Convert 0.1001(repeating) to Decimal

  • Thread starter SherlockOhms
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In summary, the fraction in my notes is in base-2, while the equation is in base-10. The trick is to convert the binary to decimal and then do the reverse.
  • #1
SherlockOhms
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Homework Statement


Not so much a problem as a general wondering. I know how to convert from decimal to binary and binary to decimal apart from when there's a recurring pattern involved. One which I really can't figure out it 0.1001 (repeating). How do you go about converting that to decimal?

Homework Equations


None really.

The Attempt at a Solution


I really don't have any attempt as I can't find much about it on the internet. It features in a worked example in our notes and it's never explained. The answer is 0.6 apparently. (Not sure if this would be considered a computing problem or a mathematical one)
 
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  • #2
SherlockOhms said:

Homework Statement


Not so much a problem as a general wondering. I know how to convert from decimal to binary and binary to decimal apart from when there's a recurring pattern involved. One which I really can't figure out it 0.1001 (repeating). How do you go about converting that to decimal?

Homework Equations


None really.

The Attempt at a Solution


I really don't have any attempt as I can't find much about it on the internet. It features in a worked example in our notes and it's never explained. The answer is 0.6 apparently. (Not sure if this would be considered a computing problem or a mathematical one)

The trick here is similar to what you do to convert a repeating decimal fraction to a rational number. For example, if you need to convert .35353535... to a fraction, you multiply by a power of 10 large enough to move the part that repeats to the other side of the decimal point, which in this case would be 102. If we write x = .353535..., then 100x = 35.353535...

Subtracting the equation with x from the equation with 100x, we get
99x = 35 ==> x = 35/99

For your binary fraction, write an equation x = .10011001...2. Get a new equation that you can subtract this one from by multiplying by 24 (or 16).
 
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  • #3
Won't you then end up with 15x = 1001, and that equals 66.733... and not 3/5 as it says in my notes. Is there a mistake in the notes or am I misunderstanding you?
 
  • #4
No, you won't. In my work I was careful to note that the fraction was in base-2 (something you neglected to do). You should have gotten 15x = 10012.

This is a bit ungainly, having a decimal number on one side, and a binary number on the other, but at least it's marked to indicate that different bases are being used.
 
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  • #5
Got it! Thanks a million.
 

1. What does it mean to convert a repeating decimal to a decimal number?

When we say "convert a repeating decimal to a decimal number," we are referring to the process of expressing a number that has an infinite number of digits after the decimal point in a finite, decimal form. In other words, we are trying to write the number in a way that does not have a repeating pattern of digits after the decimal point.

2. How do I convert 0.1001(repeating) to a decimal number?

To convert 0.1001(repeating) to a decimal number, we can use a simple algebraic equation. Let's call the repeating decimal x. We can then write the equation 1000x = 100.1001, and solve for x. This will give us the decimal equivalent of the repeating decimal.

3. Can all repeating decimals be converted to decimal numbers?

Yes, all repeating decimals can be converted to decimal numbers. This is because all repeating decimals can be expressed as a fraction, and fractions can be converted to decimal form using division.

4. Is there a shortcut or easier way to convert repeating decimals to decimal numbers?

While there is no specific shortcut, there are some techniques that can make the process easier. For example, when converting a decimal that repeats a single digit, we can write it as a fraction with the repeating digit as the numerator and a number with the same number of digits as the repeating part (minus one) as the denominator.

5. Why is it important to convert repeating decimals to decimal numbers?

Converting repeating decimals to decimal numbers is important because it allows us to express the number in a form that is easier to work with in mathematical calculations. It also helps us to better understand the value and magnitude of the number.

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