Why does the professor stop converting at the fourth digit?

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In summary, the conversation discusses the conversion of decimal numbers to binary numbers using the algorithm of multiplying by 2 and carrying the remainder. The question is raised as to why the professor stops at the fourth digit and if it is still possible to continue the algorithm. The expert suggests that the precision has been obtained or the point has been made sufficiently. The conversation also touches on the topic of binary numbers not being able to exactly represent some decimal numbers.
  • #1
beanryu
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I am sorry to bother you people again and I don't really know if this is the right place to put it.

my great professor shows us in pdf format that

0.83(decimal)=0.1101(binary) because

0.83 x 2 = 1.66 carry 1
0.66 x 2 = 1.32 carry 1
0.32 x 2 = 0.64 carry 0
0.64 x 2 = 1.28 carry 1

My question is why he stops doing it at the fourth digit?
it seems you can still do the algarithm from 1.28 onward. why he stops?

THanx alot... I am drowning.
 
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  • #2
Sometimes you just have to stop sooner or later. Each iteration contributes half as much as the previous to the final precision of the answer. Since .28 will only add a zero value to the sum either the necessary precision has been obtained or the professor is satisfied that the point has been made well enough to satisfy your need for understanding the process? Just a guess.
 
  • #3
0.1101(binary) in decimal is,

[tex] 1*2^{-1} + 1*2^{-2} + 0*2^{-3} + 1*2^{-4} = (0.8125)_{10} [/tex]

which is not 0.83(decimal)
 
  • #4
So binary # can't exactly represent a decimal number?
THANKS a lot guys!
 
  • #5
beanryu said:
So binary # can't exactly represent a decimal number?
THANKS a lot guys!

All numbers that can be represented in decimal notation can also be represented in binary notation. If a number has a terminating binary representation, then it also has a terminating decimal representation. However, some numbers, such as [itex]0.2[/itex](Decimal) have a terminating decimal representation, but have a repeating expression in binary.
 
  • #6
Perhaps your teacher thought you, as the student, might want to continue the process? Have you?
 

1. How do I convert a decimal number to binary?

To convert a decimal number to binary, you can use the repeated division-by-2 method. Start by dividing the decimal number by 2 and writing down the remainder. Then, divide the quotient by 2 and write down the remainder again. Continue this process until the quotient is 0. The binary equivalent will be the remainders in reverse order.

2. What is the maximum number of digits in a binary number?

The maximum number of digits in a binary number is dependent on the data type being used. For example, an 8-bit data type can hold a maximum of 8 digits in binary, while a 16-bit data type can hold up to 16 digits in binary. Generally, the maximum number of digits in a binary number is equal to the number of bits in the data type.

3. Can all decimal numbers be represented in binary?

No, not all decimal numbers can be represented in binary. This is because binary is a base-2 number system, while decimal is a base-10 number system. Some decimal numbers, such as 1/3, have an infinite number of digits when converted to binary, making it impossible to represent them accurately.

4. How do I convert a decimal fraction to binary?

To convert a decimal fraction (a number with a decimal point) to binary, you can use the repeated multiplication-by-2 method. Start by multiplying the decimal fraction by 2 and writing down the whole number before the decimal point. Then, multiply the decimal part by 2 again and write down the whole number before the decimal point. Continue this process until the decimal part becomes 0 or starts repeating. The binary equivalent will be the whole numbers in the same order as they were written down.

5. What are some common applications of converting decimal to binary?

Converting decimal to binary is commonly used in computer science and digital electronics. It is necessary for representing and storing numbers in binary code, which is used in computer programming and data processing. It is also used in networking and data transmission, as binary is the underlying language used for transmitting data over the internet. Additionally, converting decimal to binary is important in understanding and working with binary logic gates in digital circuitry.

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