100111 in Binary

Converting 39 from decimal to binary involves dividing the number by 2.

Since binary uses only the digits 0 and 1, each division's quotient becomes the next dividend until the quotient is 0.

The remainders, noted from bottom to top, form the binary equivalent.

For 39, this process results in 100111.

The chart below shows the binary digits (1 and 0) for 100111.

The first column lists the binary digits, the second their place values, and the third the value calculation, where binary digits are multiplied by their corresponding place values.

Adding the results verifies 100111 in binary as 39 in decimal.

The conversion of 39 to binary can be achieved using several methods. Here are the steps for two common methods:

Expansion Method:

Step 1 - Determine the place values: In binary, each place value is a power of 2. Calculate these powers:

2⁰ = 1

2¹ = 2

2² = 4

2³ = 8

2⁴ = 16

2⁵ = 32

Since 32 is less than 39, we include it.

Step 2 - Identify the largest power of 2 less than or equal to 39: Start with 2⁵ = 32, write 1 in the 2⁵ place. Subtract 32 from 39: 39 - 32 = 7

Step 3 - Next largest power for the remainder 7 is 2² = 4: Write 1 in the 2² place, subtract 4: 7 - 4 = 3

Step 4 - The next largest power for 3 is 2¹ = 2: Write 1 in the 2¹ place, subtract 2: 3 - 2 = 1

Step 5 - Finally, 2⁰ = 1 fits the remainder 1: Write 1 in the 2⁰ place. The remainder is now 0.

Step 6 - Write 0 in unused places (2⁴ and 2³): Resulting binary is 100111.

Grouping Method:

Step 1 - Divide 39 by 2: 39 / 2 = 19, remainder 1

Step 2 - Divide 19 by 2: 19 / 2 = 9, remainder 1

Step 3 - Divide 9 by 2: 9 / 2 = 4, remainder 1

Step 4 - Divide 4 by 2: 4 / 2 = 2, remainder 0

Step 5 - Divide 2 by 2: 2 / 2 = 1, remainder 0

Step 6 - Divide 1 by 2: 1 / 2 = 0, remainder 1

Step 7 - Read remainders bottom to top: Binary is 100111.

To convert any number to binary, follow these rules:

Rule 1: Place Value Method Identify the largest power of 2 less than or equal to the number. Subtract the value and find the next power of 2 for the remainder. Continue until the remainder is 0, placing 0 in unused positions.

Rule 2: Division by 2 Method Divide the number by 2, recording the remainder. The quotient becomes the new dividend. Continue until the quotient is 0, writing remainders in reverse order.

Rule 3: Representation Method Break the number into powers of 2. Allocate 1s and 0s to appropriate powers of 2. Combine the digits to form the binary result.

Rule 4: Limitation Rule Only 0s and 1s represent numbers in binary. Each binary place represents a power of 2.

Some tips and tricks for understanding binary numbers:

Memorize binary forms for numbers 1 to 39 to speed up conversions.

Recognize patterns: 1 → 1, 1 + 1 = 2 → 10, 2 + 2 = 4 → 100, 4 + 4 = 8 → 1000, 8 + 8 = 16 → 10000, 16 + 16 = 32 → 100000

Even and odd rule: Even numbers in binary end in 0, odd numbers in 1.

Cross-verify conversions by converting back to decimal.

Practice with tables to remember conversions.

• Decimal: Base 10 number system using digits 0-9.

• Binary: Base 2 number system using digits 0 and 1.

• Place Value: Value of a digit based on its position.

• Octal: Base 8 number system using digits 0-7.

• Power of 2: Each binary place represents a power of 2.