INFORMATION

BINARY AND DECIMAL NUMBERS

Binary and decimal numbering systems are commonly used in mathematics, computer science and electrical engineering.

Binary Numbering System (Base 2): In the binary system, numbers are represented using one of two digits: 0 and 1. Each digit is called a bit. For a whole number, the right-most bit has a place value of 2⁰ = 1, the next bit has a place value of 2¹ = 2, then 2² = 4, 2³ = 8, and so on, doubling with each position to the left. For a binary number with a radix point, the place value of the left-most digit in the fractional part is 2^–1, the next bit on the right has a place value 2^–2, and so on.

Decimal Numbering System (Base 10): In the decimal system, numbers are represented using ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Each digit has a place value of 10 raised to a power depending on its position in the number.

DECIMAL TO BINARY CONVERSION

WHOLE NUMBERS

We apply the following rules to convert a decimal number to binary.

• We divide the decimal number by 2 repeatedly until the quotient becomes 0.
• Starting at the least significant digit, we write the remainders in the same order of divisions.

For example, to convert decimal 6 to binary, we divide 6 by 2 repeatedly until the quotient becomes 0.

When we divide 6 by 2, the quotient is 3 and the remainder is 0. Thus, 0 is the least significant bit of the binary equivalent. We continue the algorithm with 3. When we divide 3 by 2, both quotient and remainder are 1. Then 1 is the bit at the second LSB of the binary number. Finally, we divide 1 by 2. When we do this operation, the quotient is 0 and the remainder is 1. Because the quotient is 0, we stop the procedure. Then we write the last remainder to the most significant digit of the binary number. To sum up, the binary representation of 6 is 110.

(6)₁₀ = (110)₂

DECIMAL NUMBERS

In case, the decimal number is not an integer, we can convert the whole number and fractional parts separately and add the binary equivalents up.

To convert the fractional part of a decimal number, we apply the following rules.

• We multiply the fractional part by 2 repeatedly until the product becomes an integer or the number of significant digits is sufficient for our calculations.
• At each step, we write the integer part of the rightmost digit to the fractional part of the binary number. We continue with the fractional part of the product.

For example, to convert 6.375 to binary, we multiply the fractional part by 2 repeatedly until we find an integer.

• 0.375 × 2 = 0.75
• 0.75 × 2 = 1.5
• 0.5 × 2 = 1

The fractional part of 6.375 is 0.375. When we multiply 0.375 by 2, the result is 0.75. The integer part of 0.75 is 0. Thus we write 0 to the first digit on the RHS of the radix point.

0.0

We continue with the fractional part of 0.75. When we multiply 0.75 by 2, the result is 1.5. We write the integer part of 1.5 to the next digit of the binary.

0.01

The fractional part of 1.5 is 0.5. Therefore, we continue with this number. The product of 0.5 and 2 is equal to 1. Because 1 is an integer we write it to the next digit of the binary and stop multiplying numbers.

(0.375)₁₀ = (0.011)₂

Binary representation of 6.375 is equal to the sum of binary representations of 6 and 0.375. Thus, decimal 6.375 is equal to binary 110.011.

(6.375)₁₀ = (6)₁₀ + (0.375)₁₀

= (110)₂ + (0.011)₂

= (110.011)₂

WHAT IS DECIMAL TO BINARY CONVERTER?

Decimal to binary converter,

• Computes the binary equivalent of the entered decimal number and
• Describes each step of the conversion for both whole number and fractional parts,

HOW TO USE DECIMAL TO BINARY CONVERTER?

You can use decimal to binary converter in two ways.

• USER INPUTS

  

  You can enter a decimal number to the input box and click on the "CONVERT" button. The result and explanations appaer below the calculator

• RANDOM INPUTS

  

  You can click on the DIE ICON next to the input box. If you use this property, a random decimal number is generated and entered to the calculator, automatically. You can see the result and explanations below the calculator. You can create your own examples and practice using this property.

• CLEARING THE INPUT BOX

  

  To check the binary equivalent of other decimals you can clear the input box by clicking on the CLEAR button under the input box.

• COPYING & DOWNLOADING THE SOLUTION

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    You can copy the generated solution by clicking on the "Copy Text" link, appaers under the solution panel.

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    Even you can download the solution as an image file with .jpg extension if you click on the "Download Solution" link at the bottom of the solution panel. You can share the downloaded image file.