How do we convert 1111010.11(binary) to decimal by first representing it in BCD form and then converting it to decimal
An example given in the note has divided the integer part of the given binary number by 1010 successively and multiplied the fractional part of the given binary number by 1010 successively , to convert it to BCD form , as the first step.
Then that result is converted in to decimal.
Could anyone please explain,
the reason of multiplying and dividing the given binary number by 1010?
And how to convert a binary number to a BCD form by doing that?
Many thanks!
Here is a rather lengthy video (30min) giving you a nice explanation of conversion between different bases.
The actual discussion of Binary Coded Decimals (BCD) starts near the end of the video.
https://www.youtube.com/watch?v=L2zsmYaI5ww
1111010.11
1111010 + .11
0*2*0 + 1*2^1 + 0*2^2 + 1*2^3 + 1*2^4 + 1*2^5 +1*2^6 + 1*2^-1 +1*2^-2
0 + 2 + 4 + 8 + 16 + 32 + 64 + 1/2 + 1/4
going to BCD is a chore, but once you have that, you can just read off the decimal digits.
To convert a binary number to BCD (Binary Coded Decimal) form, you need to divide the integer part by 1010 successively and multiply the fractional part by 1010 successively.
The reason for this is because BCD represents each decimal digit using a 4-bit binary code. So to convert a binary number to BCD, you need to separate it into individual decimal digits.
Here's a step-by-step explanation of the process:
1. Start with the given binary number 1111010.11.
2. Separate the integer part (left of the decimal point) and the fractional part (right of the decimal point).
- Integer part: 1111010
- Fractional part: 11
3. Convert the integer part to BCD:
- Divide each decimal digit by 1010 successively.
- Start with the leftmost digit (MSB) and proceed to the right.
For the digit '1':
- Divide 1 by 1010: 1 ÷ 1010 = 0 remainder 1.
- The first digit in BCD is 0001.
For the digit '1' (next digit to the right):
- Multiply the remaining digit (1) by 10 (in this case, 1010 in binary) and add the next digit (1).
- Multiply 11 by 1010: 11 × 1010 = 11110.
- Divide 11110 by 1010: 11110 ÷ 1010 = 10 remainder 0.
- The second digit in BCD is 0000.
For the digit '1' (next digit to the right):
- Multiply the remaining digit (0) by 10 and add the next digit (1).
- Multiply 10 by 1010: 10 × 1010 = 101000.
- Divide 101000 by 1010: 101000 ÷ 1010 = 100 remainder 0.
- The third digit in BCD is 0000.
For the digit '1' (next digit to the right):
- Multiply the remaining digit (0) by 10 and add the next digit (1).
- Multiply 0 by 1010: 0 × 1010 = 0.
- Divide 0 by 1010: 0 ÷ 1010 = 0 remainder 0.
- The fourth digit in BCD is 0000.
So, the BCD representation of the integer part is 00010000.
4. Convert the fractional part to BCD:
- Multiply each digit by 1010 successively.
- Start with the leftmost digit and proceed to the right.
For the digit '1':
- Multiply 1 by 1010: 1 × 1010 = 1010.
- The first digit in BCD is 0001.
For the digit '1' (next digit to the right):
- Multiply the previous digit (1) by 10 and add the next digit (1).
- Multiply 101 by 1010: 101 × 1010 = 1021010.
- The second digit in BCD is 0010.
So, the BCD representation of the fractional part is 00010010.
5. Combine the BCD representations of the integer and fractional parts:
- The BCD representation of 1111010.11 is 0001000000010010.
6. Convert the BCD representation to decimal:
- Each BCD digit represents a decimal digit, so the conversion is straightforward.
- Starting from the left, convert each BCD digit to decimal.
- The decimal representation of 0001000000010010 is 1690.
Therefore, the decimal representation of the binary number 1111010.11, after converting it to BCD and then to decimal, is 1690.
The process involves multiplication and division by 1010 (in binary) because each digit in BCD represents a decimal digit and is encoded using a 4-bit binary code (0000 to 1001). Multiplying by 1010 shifts the binary number to the left by a digit, and dividing by 1010 shifts it to the right. This helps in separating each decimal digit for conversion to BCD form.