: Consider \( 101 \). Assign powers: \( 1×2², 0×2¹, 1×2⁰ \). Calculate: \( 4 + 0 + 1 = 5 \). Result: \( 5 \) in decimal.
Converting Larger Numbers	Converting longer binary numbers by systematically applying powers of 2.	Write the binary number and assign powers of 2 to each digit. Multiply and add as in the basic conversion.	For \( 101101 \): Assign powers: \( 1×2⁵, 0×2⁴, 1×2³, 1×2², 0×2¹, 1×2⁰ \). Calculate: \( 32 + 0 + 8 + 4 + 0 + 1 = 45 \). Result: \( 45 \) in decimal.
Verifying Conversion	Checking the accuracy of binary-to-decimal conversion.	Convert the decimal result back to binary by repeatedly dividing by 2 and noting the remainders. Ensure the original binary number matches.	For \( 1010 \) (converted to \( 10 \) in decimal): Divide \( 10 \) by 2: \( 10 ÷ 2 = 5 \) remainder \( 0 \). Divide \( 5 \) by 2: \( 5 ÷ 2 = 2 \) remainder \( 1 \). Divide \( 2 \) by 2: \( 2 ÷ 2 = 1 \) remainder \( 0 \). Divide \( 1 \) by 2: \( 1 ÷ 2 = 0 \) remainder \( 1 \). Result: \( 1010 \) matches the original binary number.
Applications	Understanding where binary-to-decimal conversion is used.	Binary-to-decimal conversion is used in understanding data representation in computers. It is essential for decoding binary instructions in programming and digital electronics.	Example: Interpreting binary data from sensors or processors into human-readable decimal values.