thesarfo

Reference

Binary ↔ Decimal Conversion

Converting decimal to binary by repeated division, and binary to decimal by summing powers of 2, with Java implementations of both.

views 0

Decimal to Binary Conversion

(LeetCode 1017)

To convert a decimal number to binary, we repeatedly divide the number by 2 and collect the remainders. The binary representation is the remainders read from bottom to top.

Algorithm:

  1. Divide the number by 2
  2. Write down the remainder (0 or 1)
  3. Continue with the quotient
  4. Repeat until the quotient becomes 0
  5. Read the remainders from bottom to top

Example 1: Convert 13 to binary

13 ÷ 2 = 6 remainder 1  ←
6 ÷ 2 = 3 remainder 0   ←
3 ÷ 2 = 1 remainder 1   ←
1 ÷ 2 = 0 remainder 1   ←
                        ↑
Read from bottom to top: 1101


Therefore, 13 (decimal) = 1101 (binary)

Example 2: Convert 25 to binary

25 ÷ 2 = 12 remainder 1 ←
12 ÷ 2 = 6 remainder 0  ←
6 ÷ 2 = 3 remainder 0   ←
3 ÷ 2 = 1 remainder 1   ←
1 ÷ 2 = 0 remainder 1   ←
                        ↑
Read from bottom to top: 11001


Therefore, 25 (decimal) = 11001 (binary)

Binary to Decimal Conversion

To convert a binary number to decimal, we multiply each digit by its corresponding power of 2 and sum the results. Start from the rightmost digit (position 0) and move left.

Algorithm:

  1. Start from the rightmost digit (position 0)
  2. Multiply each digit by 2 raised to its position
  3. Sum all the results

Example 1: Convert 1101 to decimal

Position:  3  2  1  0
Binary:    1  1  0  1


Calculation:
1 × 2³ = 1 × 8 = 8
1 × 2² = 1 × 4 = 4
0 × 2¹ = 0 × 2 = 0
1 × 2⁰ = 1 × 1 = 1
                ────
                 13


Therefore, 1101 (binary) = 13 (decimal)

Example 2: Convert 10101 to decimal

Position:  4  3  2  1  0
Binary:    1  0  1  0  1


Calculation:
1 × 2⁴ = 1 × 16 = 16
0 × 2³ = 0 × 8  =  0
1 × 2² = 1 × 4  =  4
0 × 2¹ = 0 × 2  =  0
1 × 2⁰ = 1 × 1  =  1
                  ────
                   21


Therefore, 10101 (binary) = 21 (decimal)

Code Implementation

Decimal to binary:

String convertToBinary(int num){
    if(num == 0) return "0";


    StringBuilder res = new StringBuilder();
    while(num > 0){
        if(num % 2 == 1){
            res.append('1');
        } else {
            res.append('0');
        }
        num = num / 2;
    }


    // Reverse because we read remainders from bottom to top
    return res.reverse().toString();
}

convertToBinary(13) returns "1101". convertToBinary(25) returns "11001".

Binary to decimal:

int convertToDecimal(String binary){
    int result = 0;
    int power = 0;


    // Start from rightmost digit (position 0)
    for(int i = binary.length() - 1; i >= 0; i--){
        if(binary.charAt(i) == '1'){
            result += Math.pow(2, power);
        }
        power++;
    }


    return result;
}

convertToDecimal("1101") returns 13. convertToDecimal("10101") returns 21.